Post on 05-Jan-2016
description
Kernelization for a Hierarchy of Structural Parameters
Bart M. P. Jansen
2-4 September 2011, Vienna
2
Outline
Motivation
Hierarchy of structural parameters
Case studies
Importance of treewidth to kernelization
Conclusion and open problems
Vertex Cover / Independent Set Graph Coloring Long Path & Cycle
Problems
3
Motivations for structural parameters
• Stronger preprocessing (Vertex Cover, Two-Layer Planarization)
They can be smaller than the natural parameter
• Because it is NP-complete for fixed k (Graph Coloring)• Because it is compositional (Long Path)
The natural parameter might not admit polynomial kernels
• Change the parameter instead of the class of inputs
Alternative direction to kernels for restricted graph classes
• Guide the search for reduction rules which exploit different properties of an instance• Help explain why known heuristics work (Treewidth)
Connections to practice
• Gives a complete picture of the power of preprocessing
Fundamentals
4
A HIERARCHY OF PARAMETERS
5
Some well-known parameters
Vertex Cover
number• Size of the
smallest set intersecting each edge
6
Some well-known parameters
Vertex Cover
number• Size of the
smallest set intersecting each edge
Feedback Vertex
number• Size of the
smallest set intersecting each cycle
Odd Cycle Transversal
number• Size of the
smallest set intersecting all odd cycles
Max Leaf Spanning
tree nr• Maximum #
leaves in a spanning tree
≥ ≥
7
Structural graph parameters• Let F be a class of graphs
• Parameterize by this deletion distance for various F
• If F‘ ⊆ F then d(G, F) ≤ d(G, F’)• If graphs in F have treewidth at most c:
– TW(G) ≤ d(G, F) + c
For a graph G, the deletion distance d(G, F) to F is the minimum size of a set X such that G – X ∈ F
8
Some well-known parameters
Vertex Cover
number• Deletion
distance to an independent set
Feedback Vertex
number• Deletion
distance to a forest
Odd Cycle Transversal
number• Deletion
distance to a bipartite graph
Max Leaf Spanning
tree nr• …
≥ ≥
9
Some lesser-known parameters
Clique Deletion number
• Deletion distance to a single clique
Cluster Deletion number
• Deletion distance to a disjoint union of cliques
Linear Forest
number• Deletion
distance to a disjoint union of paths
Outerplanar Deletion number
• Distance to planar with all vertices on the outer face
≥
10
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Does problem X have a polynomial kernel when parameterized by the size of a given deletion set to a linear forest?
Assume the deletion set is given to distinguish between the complexity of
finding the deletion set ⇔ using the deletion set
Requirement that a deletion set is given can often be dropped, using an approximation algorithm
11
VERTEX COVER / INDEPENDENT SETVERTEX COVER
12
Vertex Cover parameterized by distance to F• Input: Graph G, integer l, set X⊆V s.t. G – X ∈ F• Parameter: k := |X|• Question: Does G have a vertex cover of size ≤l?
