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Clique Relaxation Models in Networks:
 Theory, Algorithms, and Applications
Sergiy Butenko (butenko@tamu.edu)
Department of Industrial and Systems Engineering
Texas A&M University
College Station, TX 77843-3131




                                                   1/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     2/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     3/93
Graphs
Graph is the mathematical term for a “network” - often visualized as
vertices (points, nodes) connected by edges (lines, arcs)




                                                                       4/93
Graph theory basics

The origins of graph theory are attributed to the Seven Bridges of
K¨nigsberg problem solved by Leonhard Euler in 1735.
 o




                                                                     5/93
Graph theory basics
A simple, undirected graph is a pair G = (V , E ), where V is a finite set of
vertices and E ⊆ V × V is a set of edges, with each edge defined on a pair
of vertices.
                                        6


                                5                1



                                    4        2


                                        3


                              V = {1, 2, 3, 4, 5, 6}

            E = {(1, 2), (1, 6), (2, 3), (2, 4), (3, 4), (4, 5), (5, 6)}
                                                                               6/93
Graph theory basics

A graph can be represented in several ways.
    We can represent a graph G = (V , E ) by its |V | × |V |
    adjacency matrix AG = [aij ], such that

                              1, if (vi , vj ) ∈ E ,
                      aij =
                              0, otherwise.

    Another way of representing a graph is by its adjacency lists,
    where for each vertex v ∈ V we record the set of vertices
    adjacent to it.



                                                                     7/93
Graph theory basics

A multigraph is a graph with repeated edges.
A directed graph, or digraph, is a graph with directions assigned
to its edges.
                                          
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                                                                    8/93
Graph theory basics


If G = (V , E ) is a graph and e = (u, v ) ∈ E , we say that u is
adjacent to v and vice-versa. We also say that u and v are
neighbors.

Neighborhood N(v ) of a vertex v is the set of all its neighbors
in G : N(v ) = {j : (i, j) ∈ E }.

If G = (V , E ) is a graph and e = (u, v ) ∈ E , we say that e is
incident to u and v .

The degree of a vertex v is the number of its incident edges.


                                                                    9/93
Graph theory basics



G = (V , E ) is a simple undirected graph, V = {1, 2, . . . , n}.

G = (V , E ), is the complement graph of G = (V , E ), where
E = {(i, j) | i, j ∈ V , i = j and (i, j) ∈ E }.
                                          /

For S ⊆ V , G (S) = (S, E ∩ S × S) the subgraph induced by S.




                                                                    10/93
Cliques and independent sets


A subset of vertices C ⊆ V is called a clique if G (C ) is a
complete graph.

A subset I ⊆ V is called an independent set (stable set, vertex
packing) if G (I ) has no edges.

                                                             ¯
C is a clique in G if and only if C is an independent set in G .




                                                                   11/93
Cliques and independent sets


A clique (independent set) is said to be
  – maximal, if it is not a subset of any larger clique (independent
    set);
  – maximum, if there is no larger clique (independent set) in the
    graph.


ω(G ) – the clique number of G .

α(G ) – the independence (stability) number of G .



                                                                       12/93
Cliques and independent sets

G           1                                    1         G
                        Complement
    5               2                    5               2



        4       3                            4       3


{1,2,5} : maximal clique             {1,2,5} : maximal
                                     independent set
    {1,4} : maximal
                                     {1,4} : maximal clique
    independent set

{2,3,4,5} : maximum clique           {2,3,4,5} : maximum
                                     independent set


                                                               13/93
Social networks


A social network is described by G = (V , E ) where V is the set of
“actors” and E is the set of “ties”.

    actors are people and a tie exists if two people know each other.
    actors are wire transfer database records and a tie exists if two
    records have the same matching field.
    actors are telephone numbers and a tie exists if calls were made
    between them.




                                                                        14/93
Social network analysis



“Popular” social networks:
    Kevin Bacon Number
    Erd¨s Number
       o
    Six degrees of separation and small world phenomenon in
    acquaintance networks




                                                              15/93
Cohesive subgroups



Cohesive subgroups are “tightly knit groups” in a social
network.
Social cohesion is often used to explain and develop sociological
theories.
Members of a cohesive subgroup are believed to share
information, have homogeneity of thought, identity, beliefs,
behavior, even food habits and illnesses.




                                                                    16/93
Applications

Acquaintance Networks - criminal network analysis
Wire Transfer Database Networks - detecting money
laundering
Call Networks - organized crime detection
Protein Interaction Networks - predicting protein functions
Gene Co-expression Networks - detecting network motifs
Metabolic Networks - identifying metabolic pathways
Stock Market Networks - stock portfolios
Internet Graphs - information search and retrieval
Wireless and telecommunication networks - clustering and
routing
                                                              17/93
Random graph models


Uniform random graphs (Erdos  Renyi):
    G (n, p): n vertices, each edge exists with a probability p;
    G (n, M): all graphs with n vertices and M edges are assigned
    the same probability
Power-law random graphs:
    the probability that a vertex has a degree k is proportional to
    k −β
Random graphs with a given degree sequence



                                                                      18/93
Scale-free graphs


Scale-free nature : A property observed in many natural and
man-made networks - biological networks, social networks,
internet graphs
Degree distribution follows a power law - P(k) ∝ k −β
Fundamentally different from classical random graph model of
Erd¨s and Renyi
   o
Principle of preferential attachment underlies the growth and
evolution of such networks (i.e. the rich get richer!)



                                                                19/93
Science - co-authorship network
Image source:
http://www.jeffkennedyassociates.com:16080/connections/concept/image.html




                                                                           20/93
Internet colored by IP addresses
Image source:
http://www.jeffkennedyassociates.com:16080/connections/concept/image.html




                                                                           21/93
H. Pylori - protein interactions
Image generated using Graphviz




                                                               22/93
Properties of cohesive subgroups


Some desirable properties of a cohesive subgroup are:
    Familiarity (degree);
    Reachability (distance, diameter);
    Robustness (connectivity);
    Density (edge density).
Earliest models of cohesive subgroups were cliques (Luce and Perry,
1949), representing an “ideal” model of a cohesive subgroup.




                                                                      23/93
Clique relaxations: k-cliques and k-clubs

A k-clique is a subset of vertices C such that for every i, j ∈ C ,
d(i, j) ≤ k.

A k-club is a subset of vertices D such that diam(G [D]) ≤ k.
         6


 5               1



     4       2


         3


                                                                      24/93
Clique relaxations: k-cliques and k-clubs

A k-clique is a subset of vertices C such that for every i, j ∈ C ,
d(i, j) ≤ k.

A k-club is a subset of vertices D such that diam(G [D]) ≤ k.
         6                           {2,3,4} is a 1-club ... the
                                     “regular” clique
 5               1



     4       2


         3



                                                                      24/93
Clique relaxations: k-cliques and k-clubs

A k-clique is a subset of vertices C such that for every i, j ∈ C ,
d(i, j) ≤ k.

A k-club is a subset of vertices D such that diam(G [D]) ≤ k.
          6                          {2,3,4} is a 1-club ... the
                                     “regular” clique
  5               1
                                     {1,2,4,5,6} is a 2-club

      4       2


          3



                                                                      24/93
Clique relaxations: k-cliques and k-clubs

A k-clique is a subset of vertices C such that for every i, j ∈ C ,
d(i, j) ≤ k.

A k-club is a subset of vertices D such that diam(G [D]) ≤ k.
          6                          {2,3,4} is a 1-club ... the
                                     “regular” clique
  5               1
                                     {1,2,4,5,6} is a 2-club
                                     {1,2,3,4,5} is a 2-clique but
      4       2                      NOT a 2-club

          3



                                                                      24/93
Clique relaxations: k-cliques and k-clubs

A k-clique is a subset of vertices C such that for every i, j ∈ C ,
d(i, j) ≤ k.

A k-club is a subset of vertices D such that diam(G [D]) ≤ k.
         6                           {2,3,4} is a 1-club ... the
                                     “regular” clique
 5               1
                                     {1,2,4,5,6} is a 2-club
                                     {1,2,3,4,5} is a 2-clique but
     4       2                       NOT a 2-club
                                     maximality of a 2-club is harder
         3                           to test

                                                                        24/93
k-plex

Definition
A subset of vertices S is said to be a k-plex if the minimum degree
in the induced subgraph δ(G [S]) ≥ |S| − k
i.e. every vertex in G [S] has degree at least |S| − k.

