1. Introduction
Algorithms
Application
Conclusion
The Maximum Clique Problem
Dam Thanh Phuong, Ngo Manh Tuong
November, 2012
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
2. Introduction
Algorithms
Application
Conclusion
Motivation
How to put as much left-over stuff as possible in a tasty meal
before everything will go off?
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
3. Introduction
Algorithms
Application
Conclusion
Motivation
Find the largest collection of food where everything goes
together! Here, we have the choice:
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
4. Introduction
Algorithms
Application
Conclusion
Motivation
Find the largest collection of food where everything goes
together! Here, we have the choice:
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
5. Introduction
Algorithms
Application
Conclusion
Motivation
Find the largest collection of food where everything goes
together! Here, we have the choice:
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
6. Introduction
Algorithms
Application
Conclusion
Motivation
Find the largest collection of food where everything goes
together! Here, we have the choice:
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
7. Introduction
Algorithms
Application
Conclusion
Outline
1 Introduction
2 Algorithms
3 Applications
4 Conclusion
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
8. Introduction
Algorithms
Application
Conclusion
Graph (G): a network of vertices (V(G)) and edges (E(G)).
Graph Complement (G ): the graph with the same vertex set
of G but whose edge set consists of the edges not present in G.
Complete Graph: every pair of vertices is connected by an
edge.
A Clique in an undirected graph G=(V,E) is a subset of the
vertex set C ⊆ V ,such that for every two vertices in C, there
exists an edge connecting the two.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
9. Introduction
Algorithms
Application
Conclusion
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
10. Introduction
Algorithms
Application
Conclusion
Maximum Clique: A Clique of the largest possible size in a
given graph. The clique number, ω (G ), is the cardinality of
the maximum clique.
Maximal Clique: A Clique that cannot be extended by
including one more adjacent vertex.
Independent Set: a subset of the vertices such that no two
vertices in the subset are connected by an edge of G.
Vertex cover: a subset of the vertices of G which contains at
least one of the two endpoints of each edge.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
11. Introduction
Algorithms
Application
Conclusion
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
12. Introduction
Algorithms
Application
Conclusion
Maximum Clique Problem
Does there exist an integer k such that G contains an clique
of cardinality k?
What is the clique in G with maximum cardinality?
What is the clique number of G?
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
13. Introduction
Algorithms
Application
Conclusion
Equivalent Problems
Maximum Independent Set Problem in G
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
14. Introduction
Algorithms
Application
Conclusion
Equivalent Problems
Minimum Vertex Cover Problem in G
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
15. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
NP-hard
A problem is NP-hard if solving it in polynomial time would
make it possible to solve all problems in the class of NP
problems in polynomial time.
All 3 versions of the Maximum Clique problem are known to
be NP-hard for general graphs.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
16. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
NP : the class of decision problem which can be solved by a
non-deterministic polynomial algorithm.
P: the class of problems which can be solved by a
deterministic polynomial algorithm.
NP-hard: the class of problems to which every NP problem
reduces.
NP-complete (NPC): the class of problems which are NP-hard
and belong to NP.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
17. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Method to Solve Maximum Clique Problem
Non polynomial-time algorithms.
Polynomial-time algorithms providing approximate solutions.
Polynomial-time algorithms providing exact solutions to
graphs of special classes.
Two effective algorithms for dealing with NP-complete
Problems: backtracking, branch and bound
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
18. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Introduction Bron Kerbosch Algorithm
The Algorithm was Designed and Published in 1973 by the
Dutch scientists Joep Kerbosch and Coenradd Bron.
Bron Kerbosch Algorithm is for Finding the Maximal Cliques
in undirected graph.
It is known to be one of the most efficient algorithm which
uses recursive backtracking to find Cliques is practically
proven.