Equivalent to: α(G) ≥ |V| - l? (parameter does not change)
Vertex cover Deletion to independent set
Feedback Vertex Set
Deletion to forest
Odd Cycle Transversal
Deletion to bipartite
13
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
14
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Distance to Outerplanar Pathwidth
NP-complete for fixed k
• Planar Vertex Cover is NP-complete• Planar graphs are 4-colorable
15
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
PathwidthFixed-Parameter Tractable
• Guess how solution intersects deletion set• Compute optimal solution in remainder• Perfect graph, so polynomial time by Grötschel,
Lovász & Schrijver 1988
16
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
17
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
PathwidthFixed-Parameter Tractable by Dynamic Programming
18
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
19
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernel
• O(k2) vertices [BussG’93]• Linear-vertex kernels
Nemhauser-Trotter theorem [NT’75] Crown reductions [ChorFJ’04, Abu-KhzamFLS’07]
20
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
21
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Distance to Outerplanar Pathwidth
Linear-vertex kernel
• Using extremal structure arguments [FellowsLMMRS’09]
22
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Distance to Outerplanar Pathwidth
23
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Cubic-vertex kernel
• Through combinatorial arguments [BodlaenderJ’11]
24
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
25
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Randomized polynomial kernel
• Using Matroid compression technique of Kratsch & Wahlström
• Unpublished result [JansenKW]
26
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
27
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Distance to Outerplanar Pathwidth
No polynomial kernel unless NP coNP/poly⊆
• Using cross-composition [BodlaenderJK’11]
28
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
29
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic NumberDistance to
Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Odd Cycle Transversal
Distance to Clique
Distance to Cluster
Pathwidth
No polynomial kernel unless NP coNP/poly⊆
• Using OR-composition for the refinement version [BodlaenderDFH’09]
30
Vertex Cover
Distance to linear forest
Distance to Cograph
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Vertex Cover / Independent Set
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Chordal
Distance to Clique
Distance to Cluster
Pathwidth
31
Vertex Cover
Distance to linear forest
Distance to Cograph
Feedback Vertex Set
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Vertex Cover / Independent Set
Distance to split graph
components
Distance to Interval
Distance to Chordal
Distance to Clique
Distance to Cluster
Distance to Outerplanar Pathwidth
No polynomial kernel unless NP coNP/poly⊆
• Unpublished, using Cross-Composition [JansenK]
32
Vertex Cover
Distance to linear forest
Distance to Cograph
Feedback Vertex Set
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Vertex Cover / Independent Set
Distance to split graph
components
Distance to Interval
Distance to Chordal
Distance to Clique
Distance to Cluster
Distance to Outerplanar Pathwidth
Polynomial kernels
NP-complete for k=4
33
Vertex Cover
Distance to linear forest
Distance to Cograph
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Cluster
Distance to Outerplanar Pathwidth
Distance to Clique
Distance to Chordal
Complexity overview for Vertex Cover parameterized by…
FPT, no polykernel unless
NP coNP/poly⊆
34
Weighted Independent Set param. by Vertex Cover number• Input: Graph G on n vertices, integer l, a vertex
cover X, and a weight function w: V→{1,2,…,n}
• Parameter: k := |X|• Question: Does G have an independent set of weight ≥
l?
• We will prove a kernel lower-bound for this problem using cross-composition [BodlaenderJ@STACS’11]
35
Cross-composition• Defined in [BodlaenderJK@STACS’11]
• A polynomial equivalence relationship ℜ is – a way of partitioning instances on at most n bits each, – into poly(n) classes,– such that equivalency can be tested in polynomial time
• Informally: an efficient way of grouping instances of size ≤n each into poly(n) groups
• Cross-composition is defined with respect to useful problem-specific polynomial equivalence relationship ℜ
poly(t · n) time
Cross-composition of à into B
x1 x2 x3 x4 x5 x6 x… xt
n
x* k*
poly(n+log t)
ℜ-equivalent instances of
NP-hard problem Ã
1 instance of param. problem B
If an NP-hard problem à cross-composes into the parameterized problem B, then B does not admit a polynomial kernel unless NP coNP/poly ⊆
[BodlaenderJK’11@STACS]
(x*,k*) B ⇔ ∈ ∃i: xi Ã∈
37
Lower-bound using cross-composition
• Polynomial equivalence relationship ℜ for the cross-composition:– Two instances are equivalent if they have the same
number of edges, vertices and target value l
• We give an algorithm to compose a sequence of instances – (G1, l), (G2, l), … , (Gt, l)
• where |V(Gi)| = n and |E(Gi)| = m for all i
Set of instances on ≤ n vertices each is partitioned into O(n · n2 · n) classes
38
Transformations for Independent Set
• Let G be a graph, and {u,v} ∈ E• By subdividing {u,v} with two new vertices, the
independence number increases by one– Reverse of the “folding” rule [ChenKJ’01]
• If G’ is obtained by subdividing all m edges of G:– a(G’) = a(G) + m
39
Second bitFirst bit
Construction of composite instance
G1 G2 G3 G4G’1 G’2 G’3 G’400 01 10 11
• Example for l =3• N:=t·n is the total # vertices in the input• Bit position vertices have weight N each• Other vertices have weight 1• Set l* := N·log t + l + m
X
Claim: Construction is polynomial-time
Claim: Parameter k’ := |X| is 2m + log t poly(n + log t)
40
Second bitFirst bit
∃i: a(Gi) ≥ l implies aw(G*) ≥ l*
G1 G2 G3 G4G’1 G’2 G’3 G’400 01 10 11
• Total weight l + m + N log t = l*
41
Second bitFirst bit
∃i: a(Gi) ≥ l follows from aw(G*) ≥ l*
G’1 G’2 G’3 G’400 01 10 11
• When a bit position is avoided:– Replace input vertices (≤N) by a position vertex
(weight N)– So assume all bit positions are used
• Independent set uses input vertices of 1 instance (complement of bitstring)
– Total weight l + m in remainder– a(G’i) ≥ l + m, so a(Gi) ≥ l
42
Results• From the cross-composition we get:
Weighted Independent Set parameterized by the size of a vertex coverdoes not have a polynomial kernel unless NP coNP/poly ⊆
Weighted Vertex Cover parameterized by the size of a vertex cover does not have a polynomial kernel unless NP coNP/poly ⊆
• By Vertex Cover Independent Set equivalence– (parameter does not change)
• Contrast: Weighted Vertex Cover parameterized by weight of a vertex cover, does admit a polynomial kernel [ChlebíkC’08]
43
The difficulty of vertex weights• Parameterized by vertex cover number:
– unweighted versions admit polynomial kernels– weighted versions do not unless NP⊆coNP/poly, but are FPT
Vertex Cover / Independent Set• [BodlaenderJ@STACS’11]
Feedback Vertex Set• [Thomasse@ACM Tr.’10], [BodlaenderJK@STACS11]
Odd Cycle Transversal• [JansenK@IPEC’11]
Treewidth• [BodlaenderJK@ICALP’11]
Chordal Deletion• Unpublished
44
GRAPH COLORINGGRAPH COLORING
45
Vertex Coloring of Graphs• Given an undirected graph G and integer q, can we assign
each vertex a color from {1, 2, …, q} such that adjacent vertices have different colors?– If q is part of the input: Chromatic Number– If q is constant: q-Coloring
• 3-Coloring is NP-complete
Chromatic Number parameterized by Vertex Cover does not admit a polynomial kernel unless NP coNP/poly ⊆
[BodlaenderJK@STACS’11]
46
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
47
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
NP-complete for k=2 [Cai’03]No kernel unless P=NP
48
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
49
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth• Fixed-Parameter Tractable by
dynamic programming
50
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
51
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
• Fixed-Parameter Tractable since yes-instances have treewidth
≤k+q
52
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
53
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Linear-vertex since vertices of degree < q can be deleted
(using Kleitman-West Theorem)
54
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
55
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
O(kq)-vertex kernel using matching (shown next) [JansenK@FCT’11]
56
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
57
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
PathwidthPolynomial kernels [JansenK@FCT’11]Polynomial kernels [JansenK@FCT’11]
58
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
59
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
No polynomial kernel unless NP coNP/poly ⊆
[JansenK@FCT’11]
60
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
q-Coloring
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernels
NP-complete for k=2
FPT, no polykernel unless
NP coNP/poly⊆
61
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Complexity overview for q-Coloring parameterized by…
62
Preprocessing algorithm parameterized by Vertex Cover Nr• Input: instance G of q-Coloring1. Compute a 2-approximate
vertex cover X of G2. For each set S of q vertices in X,
mark a vertex vS which is adjacent to all vertices of S (if one exists)
3. Delete all vertices which are not in X, and not marked
• Output the resulting graph G’ on n’ vertices
– n’ ≤ |X| + |X|q
– ≤ 2k + (2k)q
X
q=3
Claim: Algorithm runs in polynomial time
Claim: n’ is O(kq), with k = VC(G)
63
Correctness: c(G)≤q c(G’)≤q() Trivial since G’ is a subgraph
of G() Take a q-coloring of G’
– For each deleted vertex v:• If there is a color in {1, …, q}
which does not appear on a neighbor of v, give v that color
– Proof by contradiction: we cannot fail• when failing: q neighbors of v each
have a different color• let S⊆X be a set of these neighbors• look at vS we marked for set S
• all colors occur on S vS has neighbor with same color
X
64
Result• The reduction procedure gives the following:
• [JansenK@FCT’11] gives a sufficient condition for graph classes F to ensure q-Coloring has a polynomial kernel by deletion-distance to F
q-Coloring parameterized by vertex cover number has a kernel with O(kq) vertices
65
Classification theorem• Expressed using q-List Coloring:
– Given an undirected graph G and a list of allowed colors L(v) ⊆ {1,…,q} for each vertex v, can we assign each vertex v a color in L(v) such that adjacent vertices have different colors?
{1,2,3}
{2,3}
{1}
{1,2} {1,3}
66
General classification theorem• Let F be a hereditary class of graphs for which there is a function g:N→N such
that:– for any no-instance (G,L) of q-List Coloring on a graph G in F, – there is a no-subinstance (G’,L’) on at most g(q) vertices.
q-Coloring parameterized by distance to F admits a polynomial kernel with O(kq·g(q)) vertices for every fixed q [JansenK@FCT’11]
F that work
• Independent sets• Cographs• Graphs where each
component is a split graph
F that fail
• Paths
3-coloring by distance to a path does not admit a poly kernel
(unless the polynomial hierarchy collapses)
If non-list colorability on graphs in F is local, then q-Coloring admits a poly kernel by distance from F
67
LONG PATH & CYCLE PROBLEMS
68
Long Path & Cycle problems• Question: does a graph G have a simple path (cycle) on at
least l vertices?• Natural parameterization k-Path was one of the main
motivations for development of the lower-bound framework
• … not even on planar, connected graphs [ChenFM@CiE’09]
k-Path does not admit a polynomial kernel unless NP coNP/poly ⊆ [BodlaenderDFH@ICALP’08]
69
Distance to linear forest
Vertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
70
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
71
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Cubic-vertex kernel
• Through combinatorial arguments [BodlaenderJ’11]NP-complete for k=0
72
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
73
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
PathwidthFixed-Parameter Tractable by Dynamic Programming
74
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
75
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernel using matching technique
[BodlaenderJK’11]
76
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
77
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernel using a weighted problem with a Karp reduction
[BodlaenderJK’11]
78
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
79
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernel using a weighted problem with a Karp reduction
[BodlaenderJK’11]
80
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
81
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
No polynomial kernel unless NP coNP/poly⊆
• Simple Cross-Composition
82
Distance to linear forest
Long PathVertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
No polynomial kernel unless NP coNP/poly⊆
• By Cross-Composing Hamiltonian s-t Path on bipartite graphs [BodlaenderJK’11]
83
Distance to linear forest
Vertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernels
NP-complete for k=0
FPT, no polykernel unless
NP coNP/poly⊆
FPTpoly kernel?
FPT?poly kernel?
Complexity overview for Long Path parameterized by…
84
Long Cycle parameterized by Vertex Cover
• Input: Graph G, vertex cover X of G, integer l• Question: Does G have a cycle on at least l vertices?
• We will show the existence of a quadratic-vertex kernel
• First: a property of matchings
85
Property of maximum matchings• Let G = (R ∪ B, E) be a bipartite graph• Let M be a maximum matching in G• Let RM be vertices of R saturated by M
• Proof using augmenting paths
Theorem. For all B’ B: if G has a matching saturating B’,
then G[RM B] has a matching saturating B’.∪
86
Quadratic-vertex kernel for Long Cycle by Vertex Cover• Input: Graph G, vertex cover X of G, integer l• Question: Does G have a cycle on at least l vertices?
– Assume l > 4 (otherwise, solve by brute force)
• Example for l = 6
87
Reduction algorithm• Bipartite auxiliary graph H = (R ∪ B, E)
– Red vertices are V(G) \ X– Blue vertex v(p,q) for each pair p,q ∈ X
• v(p,q) adjacent to N(p)∩N(q) \ X• Compute maximum matching in H
– Let RM be the saturated red vertices
• Output (G[X ∪ RM], q) with ≤ |X| + |X|2 vertices
88
Correctness (I)• G has a cycle of length l G[X ∪ RM] has a cycle of length l
• () Trivial since cycle in subgraph gives cycle in G• () Proof using the matching property
– Suppose G has a cycle C of length l > 4
89
Correctness (II)• All (blue) vertices and edges of G[X] are still
present• Red vertices in G-X are used to connect two blue
vertices in X• Subpath (b1, r, b2) of C is an indirect connection
– r ∈ N(b1) ∩ N(b2) \ X
• Find red vertices in G[X ∪ RM] to replace all indirect connections
90
Correctness (III)• No two connections (b1, r, b2) and (b1, r’, b2) since l >
4• For each connection (b1, r, b2):
– match v(b1,b2) to r in H
– matching in H saturating all connected pairs• By matching property: exists matching in H[RM∪B]
saturating all connected pairs• Update cycle accordingly
91
The kernel• For the decision problem with a vertex cover in the input:
• Kernel does not depend on desired length of the cycle– Works for optimization problem as well
• If X is not given: – Compute a 2-approximate vertex cover, use it as X– Resulting instance has ≤ |X| + |X|2 vertices– So ≤ 2k + (2k)2 vertices for a graph with min-VC size k
Long Cycle parameterized by a vertex cover X has a kernel with |X| + |X|2 vertices.
92
Other applications
• Matching technique gives O(|X|2)-vertex kernels for all these problems on a graph G with vertex cover X
• Also applies to directed variants
• Given G and pairs of vertices (s1, t1), … , (sl, tl), are there vertex-disjoint paths connecting each si to ti?
Disjoint Paths
• Given G and an integer l, are there l vertex-disjoint simple cycles in G?
Disjoint Cycles
• Given G and an integer l, does G contain a path of length l?
Long Path
93
IMPORTANCE OF TREEWIDTH
94
Treewidth Deletion distance to constant treewidth
• Vertex Cover (r=1)• Feedback Vertex Set (r=2)
As a problem
• All MSOL problems in FPT• Some hard layout problems FPT
parameterized by Vertex Cover [FellowsLMRS’08]
Parameter for algorithms
• Polynomial kernels for some problems• Strongly related to protrusions on
graphs of truly sublinear treewidth
Parameter for kernels
• f(k)O(n) by Bodlaender’s algorithm
As a problem
• All MSOL problems FPT by treewidth (Courcelle’s Theorem)
Parameter for algorithms
• No polynomial kernels known• OR / AND composition & Improvement
versions
Parameter for kernels
95
… parameterized by deletion distance to constant treewidth[on general graphs]
TW 0 TW 1 TW 2
Vertex Cover Feedback Vertex Set Odd Cycle Transversal Treewidth ?Longest Path ? q-Coloring Clique Chromatic Number Dominating Set
• We cross a threshold going from 1 to 2 – why ?
96
… parameterized by deletion distance to constant treewidth[on H-minor-free graphs]
• Meta-theorems for kernelization on– planar, bounded-genus [BodlaenderFLPST’09]– and H-minor-free graphs [FominLRS’11]
• Work by replacing protrusions in the graph– Pieces of constant treewidth, with a constant-size
boundary
• Existence of large protrusions is governed by deletion distance to constant treewidth
Theorem. For any fixed graph H, if G is H-minor-free and has deletion distance k to constant treewidth, then G has a protrusion of size
W(n/k) [FominLRS’11]
97
CONCLUSION
98
Polynomial kernels
NP-complete for k=4
Vertex Cover
Distance to linear forest
Distance to Cograph
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Cluster
Distance to Outerplanar Pathwidth
Distance to Clique
Distance to Chordal
FPT, no polykernel unless
NP coNP/poly⊆
Polynomial kernels
NP-complete for k=2
FPT, no polykernel unless
NP coNP/poly⊆
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Distance to linear forest
Vertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernels
NP-complete for k=0
FPT, no polykernel unless
NP coNP/poly⊆
FPTpoly kernel?
FPT?poly kernel?
99
Recent results• Fellows, Lokshtanov, Misra, Mnich, Rosamond & Saurabh [CIE’07]
– The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number• Dom, Lokshtanov & Saurabh [ICALP’09]
– Incompressibility through Colors and ID’s• Johannes Uhlmann & Mathias Weller [TAMC’10]
– Two-Layer Planarization Parameterized by Feedback Edge Set• Bodlaender, Jansen & Kratsch [STACS’11]
– Cross-Composition: A New Technique for Kernelization Lower Bounds• Jansen & Bodlaender [STACS’11]
– Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter• Bodlaender, Jansen & Kratsch [ICALP‘11]
– Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization• Betzler, Bredereck, Niedermeier & Uhlmann [SOFSEM’11]
– On Making a Distinguished Vertex Minimum Degree by Vertex Deletion• Jansen & Kratsch [FCT’11]
– Data Reduction for Graph Coloring Problems• Cygan, Lokshtanov, Pilipczuk, Pilipczuk & Saurabh [IPEC’11]
– On cutwidth parameterized by vertex cover– On the hardness of losing width
• Jansen & Kratsch [IPEC’11] – On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal
• Bodlaender, Jansen & Kratsch [IPEC’11]– Kernel Bounds for Path and Cycle Problems
100
Open problemsPoly kernels parameterized by Vertex Cover for:• Bandwidth• Cliquewidth• Branchwidth
Poly kernels for Long Path parameterized by:• distance to a path• distance to a forest (feedback vertex number) • distance to a cograph
Poly kernel for Treewidth parameterized by a (given) :• deletion set to an Outerplanar graph• deletion set to constant treewidth
Is Longest Path in FPT …• parameterized by a (given) deletion set to an Interval graph?
101
Polynomial kernels
NP-complete for k=4
Vertex Cover
Distance to linear forest
Distance to Cograph
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Cluster
Distance to Outerplanar Pathwidth
Distance to Clique
Distance to Chordal
FPT, no polykernel unless
NP coNP/poly⊆
Polynomial kernels
NP-complete for k=2
FPT, no polykernel unless
NP coNP/poly⊆
Vertex Cover
Distance to linear forest
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Distance to split graph
components
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Distance to linear forest
Vertex Cover
Distance to Cograph
Distance to Chordal
Treewidth
Chromatic Number
Odd Cycle Transversal
Distance to Perfect
Max Leaf #
Distance to Co-cluster
Distance to Outerplanar
Feedback Vertex Set
Distance to Interval
Distance to Clique
Distance to Cluster
Pathwidth
Polynomial kernels
NP-complete for k=0
FPT, no polykernel unless
NP coNP/poly⊆
FPTpoly kernel?
FPT?poly kernel?
THANK YOU!
102
OLD BEYOND HERE
103
Cross-composition proof
104
Cross-composition proof
105
Outline• Motivation• A hierarchy of structural parameters• Case studies
– Vertex Cover / Independent Set• Kernelization complexity overview in the hierarchy• Lower-bound for weighted variant parameterized by size of a
vertex cover– Graph Coloring
• Kernelization complexity overview in the hierarchy• Polynomial kernel for q-Coloring parameterized by Vertex Cover
– Long Path & Cycle problems• Kernelization complexity overview in the hierarchy• Polynomial kernel for Longest Cycle parameterized by Vertex
Cover• Treewidth and kernelization complexity• Conclusion and open problems
106
Outline
Motivation
Hierarchy of structural parameters
• Vertex Cover / Independent Set• Graph Coloring• Long Path & Cycle problems
Case studies
Role of Treewidth in kernelization complexity
Conclusion and open problems
107
Motivations for structural parameterization
• Stronger preprocessing (Vertex Cover, Two-Layer Planarization)
They can be smaller than the natural parameter
• Because it is NP-complete for fixed k (Graph Coloring)• Because it is compositional (Long Path)
The natural parameter might not admit polynomial kernels
• Change the parameter instead of the class of inputs
Alternative direction to kernels for restricted graph classes
• Guide the search for reduction rules which exploit different properties of an instance• Help explain why known heuristics work (Treewidth)
Connections to practice
• Gives a complete picture of the power of preprocessing
Fundamentals
108
Motivations for structural parameterization• They can be smaller than the natural parameter
– Stronger preprocessing– [Vertex Cover]
• The natural parameter might not admit polynomial kernels– Because it is NP-complete for fixed k: Graph Coloring– Because it is compositional: Long Path
• Alternative direction to kernels for restricted graph classes– d-degenerate, H-minor-free, truly sublinear treewidth
• Other motivations– Guide the search for reduction rules which exploit different
properties of an instance, might be useful in practice– Help explain why known heuristics work [Treewidth]– Gives a complete picture of the power of preprocessing
109
Outline• Hierarchy of structural parameters• Case studies:
– Vertex Cover ( = Independent Set)– Long Path– Coloring
• Some trends and observations– Dist to constant TW not enough for kernel, but sufficient for FPT on
MSOL problems– Weighted problems: hard by VC– Connectivity problems: hard by natural param– Truly sublinear tw + dist to constant tw poly kernel? Double
check; need to guarantee that protrusion replacement does not increase deletion distance. What about FII or not?
– (Induced?)• Recent results• Open problems
110
• Case studies– Independent set / Vertex Cover [By OCT if preprint is available; mention
distance from tw 2 by Daniel et al]– Treewidth [Mention new result: dist from clique?]– Coloring– Feedback Vertex Set– Path– Odd Cycle Transversal
• Trends– Dist to constant TW not enough for kernel, but sufficient for FPT on MSOL
problems– Weighted problems: hard by VC– Connectivity problems: hard by natural param– Truly sublinear tw + dist to constant tw poly kernel? Double check;
need to guarantee that protrusion replacement does not increase deletion distance. What about FII or not?
– (Induced?)
111
Treewidth
• f(k)O(n) by Bodlaender’s algorithm
As a problem
• All MSOL problems FPT by treewidth (Courcelle’s Theorem)
Parameter for algorithms
• No polynomial kernels known• OR / AND composition &
Improvement versions
Parameter for kernels
Deletion distance to constant treewidth
• Vertex Cover (r=1), Feedback Vertex Set (r=2)
As a problem
• All MSOL problems are FPT• Some hard layout problems FPT
parameterized by Vertex Cover
Parameter for algorithms
• Polynomial kernels for some problems
• Strongly related to protrusions on graphs of truly sublinear treewidth
Parameter for kernels
112
Some well known parameters
Vertex Cover number
• Size of the smallest set intersecting each edge
Feedback Vertex
number• Size of the
smallest set intersecting each cycle
Odd Cycle Transversal
number• Size of the
smallest set intersecting all odd cycles
Max Leaf Spanning
tree nr• Maximum #
leaves in a spanning tree
113
Some lesser-known parameters
Clique Deletion number
• Deletion distance to a single clique
Cluster Deletion number
• Deletion distance to a disjoint union of cliques
Linear Forest number
• Deletion distance to a disjoint union of paths
Outerplanar Deletion number
• Distance to planar with all vertices on the outer face
≥
114
Open problemsPoly kernels parameterized by Vertex Cover for:• Bandwidth• Cliquewidth• Branchwidth
Poly kernels for Long Path parameterized by:• distance to a path• distance to a forest (feedback vertex number) • distance to a cograph
Poly kernel for Treewidth parameterized by the size of a (given) :• deletion set to an Outerplanar graph• deletion set to constant treewidth
Is Longest Path in FPT …• parameterized by the size of a (given) deletion set to an Interval graph?