 2                   1


     6           3



     5           4




                                                                      25/93
k-plex

Definition
A subset of vertices S is said to be a k-plex if the minimum degree
in the induced subgraph δ(G [S]) ≥ |S| − k
i.e. every vertex in G [S] has degree at least |S| − k.
 2                    1           {3,4,5,6} is a 1-plex ... the
                                  “regular” clique
      6           3



      5           4




                                                                      25/93
k-plex

Definition
A subset of vertices S is said to be a k-plex if the minimum degree
in the induced subgraph δ(G [S]) ≥ |S| − k
i.e. every vertex in G [S] has degree at least |S| − k.
 2                    1           {3,4,5,6} is a 1-plex ... the
                                  “regular” clique
     6           3                {1,3,4,5,6} is a 2-plex
                                  (and NOT a 1-plex)

     5           4




                                                                       25/93
k-plex

Definition
A subset of vertices S is said to be a k-plex if the minimum degree
in the induced subgraph δ(G [S]) ≥ |S| − k
i.e. every vertex in G [S] has degree at least |S| − k.

 2                   1
                                  {3,4,5,6} is a 1-plex ... the
                                  “regular” clique
     6           3                {1,3,4,5,6} is a 2-plex
                                  (and NOT a 1-plex)
                                  {1,2,3,4,5,6} is a 3-plex
     5           4
                                  (and NOT a 2-plex)


                                                                       25/93
co-k-plex
Definition
A subset of vertices S is a co-k-plex if the maximum degree in the
induced subgraph ∆(G [S]) ≤ k − 1.
i.e. degree of every vertex in G [S] is at most k − 1.
S is a co-k-plex in G if and only if S is a k-plex in the complement
       ¯
graph G .

              2                 1    2                       1


                  6         3            6               3



                  5         4            5               4


                  3-plex                     Co-3-plex
                                                                             26/93
Structural Properties of a k-Plex


If G is a k-plex then
 1. Every subgraph of G is a k-plex;
             n+2
 2. If k     2    then diam(G ) ≤ 2;
 3. κ(G ) ≥ n − 2k + 2.

    k-plexes for “small” k values, guarantee reachability and
    connectivity while relaxing familiarity.




                                                                27/93
Known clique relaxation models


k-clique (distance)
k-club (diameter)
k-plex (degree)
k-core (degree)
k-connected subgraph (connectivity)
γ-quasi-clique (density)
(γ, λ)-quasi-clique (density, degree)
R-robust k-club (connectivity, diameter)



                                                   28/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     29/93
Alternative clique definitions


Distance-based
Vertices are distance one away from each other.

Diameter-based
Vertices induce a subgraph of diameter one.

Domination-based
Every one vertex forms a dominating set.




                                                          30/93
Alternative clique definitions


Degree-based
Each vertex is connected to all other vertices.

Connectivity-based
All vertices need to be removed to disconnect the induced subgraph.

Density-based
Vertices induce a subgraph that has all possible edges.




                                                                      31/93
Order of a clique relaxation


We introduce a hierarchy of clique relaxation models by defining
their order as follows.
    Clique is the only clique relaxation of order 0.
    Clique relaxations of order 1 relax one of the elementary
    properties (distance, diameter, ...) used to define clique.
    Clique relaxations of order 2 relax two of the elementary
    clique-defining properties.
    ...



                                                                  32/93
Nature of a clique relaxation


Parametric clique relaxations
relax a parameter (“one”, “all”) describing an elementary
clique-defining property.

Partial clique relaxations
require only a (high) fraction of elements to satisfy an elementary
clique-defining property or a parametric clique-relaxation property.




                                                                      33/93
Types of parametric clique relaxations


Parametric clique relaxation of type 1
Replace “one” with “(at most) k”, k  1, in a definition of clique.

Parametric clique relaxation of type 2
Replace “one” with “(at most) all but k”, k  1, in a definition of clique.

Parametric clique relaxation of type 1a
Replace “all” with “(at least) k”, k  1, in a definition of clique.

Parametric clique relaxation of type 2a
Replace “all” with “(at least) all but k”, k  1, in a definition of clique.

                                                                              34/93
First-order parametric
                            clique relaxations of type 1

              Replace “one” with “(at most) k”, k  1.

Distance-based: k-clique
Vertices are distance at most k away from each other.

Diameter-based: k-club
Vertices induce a subgraph of diameter at most k.

Domination-based: k-plex
Every k vertices form a dominating set.


                                                           35/93
First-order parametric
                            clique relaxations of type 2

           Replace “one” with “(at most) all but k”, k  1

Distance-based: N/A
Vertices in S are distance at most |S| − k away from each other.

Diameter-based: N/A
Vertices induce a subgraph G [S] of diameter at most |S| − k.

Domination-based: N/A
Every |S| − k vertices form a dominating set.


                                                                   36/93
First-order parametric
                           clique relaxations of type 1a

                Replace “all” with “(at least) k”, k  1

Degree-based: k-core
Each vertex is connected to at least k other vertices.

Connectivity-based: k-connected subgraph
At least k vertices need to be removed to disconnect the induced
subgraph.

Density-based: N/A
Vertices induce a subgraph that has at least k edges.

                                                                   37/93
First-order parametric
                            clique relaxations of type 2a

            Replace “all” with “(at least) all but k”, k  1.

Degree-based: k-plex
Each vertex is connected to at least all but k other vertices.

Connectivity-based: N/A
At least all but k vertices need to be removed to disconnect the
induced subgraph.

Density-based: N/A
Vertices induce a subgraph that has at least all but k edges.

                                                                   38/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     39/93
Basics of the complexity theory



Given a combinatorial optimization problem, a natural question
is:
           is this problem “easy” or “hard”?
Methods for easy and hard problems are fundamentally
different.

How do we distinguish between “easy” and “hard” problems?




                                                                 40/93
“Easy” problems


By easy or tractable problems we mean the problems that can
be solved in time polynomial with respect to their size.

We also call such problems polynomially solvable and denote
the class of polynomially solvable problems by P.
    Sorting
    Minimum weight spanning tree
    Linear programming




                                                              41/93
Defining “hard” problems


How do we define “hard” problems?

How about defining hard problems as all problems that are not
easy, i.e., not in P?

Then some of the problems in such a class could be TOO hard
– we cannot even hope to be able to solve them.

We want to define a class of hard problems that we may be
able to solve, if we are lucky (say, we may be able to guess the
solution and check that it is indeed correct).


                                                                   42/93
Defining “hard” problems

Let us take the maximum clique problem as an example and try
to guess a solution.

We can pick some random clique, compute its size, but how do
we know if this solution is optimal?

Indeed, this is as difficult to verify as to solve the maximum
clique problem – it is unlikely that we can count on luck in this
case...

How about transforming our optimization problem to such a
form, where instead of finding an optimal solution the question
would be “is there a solution with objective value ≥ s, where s
is a given integer?
                                                                    43/93
Three versions of optimization problems

Consider a problem

                     min f (x) subject to x ∈ X .

    Optimization version: find x from X that maximizes f (x);
    Answer: x ∗ maximizes f (x)

    Evaluation version: find the largest possible f (x);
    Answer: the largest possible value for f (x) is f ∗

    Recognition version: Given f ∗ , does there exist an x such that
    f (x) ≥ f ∗ ?
    Answer: “yes” or “no”

                                                                       44/93
Recognition problems

Recognition problems can still be undecidable.
    Halting problem: Given a computer program with its
    input, will it ever halt?

Recall the “luck factor”: if we pick a random feasible solution
and it happens to give “yes” answer, then we solved the
problem in polynomial time.
          Max clique: Randomly pick a solution (a clique). If its
          size is ≥ s (which we can verify in polynomial time),
          then obviously the answer is “yes”. This clique can be
          viewed as a certificate proving that this is indeed a yes
          instance of max clique.

                                                                     45/93
Class N P



We only consider problems for which any yes instance there
exists a concise (polynomial-size) certificate that can be verified
in polynomial time.

We call this class of problems nondeterministic polynomial and
denote it by N P.




                                                                    46/93
P vs N P

Note that any problem from P is also in N P (i.e., P ⊆ N P),
so there are easy problems in N P.

So, are there “hard” problems in N P, and if there are, how do
we define them?

We don’t know if P = NP, but “most” people believe that
P = NP.
An “easy” way to make $1,000,000!
http://www.claymath.org/millennium/P vs NP/

We can call a problem hard if the fact that we can solve this
problem would mean that we can solve any other problem in
comparable amount of time.
                                                                 47/93
Polynomial reducibility


Reduce π1 to π2 : if we can solve π2 fast, then we can solve π1
fast, given that the reduction is “easy”.
Polynomial reduction from π1 to π2 requires existence of
polynomial-time algorithms
 1. A1 converts an input for π1 into an input for π2 ;
 2. A2 converts an output for π2 into output for π1 .