The Bron Kerbosch Algorithm uses the vertex in graph and its
neighbour with few functions to generate some effective
results.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
19. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Without Pivoting Strategy
BronKerbosch(R, P, X )
if {P = X = ∅}
Report R as the Maximal Clique
for each vertex v in P
BronKerbosch(R ∪ {v } , P ∩ N {v } , X ∩ N {v })
P := P {v }
X := X ∪ {v }
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
20. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
With Pivoting Strategy
BronKerbosch(R, P, X )
if {P = X = ∅}
Report R as the Maximal Clique
Choose Pivot Vertex u in P ∪ X
for each vertex v in P
BronKerbosch(R ∪ {v } , P ∩ N {v } , X ∩ N {v })
P := P {v }
X := X ∪ {v }
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
21. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Example
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
22. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
R = X = ∅, P = (1, 2, 3, 4, 5, 6)
Choosing the pivot element u as 4.
4 in PN(v ) = (1, 2, 3, 4, 5, 6) (1, 2, 3, 5, 6) = 4 in 4
Finds the values of Rnew , Pnew , Xnew
Pnew = P ∩ N (v ); Rnew = R ∪ v ; Xnew = X ∩ N (v )
Rnew = 4; Pnew = (1, 2, 3, 5, 6) ; Xnew = ∅
BronKerbosch(4,(1,2,3,5,6),∅)
BronKerbosch((4,1),(2,3),∅)
BronKerbosch((4,1,2),∅,∅)
Report (4,1,2) as one of the Maximal Clique
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
23. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
BronKerbosch(4,(1,2,3,5,6),∅)
BronKerbosch((4,3),(1),∅)
BronKerbosch((4,3,1),∅,∅)
Report (4,3,1) as one of the other Maximal Clique.
BronKerbosch(4,(1,2,3,5,6),∅)
BronKerbosch((4,2),(1,5),∅)
BronKerbosch((4,2,5),∅,∅)
Report (4,2,5) as an other Maximal Clique.
BronKerbosch(4,(1,2,3,5,6),∅)
BronKerbosch((4,6),∅,∅)
Report (4,6) as the Maximal Clique.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
24. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
This backtracking algorithm is a method for finding all the
sub sets in an undirected graph G.
Given a graph G with V vertices and E edges, G=(V,E)
Let us take an integer variable k.
This algorithm is used in scientific and engineering
applications.
This algorithm is a Depth First Search algorithm.
The algorithm for finding k-clique in an undirected graph is a
NP-complete problem.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
25. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Example
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
26. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
27. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
28. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
List out all the possibilities in the subgraph and check for each
and every edge.
Check for a subgraph in which every node is connected to
every other node.
Check for all possible Cliques in the graphs.
Check the size of clique whether it is equal to k or not.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
29. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
Any n-vertex graph has at most 3n/3 Maximal Cliques
The worst-case running time of the BronKerbosch algorithm
(with a pivot strategy that minimizes the number of recursive
calls made at each step) is O(3n/3 )
This Backtracking algorithm runs in polynomial time if size of
k is fixed. If k is varying then it is in exponencial time
Running time of the algorithm is O (nk), where k = O(log n)
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
30. Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound
n
f (x) = − xi → min
i=1
xi + xj ≤ 1, ∀ (i, j) ∈ E
x ∈ {0, 1}n
f (x) = x T Ax → min
x ∈ {0, 1}n , whereA = AG − I
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
31. Introduction
Algorithms
Application
Conclusion
Scheduling
Coding Theory: Hamming and Johnson Graphs
Map Labeling
Computer Vision and Pattern Recognition
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
32. Introduction
Algorithms
Application
Conclusion
Problem: finding maximum cliques of a graph efficiently
Hard task (in terms of memory and runtime)
Bron-Kerbosch algorithm is one efficient solution
Several applications
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
33. Introduction
Algorithms
Application
Conclusion
References
1. D.-Z. Du and P.M.Pardalos ”Handbook of Combinatorial
Optimization”. Kluwer Academic Publishers, 1999
2. http : //en.wikipedia.org/wiki/Clique - problem.
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem
34. Introduction
Algorithms
Application
Conclusion
Thank you for your attention!
Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem