Kernelization algorithms for graph and other structure modiﬁcation problems

Kernelization algorithms for graph and other
structure modiﬁcation problems

Anthony P EREZ
´
Supervisors: Stephane B ESSY and Christophe PAUL
(AlGCo Team)

November 14

I NTRODUCTION

(Graph) Modification problems

Input: A graph (or another structure) and a (graph) property G.
Output: A minimum number of modification of the graph in order to
satisfy G.

modification: adding edges, deleting edges, deleting vertices, ...

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I NTRODUCTION


C LUSTER E DITING
Input: A graph G = (V , E).
Output: A set F ⊆ (V × V ) of minimum size such that the graph
H = (V , E F ) is a disjoint union of cliques.

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I NTRODUCTION


Cover a broad range of NP-Hard problems:
VERTEX COVER
FEEDBACK VERTEX SET
More general: F - MINOR DELETION
EDGE - MULTICUT

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I NTRODUCTION


Find applications in various domains:
bioinformatics
machine learning
relational databases
image processing

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I NTRODUCTION

Different approaches

Most modiﬁcation problems are NP-hard.
How to solve them efﬁciently?

Approximation algorithms
Exact exponential algorithms
Preprocessing steps (heuristics)

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I NTRODUCTION




How to measure the efﬁciency of heuristics?

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I NTRODUCTION




Exploit the fact that the number of modiﬁcations needed should be
small compared to the instance size n.

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Outline of the talk

1 Parameterized complexity

Part I. Graph Modiﬁcation Problems

2 Branches and generic reduction rules

3 P ROPER I NTERVAL C OMPLETION

Part II. Different modiﬁcation problems

4 Considered problems

5 F EEDBACK A RC S ET IN TOURNAMENTS

PARAMETERIZED COMPLEXITY

Parameterized problem

G-M ODIFICATION
Input: A graph G = (V , E), k ∈ N.
Parameter: k .
Output: A set F ⊆ (V × V ) of size at most k such that the graph
H = (V , E F ) belongs to G.

Idea. Measure the complexity of a problem with respect to
some parameter k.

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Parameterized problem

G-M ODIFICATION
Parameter: k .
Output: A set F ⊆ (V × V ) of size at most k such that the graph
H = (V , E F ) belongs to G.

Parameterized algorithm
A problem parameterized by some k ∈ N admits a parameterized
algorithm if it can be solved in time f (k ) · nO(1) .

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Kernels

Given an instance (I, k ) of a parameterized problem L,
a kernelization algorithm:

runs in time Poly (|I| + k)
and outputs an instance (I , k ) such that:
(i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES
(ii) |I | h(k ) and k k

(I , k ) (I , k )
Poly (|I | + k )
|I | h(k )
k k
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Kernels


(i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES

Theorem (Folklore)
Parameterized algorithm ⇔ Kernelization algorithm

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Kernels


(i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES

Size: super-polynomial

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Kernels


(i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES

Size: super-polynomial

Do all parameterized problems admit polynomial kernels?

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Lower bounds for kernels

There exist some parameterized problems that do not admit polynomial
kernels. (under a complexity-theoretic assumption)
(i) Or-composition [Bodlaender et al., 2008 - Fortnow and Santhanam, 2008]
(ii) Polynomial time and parameter transformations
[Bodlaender et al., 2009]
(iii) Cross-composition [Bodlaender et al., 2011]

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Graph modiﬁcation problems



G-M ODIFICATION
Parameter: k.
Output: A set F ⊆ (V × V ) of size at most k s.t. the graph H = (V , E F ) belongs to G.

B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION

Generic reduction rules

Connected component.

If G is hereditary and closed under disjoint union, remove any
connected component C that belongs to G.

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Sunﬂower.
Consider a ﬁnite forbidden induced subgraph of G (obstruction).
For any pair e ⊆ (V × V ) that belongs to a set of k + 1 obstructions
pairwise intersecting exactly in e, transform G into (V , E {e}).

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Critical clique.
Assume G is hereditary and closed under true twin addition.
For any critical clique T with |T | > k + 1, remove |T | − (k + 1)
arbitrary vertices from T .

u

v

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Critical clique.
Assume G is hereditary and closed under true twin addition.
For any critical clique T with |T | > k + 1, remove |T | − (k + 1)
arbitrary vertices from T .

k =1

Lemma [Bessy, Paul and P., 2010]
There always exists an optimal edition
that preserves the critical cliques.

k =1

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Branches: a natural idea

Reduce set of vertices that induce a graph belonging to G.
The Connected Component rule is a Branch reduction rule.