Transitivity: If π1 is polynomially reducible to π2 and π2 is
polynomially reducible to π3 then π1 is polynomially reducible
to π3 .


                                                                  48/93
NP-complete problems
A problem π is called NP-complete if
 1. π ∈ NP;
 2. Any problem from NP can be reduced to π in polynomial time.
A problem π is called NP-hard if any problem from NP can be
reduced to π in polynomial time. (no π ∈ NP requirement)

Due to transitivity of polynomial reducibility, in order to show
that a problem π is N P-complete, it is sufficient to show that
 1. π ∈ NP;
 2. There is an NP-complete problem π that can be reduced to π
    in polynomial time.

To use this observation, we need to know at least one
N P-complete problem...
                                                                   49/93
Satisfiability (SAT) problem

A Boolean variable x is a variable that can assume only the values
true and false.
Boolean variables can be combined to form Boolean formulas using
the following logical operations:
  1. Logical AND (∧ or ·) -conjunction
  2. Logical OR (∨ or +) - disjunction
  3. Logical NOT (¯)
                  x
                   kj
A clause is Cj =         yjp , where a literal yjp is xr or xr for some r .
                                                            ¯
                   p=1
                                             m
Conjunctive normal form (CNF):F =                 Cj , where Cj is a clause.
                                            j=1



                                                                               50/93
Satisfiability problem

    A CNF F is called satisfiable if there is an assignment of
    variables such that F = 1 (TRUE ).
    Satisfiability (SAT) problem: Given m clauses C1 , . . . , Cm
    involving the variables x1 , . . . , xn , is the CNF
                                     m
                               F =         Cj ,
                                     j=1

    satisfiable?

Theorem (Cook, 1971)
SAT is NP-complete.

                                                                   51/93
“Best” approximation algorithms and
                                   heuristics


For some problems there are hardness of approximation results
stating that the problem is hard to approximate within a certain
factor.
For example, the k-center problem is hard to approximate
within a factor better than 2.
Then any polynomial-time algorithm approximating the
k-center problem within the factor of 2 can be considered the
“best” approximation algorithm for this problem.



                                                                   52/93
“Best” approximation algorithms and
                                   heuristics


However, some problems are even harder to approximate. For
example, the maximum clique is hard to approximate within a
factor n1− for any positive .
In this case, by the “best” heuristic we could mean a heuristic
that cannot be provably outperformed by any other
polynomial-time algorithm (unless P = NP).




                                                                  53/93
Recognizing the gap between k-club and
                                l -club numbers


Theorem
Let positive integer constants k and l, l  k be given. The problem
of checking whether ωl (G ) = ωk (G ) is NP-hard.
                      ¯        ¯

Note that
                    ω(G ) ≤ ∆(G ) + 1 ≤ ωk (G )
                                        ¯
and observe that we can easily check whether ω(G ) = ∆(G ) + 1.
Hence, it is NP-hard to check whether ωk (G ) = ∆(G ) + 1.
                                      ¯


                                                                      54/93
“Best” heuristics for k-club/clique




Corollary
Let k be a fixed integer, k ≥ 2. Unless P = NP, there cannot be a
polynomial time algorithm that finds a k-club of size greater than
∆(G ) + 1 whenever such a k-club exists in the graph.




                                                                    55/93
Protein Interaction Networks




                               56/93
A max 2-club/clique of S. Cerevisiae.




                                        57/93
A max 2-club/clique of H. Pylori.




                                    58/93
A max 3-clique/club of S. Cerevisiae




                                       59/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     60/93
Size of clique relaxations in a graph

Question: How large could a k-clique (k-club, k-plex) be for a
graph with a small clique number?




                                                                 61/93
Size of clique relaxations in a graph

Question: How large could a k-clique (k-club, k-plex) be for a
graph with a small clique number?
Answers:
    k-club: ω(K1,n ) = 2, but ωk (K1,n ) = n + 1, could be arbitrary
                              ¯
    large
    k-clique: same as k-club




                                                                       61/93
Size of clique relaxations in a graph

Question: How large could a k-clique (k-club, k-plex) be for a
graph with a small clique number?
Answers:
    k-club: ω(K1,n ) = 2, but ωk (K1,n ) = n + 1, could be arbitrary
                              ¯
    large
    k-clique: same as k-club
    k-plex:
Lemma
For any graph G and a positive integer k,

               ωk (G ) ≤ kω(G ) and αk (G ) ≤ kα(G )

                                                                       61/93
Upper bound on the quasi-clique number

Proposition
The γ-clique number ωγ (G ) of a graph G with n vertices and m
edges satisfies the following inequality:

                                  γ+   γ 2 + 8γm
                      ωγ (G ) ≤                  .               (1)
                                       2γ
Moreover, if a graph G is connected then

                          γ+2+     (γ + 2)2 + 8(m − n)γ
              ωγ (G ) ≤                                 .        (2)
                                       2γ


                                                                       62/93
Relation between ωγ (G ) and ω(G )

Proposition
The γ-clique number ωγ (G ) and the clique number ω(G ) of graph
G satisfy the following inequalities:

                 ω(G ) − 1   ωγ (G ) − 1   1 ω(G ) − 1
                           ≤             ≤             .      (3)
                   ω(G )       ωγ (G )     γ ω(G )

Corollary
               1
If γ  1 −   ω(G )   then

                                         ω(G )γ
                       ωγ (G ) ≤                      .       (4)
                                   1 − ω(G ) + ω(G )γ

                                                                    63/93
Relation between omegaγ (G ) and ω(G )

Table: The value of upper bound (4) on γ-clique number with
γ = 0.95, 0.9, 0.85 for graphs with small clique number.
                              1
               ω(G )   1 − ω(G )    0.95    0.9   0.85
                 3       0.667      3.35   3.86   4.64
                 4       0.75       4.75     6     8.5
                 5        0.8      6.33      9     17
                 6       0.83       8.14   13.5    51
                 7       0.86      10.23    21      –
                 8       0.88      12.67    36      –
                 9       0.89      15.55    81      –
                10        0.9        19      –      –

                                                              64/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     65/93
Math programming formulations



Consider a simple undirected graph G = (V , E ) with n vertices
Let A = [aij ]n
              i,j=1 be the adjacency matrix of G
Let x = (x1 , . . . , xn ) be a 0-1 vector with xi = 1 if node i
belongs to Gs , and xi = 0 otherwise.




                                                                   66/93
Math programming formulations

GS is a γ-clique if it has at least γ|GS |(|GS | − 1)/2 edges
                         n
We have: |GS | =              xi
                        i=1
This number of edges can be expressed in terms of vector x as:
              n          n                                n                n
        1
        2γ         xi         xi − 1     = 1γ
                                           2                    xi xj −         xi
             i=1        i=1                             i,j=1             i=1
                   n                n              n                           n
      1                                                           1
    = 2γ                 xi xj +         xi2 −          xi      = 2γ                 xi xj
             i,j=1;i=j             i=1           i=1                   i,j=1;i=j
                                               n
                                          1
The number of edges in GS is              2            aij xi xj
                                              i,j=1


                                                                                             67/93
Math programming formulations


We obtain the following 0-1 problem with one quadratic constraint:
                                       n
                                max          xi
                                      i=1

subject to:
                     n                            n
                            aij xi xj ≥ γ               xi xj
                    i,j=1                   i,j=1;i=j




                                                                     68/93
Math programming formulations

Define wij = xi xj . The constraint wij = xi xj is equivalent to

                wij ≤ xi , wij ≤ xj , wij ≥ xi + xj − 1 .
                                n
Linearized formulation: max           xi
                               i=1
                                n                      n
                subjectto :           aij wij ≥ γ               wij
                              i,j=1                 i,j=1;i=j

                wij ≤ xi , wij ≤ xj , wij ≥ xi + xj − 1 .
                  wij , xi ∈ {0, 1}, ∀i  j = 1, . . . , n
               O(n2 ) 0-1 variables, O(n2 ) constraints

                                                                      69/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     70/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     71/93
Clique relaxation and node deletion problems


       Node deletion problems


        Clique      Node deletion
      relaxation      problems
      problems      with heredity




                                               72/93
Node deletion problems


A graph property Π is said to be hereditary on induced
subgraphs, if G is a graph with property Π and the deletion of
any subset of vertices does not produce a graph violating Π
Π is said to be nontrivial if it is true for a single vertex graph
and is not satisfied by every graph
Π is said to be interesting if there are arbitrarily large graphs
satisfying Π




                                                                     73/93
Yannakakis theorem



The maximum Π problem is to find the largest order induced
subgraph that does not violate property Π

Theorem (Yannakakis, 1978)
The maximum Π problem for nontrivial, interesting graph properties
that are hereditary on induced subgraphs is NP-hard.




                                                                     74/93
Max weight node deletion problem

Given
       a simple, undirected graph G = (V , E ),
       positive weights w (v ) for each v ∈ V ,
       and a nontrivial, interesting and hereditary (on induced
       subgraphs) property Π.
Find

            ν(G ) = max{w (P) : P ⊆ V , G [P] satisfies Π},
where w (P) =       v ∈P   w (v ) and G [P] is the subgraph induced by P.


                                                                            75/93
Generalizing max clique algorithms


Some of the most practically effective algorithms for the maximum
clique problem rely on the fact that clique is hereditary on induced
subgraphs.
    Carraghan and Pardalos (1990) - used as DIMACS benchmark
    ¨
    Osterg˚rd (2002)
          a
We use this observation to develop a generalized algorithm for the
minimum weight node deletion problem (GAMWNDP).




                                                                       76/93
GAMWNDP

Order vertices V = {v1 , v2 , · · · , vn }, define Si = {vi , vi+1 , · · · , vn }
We compute the function c(i) that is the weight of the maximum
induced subgraph with property Π in G [Si ].
Obviously, c(n) = w (vn ) and c(1) = ν(G ).
For the unweighted case with w (vi ) = 1, i = 1, . . . , n we have

                   c(i + 1) + 1,      if the solution must contain vi
       c(i) =
                       c(i + 1),      otherwise

For the weighted case, c(i)  c(i + 1) implies that vi belongs to
every optimal solution, and c(i) ≤ c(i + 1) + w (vi ).


                                                                                   77/93
GAMWNDP


We compute the value of c(i) starting from c(n) and down to
c(1), and in each major iteration work with a current feasible
solution P, a candidate set C , and an incumbent solution S.
Pruning occurs when
    w (C ) + w (P)  w (S) or
    c(i) + w (P)  w (S), where i = min{j : vj ∈ C }.
Problem-specific features:
    candidate set generation;
    vertex ordering.




                                                                 78/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     79/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     80/93
k-Core and k-community


k-Core
A sub-graph G of G such that each node has degree greater than or
equal to k.


k-Community
A subgraph G of G such that each (i, j) ∈ E if (i, j) ∈ E and i, j
have at least k common neighbors in G .




                                                                     81/93
Example - k-Core                             k-Core example




          (a) Original Graph                 (b) Max 3-core


  Each node in the max 3-core is connected to at least 3 other
                                                                 82/93
  vertices.
Example - k-Community           k-Community example




          (c) Original Graph             (d) Max 2-community


  Any two vertices in the max 2-community are connected only if   83/93
  they have at least 2 mutual neighbors.
Scale reduction approaches



Reduce problem size by ignoring certain parts of the graph
which we are certain would not be in the graph.
Most commonly used technique: Peeling
 1. Remove all the vertices with degree ≤ k − 1.
 2. If no vertices removed, stop. Else, redo step 1 using the
    obtained graph.
Peeling is nothing but finding the k-core!




                                                                84/93
Scale reduction approaches


Property 1
A clique of size k is both a (k − 1)-core and (k − 2)-community.

Property 2
If the k-Core of G is empty, then ω(G )  k + 1.

Property 3
If the k-Community of G is empty, then ω(G )  k + 2.



                                                                   85/93
k-Core Upper Bound
      Find smallest k1 such that k1 -Core is empty. UB = k1 = 5.k-Core upper bound




                            (e) Original Graph                      (f) 4-core




                            (g) Original Graph                      (h) 5-core
      Anurag Verma, Sergiy Butenko, Texas AM University   10/ 20

Find smallest k1 such that k1 -Core is empty. UB = k1 = 5.
                                                                                     86/93
k-Community Upper Bound
                            k-Community upper bound
  Find smallest k2 such that k2 -Comm is empty.




     (i) Original Graph     (j) Intermediate      (k) 2-comm


 Find smallest k2 such that k2 -Comm 1 = 4.
  3-comm is empty. Thus UB = k2 + is empty. UB = k2 + 1 = 4.


                                                               87/93
Finding the least k2



A simple binary search strategy:
       Step 0 Set ku = n − 2, kl = 0, k2 = n − 2
       Step 1 If k2 -Comm(G) is empty, ku = k2 . Else, kl = k2 .
       Step 2 If ku − kl ≤ 1, set k2 = ku , STOP. Else, set
              k2 = (ku + kl )/2, go to Step 1.




                                                                   88/93
The call graph


Abello, Pardalos, and Resende, 1999:
    20-day period: 290 million vertices and 4 billion edges.
    one-day call graph: 53,767,087 vertices and over 170 millions of
    edges.
         3,667,448 connected components; 302,468 of them with more
         than 3 vertices.
         A giant connected component with 44,989,297 vertices.




                                                                       89/93
Upper Bounds obtained bounds on SNAP
              Upper on SNAP Graphs                                       database

         Graph                 n          m      k-Core   UB        k-Comm   UB
                                               UB            n’   UB           n”
         WikiTalk         2394385    4659565   132          700    52         237
         cit-Patents      3774768   16518947    65          106    35          83
         Email-EuAll       265214     364481    38          292    19          62
         Cit-HepPh          34546     420877    31           40    24          36
         Cit-HepTh          27770     352285    38           52    29          48
         Slashdot0811       77360     469180    55          129    34          87
         Slashdot0902       82168     504230    56          134    35          96
         soc-Epinions1      75879     405740    68          486    32          61
         Wiki-Vote           7115     100762    54          336    22          50
         p2p-Gnutella31     62586     147892     7         1004     4          57
         p2p-Gnutella04     10876      39994     8          365     4          12
         p2p-Gnutella24     26518      65369     6         7480     4          41
         p2p-Gnutella25     22687      54705     6         6091     4          25
         p2p-Gnutella30     36682      88328     8           14     4          42
         web-Stanford      281903    1992636    72          387    61         126
         web-NotreDame     325729    1090108   156         1367   155         155
         web-Google        875713    4322051    45           48    44          48
         web-BerkStan      685230    6649470   202          392   201         392
         Amazon0601        403394    2443408    11        32886    11        5361
         Amazon0505        410236    2439437    11        32632    11        4878
         Amazon0302        262111     899792     7          286     7         105
         Amazon0312        400727    2349869    11        27046    11        4534


  Comparison of the upper bounds obtained by a k-core scheme vs a                   90/93

  k-comm scheme.
Upper Bounds obtained on SNAPon SNAP
              Upper bounds Graphs                                        database

         Graph                 n          m      k-Core   UB        k-Comm   UB
                                               UB            n’   UB           n”
         WikiTalk         2394385    4659565   132          700    52         237
         cit-Patents      3774768   16518947    65          106    35          83
         Email-EuAll       265214     364481    38          292    19          62
         Cit-HepPh          34546     420877    31           40    24          36
         Cit-HepTh          27770     352285    38           52    29          48
         Slashdot0811       77360     469180    55          129    34          87
         Slashdot0902       82168     504230    56          134    35          96
         soc-Epinions1      75879     405740    68          486    32          61
         Wiki-Vote           7115     100762    54          336    22          50
         p2p-Gnutella31     62586     147892     7         1004     4          57
         p2p-Gnutella04     10876      39994     8          365     4          12
         p2p-Gnutella24     26518      65369     6         7480     4          41
         p2p-Gnutella25     22687      54705     6         6091     4          25
         p2p-Gnutella30     36682      88328     8           14     4          42
         web-Stanford      281903    1992636    72          387    61         126
         web-NotreDame     325729    1090108   156         1367   155         155
         web-Google        875713    4322051    45           48    44          48
         web-BerkStan      685230    6649470   202          392   201         392
         Amazon0601        403394    2443408    11        32886    11        5361
         Amazon0505        410236    2439437    11        32632    11        4878
         Amazon0302        262111     899792     7          286     7         105
         Amazon0312        400727    2349869    11        27046    11        4534


  Comparison of the number of nodes in the residual graph after the                 91/93

  scale reduction.
Upper bounds on random graphs




Uniform random graphs with 1000 nodes.
Bollobas-Erd¨ s estimate: 2 log(n)/ log(1/p).
            o
                                                      92/93
Outline
Introduction
    Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
   Complexity Issues
   Analytical bounds
   Mathematical programming formulations

Algorithms
   Node Deletion Problems with Heredity
   Scale Reduction Approaches

Conclusion and References
                                                     93/93
Conclusion

We propose a taxonomy of clique relaxation models
    opens new avenues for systematic study of clique relaxations
    covers known clique relaxation models
    unveils new models of potential interest
An effective algorithm is provided for node deletion problems
that are nontrivial, interesting, and hereditary on induced
subgraphs
    state-of-the-art exact approach to max k-plex
    can be used to solve many other important problems
A scale-reduction approach based on the concept of
k-community is proposed
    outperforms the peeling approach based on k-core
                                                                   94/93
References
B. Balasundaram, S. Butenko, I. V. Hicks.
Clique relaxations in social network analysis: The maximum k-plex
problem.
Operations Research, 59: 133–142, 2011.
S. Trukhanov, B. Balasundaram, and S. Butenko.
Exact algorithms for hard node deletion problems in networks.
Submitted.
S. Butenko, J. Pattillo, and N. Youssef.
A taxonomy of clique relaxation models in networks.
In preparation.
A. Verma and S. Butenko.
A new scale-reduction technique for the maximum clique and related
problems.
In preparation.
                                                                     95/93
References



J. Pattillo, A. Veremyev, S. Butenko, and V. Boginski.
On the maximum quasi-clique problem.
Submitted.
S. Butenko, S. Kahruman, and O. Prokopyev.
On “provably best” heuristics for hard optimization problems.
In preparation.




                                                                 96/93
hank you!
T
              97/93

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Clique Relaxation Models in Networks: Theory, Algorithms, and Applications

  • 1. Clique Relaxation Models in Networks: Theory, Algorithms, and Applications Sergiy Butenko (butenko@tamu.edu) Department of Industrial and Systems Engineering Texas A&M University College Station, TX 77843-3131 1/93
  • 2. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 2/93
  • 3. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 3/93
  • 4. Graphs Graph is the mathematical term for a “network” - often visualized as vertices (points, nodes) connected by edges (lines, arcs) 4/93
  • 5. Graph theory basics The origins of graph theory are attributed to the Seven Bridges of K¨nigsberg problem solved by Leonhard Euler in 1735. o 5/93
  • 6. Graph theory basics A simple, undirected graph is a pair G = (V , E ), where V is a finite set of vertices and E ⊆ V × V is a set of edges, with each edge defined on a pair of vertices. 6 5 1 4 2 3 V = {1, 2, 3, 4, 5, 6} E = {(1, 2), (1, 6), (2, 3), (2, 4), (3, 4), (4, 5), (5, 6)} 6/93
  • 7. Graph theory basics A graph can be represented in several ways. We can represent a graph G = (V , E ) by its |V | × |V | adjacency matrix AG = [aij ], such that 1, if (vi , vj ) ∈ E , aij = 0, otherwise. Another way of representing a graph is by its adjacency lists, where for each vertex v ∈ V we record the set of vertices adjacent to it. 7/93
  • 8. Graph theory basics A multigraph is a graph with repeated edges. A directed graph, or digraph, is a graph with directions assigned to its edges. 2 €€ 7 €€ b €€ I  d q    5 € d        d I €€ d d €€      € q ~  1 E 3 €€ d  d  8 E 10 d d    d €€ ‚   I b    €€ q   d 6 €    d   dd   I €€ €€ ‚  ~    q € 4 9 8/93
  • 9. Graph theory basics If G = (V , E ) is a graph and e = (u, v ) ∈ E , we say that u is adjacent to v and vice-versa. We also say that u and v are neighbors. Neighborhood N(v ) of a vertex v is the set of all its neighbors in G : N(v ) = {j : (i, j) ∈ E }. If G = (V , E ) is a graph and e = (u, v ) ∈ E , we say that e is incident to u and v . The degree of a vertex v is the number of its incident edges. 9/93
  • 10. Graph theory basics G = (V , E ) is a simple undirected graph, V = {1, 2, . . . , n}. G = (V , E ), is the complement graph of G = (V , E ), where E = {(i, j) | i, j ∈ V , i = j and (i, j) ∈ E }. / For S ⊆ V , G (S) = (S, E ∩ S × S) the subgraph induced by S. 10/93
  • 11. Cliques and independent sets A subset of vertices C ⊆ V is called a clique if G (C ) is a complete graph. A subset I ⊆ V is called an independent set (stable set, vertex packing) if G (I ) has no edges. ¯ C is a clique in G if and only if C is an independent set in G . 11/93
  • 12. Cliques and independent sets A clique (independent set) is said to be – maximal, if it is not a subset of any larger clique (independent set); – maximum, if there is no larger clique (independent set) in the graph. ω(G ) – the clique number of G . α(G ) – the independence (stability) number of G . 12/93
  • 13. Cliques and independent sets G 1 1 G Complement 5 2 5 2 4 3 4 3 {1,2,5} : maximal clique {1,2,5} : maximal independent set {1,4} : maximal {1,4} : maximal clique independent set {2,3,4,5} : maximum clique {2,3,4,5} : maximum independent set 13/93
  • 14. Social networks A social network is described by G = (V , E ) where V is the set of “actors” and E is the set of “ties”. actors are people and a tie exists if two people know each other. actors are wire transfer database records and a tie exists if two records have the same matching field. actors are telephone numbers and a tie exists if calls were made between them. 14/93
  • 15. Social network analysis “Popular” social networks: Kevin Bacon Number Erd¨s Number o Six degrees of separation and small world phenomenon in acquaintance networks 15/93
  • 16. Cohesive subgroups Cohesive subgroups are “tightly knit groups” in a social network. Social cohesion is often used to explain and develop sociological theories. Members of a cohesive subgroup are believed to share information, have homogeneity of thought, identity, beliefs, behavior, even food habits and illnesses. 16/93
  • 17. Applications Acquaintance Networks - criminal network analysis Wire Transfer Database Networks - detecting money laundering Call Networks - organized crime detection Protein Interaction Networks - predicting protein functions Gene Co-expression Networks - detecting network motifs Metabolic Networks - identifying metabolic pathways Stock Market Networks - stock portfolios Internet Graphs - information search and retrieval Wireless and telecommunication networks - clustering and routing 17/93
  • 18. Random graph models Uniform random graphs (Erdos Renyi): G (n, p): n vertices, each edge exists with a probability p; G (n, M): all graphs with n vertices and M edges are assigned the same probability Power-law random graphs: the probability that a vertex has a degree k is proportional to k −β Random graphs with a given degree sequence 18/93
  • 19. Scale-free graphs Scale-free nature : A property observed in many natural and man-made networks - biological networks, social networks, internet graphs Degree distribution follows a power law - P(k) ∝ k −β Fundamentally different from classical random graph model of Erd¨s and Renyi o Principle of preferential attachment underlies the growth and evolution of such networks (i.e. the rich get richer!) 19/93
  • 20. Science - co-authorship network Image source: http://www.jeffkennedyassociates.com:16080/connections/concept/image.html 20/93
  • 21. Internet colored by IP addresses Image source: http://www.jeffkennedyassociates.com:16080/connections/concept/image.html 21/93
  • 22. H. Pylori - protein interactions Image generated using Graphviz 22/93
  • 23. Properties of cohesive subgroups Some desirable properties of a cohesive subgroup are: Familiarity (degree); Reachability (distance, diameter); Robustness (connectivity); Density (edge density). Earliest models of cohesive subgroups were cliques (Luce and Perry, 1949), representing an “ideal” model of a cohesive subgroup. 23/93
  • 24. Clique relaxations: k-cliques and k-clubs A k-clique is a subset of vertices C such that for every i, j ∈ C , d(i, j) ≤ k. A k-club is a subset of vertices D such that diam(G [D]) ≤ k. 6 5 1 4 2 3 24/93
  • 25. Clique relaxations: k-cliques and k-clubs A k-clique is a subset of vertices C such that for every i, j ∈ C , d(i, j) ≤ k. A k-club is a subset of vertices D such that diam(G [D]) ≤ k. 6 {2,3,4} is a 1-club ... the “regular” clique 5 1 4 2 3 24/93
  • 26. Clique relaxations: k-cliques and k-clubs A k-clique is a subset of vertices C such that for every i, j ∈ C , d(i, j) ≤ k. A k-club is a subset of vertices D such that diam(G [D]) ≤ k. 6 {2,3,4} is a 1-club ... the “regular” clique 5 1 {1,2,4,5,6} is a 2-club 4 2 3 24/93
  • 27. Clique relaxations: k-cliques and k-clubs A k-clique is a subset of vertices C such that for every i, j ∈ C , d(i, j) ≤ k. A k-club is a subset of vertices D such that diam(G [D]) ≤ k. 6 {2,3,4} is a 1-club ... the “regular” clique 5 1 {1,2,4,5,6} is a 2-club {1,2,3,4,5} is a 2-clique but 4 2 NOT a 2-club 3 24/93
  • 28. Clique relaxations: k-cliques and k-clubs A k-clique is a subset of vertices C such that for every i, j ∈ C , d(i, j) ≤ k. A k-club is a subset of vertices D such that diam(G [D]) ≤ k. 6 {2,3,4} is a 1-club ... the “regular” clique 5 1 {1,2,4,5,6} is a 2-club {1,2,3,4,5} is a 2-clique but 4 2 NOT a 2-club maximality of a 2-club is harder 3 to test 24/93
  • 29. k-plex Definition A subset of vertices S is said to be a k-plex if the minimum degree in the induced subgraph δ(G [S]) ≥ |S| − k i.e. every vertex in G [S] has degree at least |S| − k. 2 1 6 3 5 4 25/93
  • 30. k-plex Definition A subset of vertices S is said to be a k-plex if the minimum degree in the induced subgraph δ(G [S]) ≥ |S| − k i.e. every vertex in G [S] has degree at least |S| − k. 2 1 {3,4,5,6} is a 1-plex ... the “regular” clique 6 3 5 4 25/93
  • 31. k-plex Definition A subset of vertices S is said to be a k-plex if the minimum degree in the induced subgraph δ(G [S]) ≥ |S| − k i.e. every vertex in G [S] has degree at least |S| − k. 2 1 {3,4,5,6} is a 1-plex ... the “regular” clique 6 3 {1,3,4,5,6} is a 2-plex (and NOT a 1-plex) 5 4 25/93
  • 32. k-plex Definition A subset of vertices S is said to be a k-plex if the minimum degree in the induced subgraph δ(G [S]) ≥ |S| − k i.e. every vertex in G [S] has degree at least |S| − k. 2 1 {3,4,5,6} is a 1-plex ... the “regular” clique 6 3 {1,3,4,5,6} is a 2-plex (and NOT a 1-plex) {1,2,3,4,5,6} is a 3-plex 5 4 (and NOT a 2-plex) 25/93
  • 33. co-k-plex Definition A subset of vertices S is a co-k-plex if the maximum degree in the induced subgraph ∆(G [S]) ≤ k − 1. i.e. degree of every vertex in G [S] is at most k − 1. S is a co-k-plex in G if and only if S is a k-plex in the complement ¯ graph G . 2 1 2 1 6 3 6 3 5 4 5 4 3-plex Co-3-plex 26/93
  • 34. Structural Properties of a k-Plex If G is a k-plex then 1. Every subgraph of G is a k-plex; n+2 2. If k 2 then diam(G ) ≤ 2; 3. κ(G ) ≥ n − 2k + 2. k-plexes for “small” k values, guarantee reachability and connectivity while relaxing familiarity. 27/93
  • 35. Known clique relaxation models k-clique (distance) k-club (diameter) k-plex (degree) k-core (degree) k-connected subgraph (connectivity) γ-quasi-clique (density) (γ, λ)-quasi-clique (density, degree) R-robust k-club (connectivity, diameter) 28/93
  • 36. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 29/93
  • 37. Alternative clique definitions Distance-based Vertices are distance one away from each other. Diameter-based Vertices induce a subgraph of diameter one. Domination-based Every one vertex forms a dominating set. 30/93
  • 38. Alternative clique definitions Degree-based Each vertex is connected to all other vertices. Connectivity-based All vertices need to be removed to disconnect the induced subgraph. Density-based Vertices induce a subgraph that has all possible edges. 31/93
  • 39. Order of a clique relaxation We introduce a hierarchy of clique relaxation models by defining their order as follows. Clique is the only clique relaxation of order 0. Clique relaxations of order 1 relax one of the elementary properties (distance, diameter, ...) used to define clique. Clique relaxations of order 2 relax two of the elementary clique-defining properties. ... 32/93
  • 40. Nature of a clique relaxation Parametric clique relaxations relax a parameter (“one”, “all”) describing an elementary clique-defining property. Partial clique relaxations require only a (high) fraction of elements to satisfy an elementary clique-defining property or a parametric clique-relaxation property. 33/93
  • 41. Types of parametric clique relaxations Parametric clique relaxation of type 1 Replace “one” with “(at most) k”, k 1, in a definition of clique. Parametric clique relaxation of type 2 Replace “one” with “(at most) all but k”, k 1, in a definition of clique. Parametric clique relaxation of type 1a Replace “all” with “(at least) k”, k 1, in a definition of clique. Parametric clique relaxation of type 2a Replace “all” with “(at least) all but k”, k 1, in a definition of clique. 34/93
  • 42. First-order parametric clique relaxations of type 1 Replace “one” with “(at most) k”, k 1. Distance-based: k-clique Vertices are distance at most k away from each other. Diameter-based: k-club Vertices induce a subgraph of diameter at most k. Domination-based: k-plex Every k vertices form a dominating set. 35/93
  • 43. First-order parametric clique relaxations of type 2 Replace “one” with “(at most) all but k”, k 1 Distance-based: N/A Vertices in S are distance at most |S| − k away from each other. Diameter-based: N/A Vertices induce a subgraph G [S] of diameter at most |S| − k. Domination-based: N/A Every |S| − k vertices form a dominating set. 36/93
  • 44. First-order parametric clique relaxations of type 1a Replace “all” with “(at least) k”, k 1 Degree-based: k-core Each vertex is connected to at least k other vertices. Connectivity-based: k-connected subgraph At least k vertices need to be removed to disconnect the induced subgraph. Density-based: N/A Vertices induce a subgraph that has at least k edges. 37/93
  • 45. First-order parametric clique relaxations of type 2a Replace “all” with “(at least) all but k”, k 1. Degree-based: k-plex Each vertex is connected to at least all but k other vertices. Connectivity-based: N/A At least all but k vertices need to be removed to disconnect the induced subgraph. Density-based: N/A Vertices induce a subgraph that has at least all but k edges. 38/93
  • 46. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 39/93
  • 47. Basics of the complexity theory Given a combinatorial optimization problem, a natural question is: is this problem “easy” or “hard”? Methods for easy and hard problems are fundamentally different. How do we distinguish between “easy” and “hard” problems? 40/93
  • 48. “Easy” problems By easy or tractable problems we mean the problems that can be solved in time polynomial with respect to their size. We also call such problems polynomially solvable and denote the class of polynomially solvable problems by P. Sorting Minimum weight spanning tree Linear programming 41/93
  • 49. Defining “hard” problems How do we define “hard” problems? How about defining hard problems as all problems that are not easy, i.e., not in P? Then some of the problems in such a class could be TOO hard – we cannot even hope to be able to solve them. We want to define a class of hard problems that we may be able to solve, if we are lucky (say, we may be able to guess the solution and check that it is indeed correct). 42/93
  • 50. Defining “hard” problems Let us take the maximum clique problem as an example and try to guess a solution. We can pick some random clique, compute its size, but how do we know if this solution is optimal? Indeed, this is as difficult to verify as to solve the maximum clique problem – it is unlikely that we can count on luck in this case... How about transforming our optimization problem to such a form, where instead of finding an optimal solution the question would be “is there a solution with objective value ≥ s, where s is a given integer? 43/93
  • 51. Three versions of optimization problems Consider a problem min f (x) subject to x ∈ X . Optimization version: find x from X that maximizes f (x); Answer: x ∗ maximizes f (x) Evaluation version: find the largest possible f (x); Answer: the largest possible value for f (x) is f ∗ Recognition version: Given f ∗ , does there exist an x such that f (x) ≥ f ∗ ? Answer: “yes” or “no” 44/93
  • 52. Recognition problems Recognition problems can still be undecidable. Halting problem: Given a computer program with its input, will it ever halt? Recall the “luck factor”: if we pick a random feasible solution and it happens to give “yes” answer, then we solved the problem in polynomial time. Max clique: Randomly pick a solution (a clique). If its size is ≥ s (which we can verify in polynomial time), then obviously the answer is “yes”. This clique can be viewed as a certificate proving that this is indeed a yes instance of max clique. 45/93
  • 53. Class N P We only consider problems for which any yes instance there exists a concise (polynomial-size) certificate that can be verified in polynomial time. We call this class of problems nondeterministic polynomial and denote it by N P. 46/93
  • 54. P vs N P Note that any problem from P is also in N P (i.e., P ⊆ N P), so there are easy problems in N P. So, are there “hard” problems in N P, and if there are, how do we define them? We don’t know if P = NP, but “most” people believe that P = NP. An “easy” way to make $1,000,000! http://www.claymath.org/millennium/P vs NP/ We can call a problem hard if the fact that we can solve this problem would mean that we can solve any other problem in comparable amount of time. 47/93
  • 55. Polynomial reducibility Reduce π1 to π2 : if we can solve π2 fast, then we can solve π1 fast, given that the reduction is “easy”. Polynomial reduction from π1 to π2 requires existence of polynomial-time algorithms 1. A1 converts an input for π1 into an input for π2 ; 2. A2 converts an output for π2 into output for π1 . Transitivity: If π1 is polynomially reducible to π2 and π2 is polynomially reducible to π3 then π1 is polynomially reducible to π3 . 48/93
  • 56. NP-complete problems A problem π is called NP-complete if 1. π ∈ NP; 2. Any problem from NP can be reduced to π in polynomial time. A problem π is called NP-hard if any problem from NP can be reduced to π in polynomial time. (no π ∈ NP requirement) Due to transitivity of polynomial reducibility, in order to show that a problem π is N P-complete, it is sufficient to show that 1. π ∈ NP; 2. There is an NP-complete problem π that can be reduced to π in polynomial time. To use this observation, we need to know at least one N P-complete problem... 49/93
  • 57. Satisfiability (SAT) problem A Boolean variable x is a variable that can assume only the values true and false. Boolean variables can be combined to form Boolean formulas using the following logical operations: 1. Logical AND (∧ or ·) -conjunction 2. Logical OR (∨ or +) - disjunction 3. Logical NOT (¯) x kj A clause is Cj = yjp , where a literal yjp is xr or xr for some r . ¯ p=1 m Conjunctive normal form (CNF):F = Cj , where Cj is a clause. j=1 50/93
  • 58. Satisfiability problem A CNF F is called satisfiable if there is an assignment of variables such that F = 1 (TRUE ). Satisfiability (SAT) problem: Given m clauses C1 , . . . , Cm involving the variables x1 , . . . , xn , is the CNF m F = Cj , j=1 satisfiable? Theorem (Cook, 1971) SAT is NP-complete. 51/93
  • 59. “Best” approximation algorithms and heuristics For some problems there are hardness of approximation results stating that the problem is hard to approximate within a certain factor. For example, the k-center problem is hard to approximate within a factor better than 2. Then any polynomial-time algorithm approximating the k-center problem within the factor of 2 can be considered the “best” approximation algorithm for this problem. 52/93
  • 60. “Best” approximation algorithms and heuristics However, some problems are even harder to approximate. For example, the maximum clique is hard to approximate within a factor n1− for any positive . In this case, by the “best” heuristic we could mean a heuristic that cannot be provably outperformed by any other polynomial-time algorithm (unless P = NP). 53/93
  • 61. Recognizing the gap between k-club and l -club numbers Theorem Let positive integer constants k and l, l k be given. The problem of checking whether ωl (G ) = ωk (G ) is NP-hard. ¯ ¯ Note that ω(G ) ≤ ∆(G ) + 1 ≤ ωk (G ) ¯ and observe that we can easily check whether ω(G ) = ∆(G ) + 1. Hence, it is NP-hard to check whether ωk (G ) = ∆(G ) + 1. ¯ 54/93
  • 62. “Best” heuristics for k-club/clique Corollary Let k be a fixed integer, k ≥ 2. Unless P = NP, there cannot be a polynomial time algorithm that finds a k-club of size greater than ∆(G ) + 1 whenever such a k-club exists in the graph. 55/93
  • 64. A max 2-club/clique of S. Cerevisiae. 57/93
  • 65. A max 2-club/clique of H. Pylori. 58/93
  • 66. A max 3-clique/club of S. Cerevisiae 59/93
  • 67. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 60/93
  • 68. Size of clique relaxations in a graph Question: How large could a k-clique (k-club, k-plex) be for a graph with a small clique number? 61/93
  • 69. Size of clique relaxations in a graph Question: How large could a k-clique (k-club, k-plex) be for a graph with a small clique number? Answers: k-club: ω(K1,n ) = 2, but ωk (K1,n ) = n + 1, could be arbitrary ¯ large k-clique: same as k-club 61/93
  • 70. Size of clique relaxations in a graph Question: How large could a k-clique (k-club, k-plex) be for a graph with a small clique number? Answers: k-club: ω(K1,n ) = 2, but ωk (K1,n ) = n + 1, could be arbitrary ¯ large k-clique: same as k-club k-plex: Lemma For any graph G and a positive integer k, ωk (G ) ≤ kω(G ) and αk (G ) ≤ kα(G ) 61/93
  • 71. Upper bound on the quasi-clique number Proposition The γ-clique number ωγ (G ) of a graph G with n vertices and m edges satisfies the following inequality: γ+ γ 2 + 8γm ωγ (G ) ≤ . (1) 2γ Moreover, if a graph G is connected then γ+2+ (γ + 2)2 + 8(m − n)γ ωγ (G ) ≤ . (2) 2γ 62/93
  • 72. Relation between ωγ (G ) and ω(G ) Proposition The γ-clique number ωγ (G ) and the clique number ω(G ) of graph G satisfy the following inequalities: ω(G ) − 1 ωγ (G ) − 1 1 ω(G ) − 1 ≤ ≤ . (3) ω(G ) ωγ (G ) γ ω(G ) Corollary 1 If γ 1 − ω(G ) then ω(G )γ ωγ (G ) ≤ . (4) 1 − ω(G ) + ω(G )γ 63/93
  • 73. Relation between omegaγ (G ) and ω(G ) Table: The value of upper bound (4) on γ-clique number with γ = 0.95, 0.9, 0.85 for graphs with small clique number. 1 ω(G ) 1 − ω(G ) 0.95 0.9 0.85 3 0.667 3.35 3.86 4.64 4 0.75 4.75 6 8.5 5 0.8 6.33 9 17 6 0.83 8.14 13.5 51 7 0.86 10.23 21 – 8 0.88 12.67 36 – 9 0.89 15.55 81 – 10 0.9 19 – – 64/93
  • 74. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 65/93
  • 75. Math programming formulations Consider a simple undirected graph G = (V , E ) with n vertices Let A = [aij ]n i,j=1 be the adjacency matrix of G Let x = (x1 , . . . , xn ) be a 0-1 vector with xi = 1 if node i belongs to Gs , and xi = 0 otherwise. 66/93
  • 76. Math programming formulations GS is a γ-clique if it has at least γ|GS |(|GS | − 1)/2 edges n We have: |GS | = xi i=1 This number of edges can be expressed in terms of vector x as: n n n n 1 2γ xi xi − 1 = 1γ 2 xi xj − xi i=1 i=1 i,j=1 i=1 n n n n 1 1 = 2γ xi xj + xi2 − xi = 2γ xi xj i,j=1;i=j i=1 i=1 i,j=1;i=j n 1 The number of edges in GS is 2 aij xi xj i,j=1 67/93
  • 77. Math programming formulations We obtain the following 0-1 problem with one quadratic constraint: n max xi i=1 subject to: n n aij xi xj ≥ γ xi xj i,j=1 i,j=1;i=j 68/93
  • 78. Math programming formulations Define wij = xi xj . The constraint wij = xi xj is equivalent to wij ≤ xi , wij ≤ xj , wij ≥ xi + xj − 1 . n Linearized formulation: max xi i=1 n n subjectto : aij wij ≥ γ wij i,j=1 i,j=1;i=j wij ≤ xi , wij ≤ xj , wij ≥ xi + xj − 1 . wij , xi ∈ {0, 1}, ∀i j = 1, . . . , n O(n2 ) 0-1 variables, O(n2 ) constraints 69/93
  • 79. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 70/93
  • 80. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 71/93
  • 81. Clique relaxation and node deletion problems Node deletion problems Clique Node deletion relaxation problems problems with heredity 72/93
  • 82. Node deletion problems A graph property Π is said to be hereditary on induced subgraphs, if G is a graph with property Π and the deletion of any subset of vertices does not produce a graph violating Π Π is said to be nontrivial if it is true for a single vertex graph and is not satisfied by every graph Π is said to be interesting if there are arbitrarily large graphs satisfying Π 73/93
  • 83. Yannakakis theorem The maximum Π problem is to find the largest order induced subgraph that does not violate property Π Theorem (Yannakakis, 1978) The maximum Π problem for nontrivial, interesting graph properties that are hereditary on induced subgraphs is NP-hard. 74/93