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Context: can be used on problems where G admits a so-called
adjacency decomposition.

Branch: set of vertices B ⊆ V
such that:

(i) G[B] ∈ G and,
(ii) B is connected properly
to the rest of the graph.

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Context: can be used on problems where G admits a so-called
adjacency decomposition.
G [B ] ∈ G

Branch: set of vertices B ⊆ V B
such that:

(i) G[B] ∈ G and,
(ii) B is connected properly
to the rest of the graph.
GB

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Outline

Branches

Deﬁnition and known results
Branches
Reducing the branches


Deﬁnition
P ROPER I NTERVAL C OMPLETION
Parameter: k .
Output: A set F ⊆ (V × V ) E of size at most k such that
H = (V , E ∪ F ) is a proper interval graph.

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Deﬁnition
Parameter: k .

NP-Complete [Golumbic et al., 1994]
FPT : O(24k m) (motivated by applications in genomic research)
[Kaplan, Shamir and Tarjan, 1994]
Polynomial kernel?

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Deﬁnition
Parameter: k .

Theorem [Bessy and P., 2011]
The P ROPER I NTERVAL C OMPLETION problem admits a kernel with
O(k 4 ) vertices.

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Some useful results

A graph is a proper interval graph if and only if:

it does not contain any of the following graphs as an induced
subgraph.

claw 3-sun net p-cycle (p ≥ 4)

[Wegner, 1967]

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Some useful results

A graph is a proper interval graph if and only if:

its vertices admit an ordering v1 . . . vn such that:

vi vj ∈ E i < j ⇒ vi vl , vl vj ∈ E, i < l < j

[Looges and Olartu, 1993]

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Remarks. Proper interval graphs are hereditary and:
(i) closed under disjoint union:
the Connected Component rule can be applied.
(ii) do not admit any claw or C4 as an induced subgraph:
the Sunﬂower rule can be applied.
(iii) closed under true twin addition:
the Critical Clique rule can be applied.

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Remarks. Proper interval graphs are hereditary and:
(i) closed under disjoint union:
the Connected Component rule can be applied.
(ii) do not admit any claw or C4 as an induced subgraph:
the Sunﬂower rule can be applied.
(iii) closed under true twin addition:
the Critical Clique rule can be applied.

What about branches?

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Adjacency decomposition

1 3

6 8
(a) 2

4 5 7 9

4
6 7
(b) 3
8
2 1 2 3 4 5 6 7 8 9
5 9
1

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Adjacency decomposition

1 3

6 8
(a) 2

4 5 7 9

4
6 7
(b) 3
8
2 1 2 3 4 5 6 7 8 9
5 9
1

Branches can be used on P ROPER I NTERVAL C OMPLETION.

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How to deﬁne a branch?

Consider the structure of a solution.
Look at unaffected vertices.

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Branches
B
B1 BR B2
b1 bl
L R C

A subset B of V is a branch if:
(i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| ,
(ii) The vertex set V B can be partitioned into sets L, R and C with:
no edges between B and C
every vertex in L (resp. R) has a neighbor in B
NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl }
NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| }
NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for
every l ≤ i < |B|

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Branches
B
B1 BR B2
b1 b|B|
L R C

NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl }
NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| }
every l ≤ i < |B|

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Branches
B
B1 BR B2
b1 bl bl b|B|
L R C

NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl }
NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| }
every l ≤ i < |B|

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Branches

B
B1 BR B2
b1 bl
L R C

If L = ∅ (or R = ∅), B is a 1-branch, otherwise B is a 2-branch
If B is a clique, we call B a K-join

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Reducing the K -joins

Cannot be done directly.

x y z t

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A clean K -join does not intersect any claw or C4 .

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A clean K -join does not intersect any claw or C4 .

Assuming the graph is reduced by the generic rules, we can remove
O(k 3 ) vertices from any K -join to obtain a clean K -join.

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Reducing the clean K -joins

Let B be a clean K -join of size at least 2k + 2. Let Bf be the k + 1 ﬁrst
vertices of B, Bl be its k + 1 last vertices and M = B (Bf ∪ Bl ).
Remove the set of vertices M from G.