  • 84. Max weight node deletion problem Given a simple, undirected graph G = (V , E ), positive weights w (v ) for each v ∈ V , and a nontrivial, interesting and hereditary (on induced subgraphs) property Π. Find ν(G ) = max{w (P) : P ⊆ V , G [P] satisfies Π}, where w (P) = v ∈P w (v ) and G [P] is the subgraph induced by P. 75/93
  • 85. Generalizing max clique algorithms Some of the most practically effective algorithms for the maximum clique problem rely on the fact that clique is hereditary on induced subgraphs. Carraghan and Pardalos (1990) - used as DIMACS benchmark ¨ Osterg˚rd (2002) a We use this observation to develop a generalized algorithm for the minimum weight node deletion problem (GAMWNDP). 76/93
  • 86. GAMWNDP Order vertices V = {v1 , v2 , · · · , vn }, define Si = {vi , vi+1 , · · · , vn } We compute the function c(i) that is the weight of the maximum induced subgraph with property Π in G [Si ]. Obviously, c(n) = w (vn ) and c(1) = ν(G ). For the unweighted case with w (vi ) = 1, i = 1, . . . , n we have c(i + 1) + 1, if the solution must contain vi c(i) = c(i + 1), otherwise For the weighted case, c(i) c(i + 1) implies that vi belongs to every optimal solution, and c(i) ≤ c(i + 1) + w (vi ). 77/93
  • 87. GAMWNDP We compute the value of c(i) starting from c(n) and down to c(1), and in each major iteration work with a current feasible solution P, a candidate set C , and an incumbent solution S. Pruning occurs when w (C ) + w (P) w (S) or c(i) + w (P) w (S), where i = min{j : vj ∈ C }. Problem-specific features: candidate set generation; vertex ordering. 78/93
  • 88. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 79/93
  • 89. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 80/93
  • 90. k-Core and k-community k-Core A sub-graph G of G such that each node has degree greater than or equal to k. k-Community A subgraph G of G such that each (i, j) ∈ E if (i, j) ∈ E and i, j have at least k common neighbors in G . 81/93
  • 91. Example - k-Core k-Core example (a) Original Graph (b) Max 3-core Each node in the max 3-core is connected to at least 3 other 82/93 vertices.
  • 92. Example - k-Community k-Community example (c) Original Graph (d) Max 2-community Any two vertices in the max 2-community are connected only if 83/93 they have at least 2 mutual neighbors.
  • 93. Scale reduction approaches Reduce problem size by ignoring certain parts of the graph which we are certain would not be in the graph. Most commonly used technique: Peeling 1. Remove all the vertices with degree ≤ k − 1. 2. If no vertices removed, stop. Else, redo step 1 using the obtained graph. Peeling is nothing but finding the k-core! 84/93
  • 94. Scale reduction approaches Property 1 A clique of size k is both a (k − 1)-core and (k − 2)-community. Property 2 If the k-Core of G is empty, then ω(G ) k + 1. Property 3 If the k-Community of G is empty, then ω(G ) k + 2. 85/93
  • 95. k-Core Upper Bound Find smallest k1 such that k1 -Core is empty. UB = k1 = 5.k-Core upper bound (e) Original Graph (f) 4-core (g) Original Graph (h) 5-core Anurag Verma, Sergiy Butenko, Texas AM University 10/ 20 Find smallest k1 such that k1 -Core is empty. UB = k1 = 5. 86/93
  • 96. k-Community Upper Bound k-Community upper bound Find smallest k2 such that k2 -Comm is empty. (i) Original Graph (j) Intermediate (k) 2-comm Find smallest k2 such that k2 -Comm 1 = 4. 3-comm is empty. Thus UB = k2 + is empty. UB = k2 + 1 = 4. 87/93
  • 97. Finding the least k2 A simple binary search strategy: Step 0 Set ku = n − 2, kl = 0, k2 = n − 2 Step 1 If k2 -Comm(G) is empty, ku = k2 . Else, kl = k2 . Step 2 If ku − kl ≤ 1, set k2 = ku , STOP. Else, set k2 = (ku + kl )/2, go to Step 1. 88/93
  • 98. The call graph Abello, Pardalos, and Resende, 1999: 20-day period: 290 million vertices and 4 billion edges. one-day call graph: 53,767,087 vertices and over 170 millions of edges. 3,667,448 connected components; 302,468 of them with more than 3 vertices. A giant connected component with 44,989,297 vertices. 89/93
  • 99. Upper Bounds obtained bounds on SNAP Upper on SNAP Graphs database Graph n m k-Core UB k-Comm UB UB n’ UB n” WikiTalk 2394385 4659565 132 700 52 237 cit-Patents 3774768 16518947 65 106 35 83 Email-EuAll 265214 364481 38 292 19 62 Cit-HepPh 34546 420877 31 40 24 36 Cit-HepTh 27770 352285 38 52 29 48 Slashdot0811 77360 469180 55 129 34 87 Slashdot0902 82168 504230 56 134 35 96 soc-Epinions1 75879 405740 68 486 32 61 Wiki-Vote 7115 100762 54 336 22 50 p2p-Gnutella31 62586 147892 7 1004 4 57 p2p-Gnutella04 10876 39994 8 365 4 12 p2p-Gnutella24 26518 65369 6 7480 4 41 p2p-Gnutella25 22687 54705 6 6091 4 25 p2p-Gnutella30 36682 88328 8 14 4 42 web-Stanford 281903 1992636 72 387 61 126 web-NotreDame 325729 1090108 156 1367 155 155 web-Google 875713 4322051 45 48 44 48 web-BerkStan 685230 6649470 202 392 201 392 Amazon0601 403394 2443408 11 32886 11 5361 Amazon0505 410236 2439437 11 32632 11 4878 Amazon0302 262111 899792 7 286 7 105 Amazon0312 400727 2349869 11 27046 11 4534 Comparison of the upper bounds obtained by a k-core scheme vs a 90/93 k-comm scheme.
  • 100. Upper Bounds obtained on SNAPon SNAP Upper bounds Graphs database Graph n m k-Core UB k-Comm UB UB n’ UB n” WikiTalk 2394385 4659565 132 700 52 237 cit-Patents 3774768 16518947 65 106 35 83 Email-EuAll 265214 364481 38 292 19 62 Cit-HepPh 34546 420877 31 40 24 36 Cit-HepTh 27770 352285 38 52 29 48 Slashdot0811 77360 469180 55 129 34 87 Slashdot0902 82168 504230 56 134 35 96 soc-Epinions1 75879 405740 68 486 32 61 Wiki-Vote 7115 100762 54 336 22 50 p2p-Gnutella31 62586 147892 7 1004 4 57 p2p-Gnutella04 10876 39994 8 365 4 12 p2p-Gnutella24 26518 65369 6 7480 4 41 p2p-Gnutella25 22687 54705 6 6091 4 25 p2p-Gnutella30 36682 88328 8 14 4 42 web-Stanford 281903 1992636 72 387 61 126 web-NotreDame 325729 1090108 156 1367 155 155 web-Google 875713 4322051 45 48 44 48 web-BerkStan 685230 6649470 202 392 201 392 Amazon0601 403394 2443408 11 32886 11 5361 Amazon0505 410236 2439437 11 32632 11 4878 Amazon0302 262111 899792 7 286 7 105 Amazon0312 400727 2349869 11 27046 11 4534 Comparison of the number of nodes in the residual graph after the 91/93 scale reduction.
  • 101. Upper bounds on random graphs Uniform random graphs with 1000 nodes. Bollobas-Erd¨ s estimate: 2 log(n)/ log(1/p). o 92/93
  • 102. Outline Introduction Graph Theory Basics Taxonomy of Clique Relaxations Theory Complexity Issues Analytical bounds Mathematical programming formulations Algorithms Node Deletion Problems with Heredity Scale Reduction Approaches Conclusion and References 93/93
  • 103. Conclusion We propose a taxonomy of clique relaxation models opens new avenues for systematic study of clique relaxations covers known clique relaxation models unveils new models of potential interest An effective algorithm is provided for node deletion problems that are nontrivial, interesting, and hereditary on induced subgraphs state-of-the-art exact approach to max k-plex can be used to solve many other important problems A scale-reduction approach based on the concept of k-community is proposed outperforms the peeling approach based on k-core 94/93
  • 104. References B. Balasundaram, S. Butenko, I. V. Hicks. Clique relaxations in social network analysis: The maximum k-plex problem. Operations Research, 59: 133–142, 2011. S. Trukhanov, B. Balasundaram, and S. Butenko. Exact algorithms for hard node deletion problems in networks. Submitted. S. Butenko, J. Pattillo, and N. Youssef. A taxonomy of clique relaxation models in networks. In preparation. A. Verma and S. Butenko. A new scale-reduction technique for the maximum clique and related problems. In preparation. 95/93
  • 105. References J. Pattillo, A. Veremyev, S. Butenko, and V. Boginski. On the maximum quasi-clique problem. Submitted. S. Butenko, S. Kahruman, and O. Prokopyev. On “provably best” heuristics for hard optimization problems. In preparation. 96/93
  • 106. hank you! T 97/93