Bf (k + 1 vertices) M Bl (k + 1 vertices)

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Reducing the clean K -joins

Let B be a clean K -join of size at least 2k + 2. Let Bf be the k + 1 ﬁrst
vertices of B, Bl be its k + 1 last vertices and M = B (Bf ∪ Bl ).
Remove the set of vertices M from G.
Can be carried out in polynomial time!

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In polynomial time, the 1- and 2-branches can be reduced to O(k 3 )
vertices.

Remove 2k + 1 vertices

BR B1
R G (B ∪ R )
B

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In polynomial time, the 1- and 2-branches can be reduced to O(k 3 )
vertices.

Remove 2k + 1 vertices

BR B1
R G (B ∪ R )
B
2k + 1 vertices Remove 2k + 1 vertices

B1 B1 BR B2 B2
L R
B

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Main result

O(k 4 ) vertices.

1-branch K -join K -join 2-branch K -join 1-branch

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Main result

O(k 4 ) vertices.

1-branch K -join K -join 2-branch K -join 1-branch
O (k 3 ) O (k 3 ) O (k 3 ) O (k 3 ) O (k 3 ) O (k 3 )

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Main result

O(k 4 ) vertices.

Related result [Bessy, Paul and P., 2010]
The C LOSEST 3-L EAF P OWER problem admits a kernel with O(k 3 )
vertices.

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Different modification problems



Π-E DITION
Input: A dense set R of p-sized relations defined over an universe V , an integer k ∈ N.
Parameter: k.
Output: A set F ⊆ R of size at most k whose modification satisfies Π.

C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS

F EEDBACK A RC S ET IN TOURNAMENTS (FAST)

Input: A tournament T = (V , A) and an integer k ∈ N.
Parameter: k .
Output: A set at most k arcs whose reversal results in an acyclic
tournament.
1 4

3 1 4 2
2 3

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F EEDBACK A RC S ET IN TOURNAMENTS (FAST)

Input: A tournament T = (V , A) and an integer k ∈ N.
Parameter: k .
Output: A set at most k arcs whose reversal results in an acyclic
tournament.

NP-Complete [Charbit et al., 2007]
Admits constant-factor approximation algorithms [Kenyon-Mathieu and
Schudy, 2007]

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D ENSE R OOTED T RIPLET I NCONSISTENCY (RTI)
Input: A set of leaves V and a dense collection R of rooted binary
trees over three leaves of V .
Parameter: k .
Output: A set of at most k triplets whose modiﬁcation leads to a
collection admitting a consistent rooted binary tree deﬁned over V .
t1 t2 t3 t4

a b c c d b a b d a c d

R := {t1 , t2 , t3 , t4 }
R := {ab|c, cd|b, ab|d, ac|d}

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Parameter: k .
t1 t2 t3 t4

a b c c d b a b d a c d

R := {t1 , t2 , t3 , t4 }
R := {ab|c, cd|b, ab|d, ac|d}

T is not consistent with R
a b c d
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Parameter: k .
t1 t2 t3 t4

a b c c d b a b d c d a

R := {t1 , t2 , t3 , t4 }
R := {ab|c, cd|b, ab|d, cd|a}

T is consistent with R
a b c d
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Parameter: k .

NP-Complete [Barky et al., 2010]
Does not admit a constant-factor approximation algorithm yet

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Outline

F EEDBACK A RC S ET IN TOURNAMENTS
D ENSE R OOTED T RIPLET I NCONSISTENCY
Conﬂict Packing

Reduction rules
Conﬂict Packing


Consistency

FAST (folklore)
The following properties are equivalent:
(i) T is acyclic
(ii) T does not contain any directed triangle

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Consistency

RTI [Guillemot and Mnich, 2010]
The following properties are equivalent:
(i) R is consistent
(ii) R does not contain any conﬂict on four leaves

Conﬂict. Set of vertices C ⊆ V that does not admit a consistent rooted binary tree.

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Parameterized complexity

√
FAST RTI
1/3
FPT O ∗ (2 k log k ) a FPT O ∗ (2k log k ) b
Kernel with O(k 2 ) verticesa Kernel with O(k 2 ) verticesb
Linear vertex-kernel c No such result known before.
a
[Alon et al., 2009]
b
[Guillemot and Mnich, 2010]
c
[Bessy et al., 2009]

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Parameterized complexity

√
FAST RTI
1/3
FPT O ∗ (2 k log k ) a FPT O ∗ (2k log k ) b
Kernel with O(k 2 ) verticesa Kernel with O(k 2 ) verticesb
Linear vertex-kernel c No such result known before.
a
[Alon et al., 2009]
b
[Guillemot and Mnich, 2010]
c

The linear vertex-kernel for FAST described by [Bessy et al., 2009]
uses a constant-factor approximation algorithm.
Their reduction rules can be adapted to RTI.
But no constant-factor approximation!

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Conflict Packing

´
[Paul, P. and Thomasse, 2011]
works on problems characterized by some finite conflicts.
maximal collection of p-uplets disjoint conflits C.
provides a lower bound on the number of modification required.
implies that the instance induced by V V (C) is consistent.

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Reduction rules

Remove any vertex that is not part of any directed triangle. a .
a
can be carried out in polynomial time.

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Reduction rules

Safe partition
Assume V (T ) is ordered under some ordering σ, and let P be a
partition of σ into intervals.

V1 V2 Vl

AI := {uv ∈ A | ∃ i u , v ∈ Vi }
AO := A AI

B is the set of backward arcs of AO (arcs vi vj with i > j).

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Reduction rules

Safe partition
P is safe if there exist |B| arc-disjoint conﬂicts using arcs of AO
only.

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Safe Partition Reduction Rule

Let P be a safe partition of an ordered tournament T = (V , A, σ).
Reverse every arc of B and decrease k accordingly.

Use constant-factor approximation algorithm.
Use Conﬂict Packing.

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Safe Partition Reduction Rule

Let P be a safe partition of an ordered tournament T = (V , A, σ).
Reverse every arc of B and decrease k accordingly.

Main question
How to compute a safe partition in polynomial time?
Use constant-factor approximation algorithm.
Use Conﬂict Packing.

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Conﬂict Packing

A conﬂict packing of a tournament is a maximal collection of
arc-disjoint directed triangles.

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Conﬂict Packing


Can be computed greedily (i.e. in polynomial time).
Let C be a conﬂict packing. If T = (V , A) is a positive instance then
|V (C)| 3k.

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Conﬂict Packing


Conﬂict Packing Lemma [Paul, P. and Thomasse, 2011]
´
Let T = (V , A) be a tournament. There exists an ordering of T whose
backward arcs uv are such that u, v ∈ V (C).

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Conﬂict Packing


Lemma [Paul, P. and Thomasse, 2011]
´
Let T = (V , A) be a tournament such that |V | > 4k. There exists a safe
partition that can be computed in polynomial time.

proof

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Conﬂict Packing


Corollary [Paul, P. and Thomasse, 2011]
´
F EEDBACK A RC S ET IN TOURNAMENTS admits a kernel with at most 4k
vertices.

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Application to the RTI problem

Remove vertices that do not belong to any conflict
Safe Partition reduction rule
Conflict Packing allows to find a Safe Partition

Theorem [Paul, P. and Thomasse, 2011]
´
D ENSE R OOTED T RIPLET I NCONSISTENCY admits a kernel with at most
5k vertices.

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Conclusion

6 Our results

7 Open problems

O UR RESULTS O PEN PROBLEMS

Main results

Polynomial kernels
First polynomial kernels:
(i) C LOSEST 3-L EAF P OWER
(ii) P ROPER I NTERVAL C OMPLETION
(iii) C OGRAPH E DGE -E DITION
Improved polynomial kernels:
(i) F EEDBACK A RC S ET IN TOURNAMENTS
(ii) D ENSE R OOTED T RIPLET I NCONSISTENCY
(iii) D ENSE B ETWEENNESS and D ENSE C IRCULAR O RDERING

joint works with: S. Bessy, F. Fomin, S. Gaspers, S. Guillemot, F. Havet, C. Paul,
S. Saurabh and S. Thomasse. ´

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Main results

Lower bounds on kernelization:
(i) For any l 7, the Pl -F REE E DGE -D ELETION problem
does not admit a polynomial kernel.
(ii) For any l 4, the Cl -F REE E DGE -D ELETION problem
does not admit a polynomial kernel.

joint work with: S. Guillemot, F. Havet and C. Paul.

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Open problems

Do the F EEDBACK V ERTEX S ET IN TOURNAMENTS and C LUSTER
V ERTEX D ELETION problems admit linear vertex-kernels?
Characterize lower bounds for modification problems. details

Can we use branches on other problems?
(e.g. C HORDAL D ELETION)
Can we use Conflict Packing on other problems?
(e.g. (weakly)-fragile constraint modification problems)

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Kernelization algorithms for graph and other structure modiﬁcation problems

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