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Kernelization Using Structural Parameters on Sparse Graph Classes - - PowerPoint PPT Presentation

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion Kernelization Using Structural Parameters on Sparse Graph Classes Jakub Gajarsk 1 Petr Hlinn 1 Jan Obdrlek 1 Sebastian Ordyniak 1 Felix Reidl 2 Peter

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Kernelization Using Structural Parameters on Sparse Graph Classes

Jakub Gajarský 1 Petr Hliněný 1 Jan Obdržálek 1 Sebastian Ordyniak 1 Felix Reidl 2 Peter Rossmanith 2 Fernando Sánchez Villaamil 2 Somnath Sikdar 2

1Faculty of Informatics

Masaryk University Brno, Czech Republic

2Theoretical Computer Science

RWTH Aachen University Aachen, Germany Workshop on Kernelization University of Warsaw 10th April 2013

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Contents

1

The story so far

2

Sparse graph classes

3

The problem with natural parameters

4

Structural parameterization to the rescue

5

Conclusion

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Brief history

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Kernelization

A parameterized problem is fixed-parameter tractable iff it has a kernelization algorithm. Goal: obtain polynomial or linear kernels (whenever possible).

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Kernelization

A parameterized problem is fixed-parameter tractable iff it has a kernelization algorithm. Goal: obtain polynomial or linear kernels (whenever possible). Basic technique Devise reduction rules that preserve equivalence of instances; apply them exhaustively; prove kernel size.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Kernelization

A parameterized problem is fixed-parameter tractable iff it has a kernelization algorithm. Goal: obtain polynomial or linear kernels (whenever possible). Basic technique Devise reduction rules that preserve equivalence of instances; apply them exhaustively; prove kernel size. Algorithmic meta-theorems: algorithms for problem classes

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Previous work

Framework for planar graphs.

Guo and Niedermeier: Linear problem kernels for NP-hard problems on planar graphs

Meta result for graphs . . .

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Previous work

Framework for planar graphs.

Guo and Niedermeier: Linear problem kernels for NP-hard problems on planar graphs

Meta result for graphs . . . . . . of bounded genus.

Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh and Thilikos: (Meta) Kernelization

. . . excluding a fixed graph as a minor.

Fomin, Lokshtanov, Saurabh and Thilikos: Bidimensionality and kernels

. . . excluding a fixed graph as a topological minor.

Kim, Langer, Paul, Reidl, Rossmanith, Sau and S.: Linear kernels and single-exponential algorithms via protrusion decompositions

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Previous work

Framework for planar graphs.

Guo and Niedermeier: Linear problem kernels for NP-hard problems on planar graphs

Meta result for graphs . . . . . . of bounded genus.

Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh and Thilikos: (Meta) Kernelization

. . . excluding a fixed graph as a minor.

Fomin, Lokshtanov, Saurabh and Thilikos: Bidimensionality and kernels

. . . excluding a fixed graph as a topological minor.

Kim, Langer, Paul, Reidl, Rossmanith, Sau and S.: Linear kernels and single-exponential algorithms via protrusion decompositions

. . . of bounded expansion, locally bounded expansion and nowhere-dense graphs using structural parameterization.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Sparse graphs

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

The big picture

Bounded treedepth Bounded treewidth Excluding a minor Excluding a topological minor Bounded expansion Locally bounded expansion Nowhere dense Outerplanar Planar Bounded genus Bounded degree Locally bounded treewidth Locally excluding a minor Forest

Natural parameter Structural parameter

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Minors and topological minors

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Shallow minors and shallow topological minors

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Bounded expansion

G ▽ r denotes the set of the r-shallow minors of G.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Bounded expansion

G ▽ r denotes the set of the r-shallow minors of G. Definition (Grad, Expansion) The greatest reduced average density of a graph G is defined as ∇r(G) = max

H∈G ▽ r

|E(H)| |V (H)|.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Bounded expansion

G ▽ r denotes the set of the r-shallow minors of G. Definition (Grad, Expansion) The greatest reduced average density of a graph G is defined as ∇r(G) = max

H∈G ▽ r

|E(H)| |V (H)|. The expansion of a graph class G is defined as ∇r(G) = sup

G∈G

∇r(G).

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Bounded expansion

G ▽ r denotes the set of the r-shallow minors of G. Definition (Grad, Expansion) The greatest reduced average density of a graph G is defined as ∇r(G) = max

H∈G ▽ r

|E(H)| |V (H)|. The expansion of a graph class G is defined as ∇r(G) = sup

G∈G

∇r(G). A graph class G has bounded expansion if for some function f and all r ∈ N ∇r(G) ≤ f(r).

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Excluded minors vs Bounded Expansion

Excluded minors Bounded Expansion d-degenerate (depends on ex- cluded minor). f(0)-degenerate (depends on expansion).

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Excluded minors vs Bounded Expansion

Excluded minors Bounded Expansion d-degenerate (depends on ex- cluded minor). f(0)-degenerate (depends on expansion). Linear number of edges. Linear number of edges.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Excluded minors vs Bounded Expansion

Excluded minors Bounded Expansion d-degenerate (depends on ex- cluded minor). f(0)-degenerate (depends on expansion). Linear number of edges. Linear number of edges. No large cliques. No large cliques.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Excluded minors vs Bounded Expansion

Excluded minors Bounded Expansion d-degenerate (depends on ex- cluded minor). f(0)-degenerate (depends on expansion). Linear number of edges. Linear number of edges. No large cliques. No large cliques. No large clique-minors. Can contain large clique minors.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Excluded minors vs Bounded Expansion

Excluded minors Bounded Expansion d-degenerate (depends on ex- cluded minor). f(0)-degenerate (depends on expansion). Linear number of edges. Linear number of edges. No large cliques. No large cliques. No large clique-minors. Can contain large clique minors. Closed under taking minors. “Closed” under taking shallow minors.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Excluded minors vs Bounded Expansion

Excluded minors Bounded Expansion d-degenerate (depends on ex- cluded minor). f(0)-degenerate (depends on expansion). Linear number of edges. Linear number of edges. No large cliques. No large cliques. No large clique-minors. Can contain large clique minors. Closed under taking minors. “Closed” under taking shallow minors. Degeneracy of every minor is d. Degeneracy of r-shallow minors at most 2 · f(r).

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

Excluded minors vs Bounded Expansion

Excluded minors Bounded Expansion d-degenerate (depends on ex- cluded minor). f(0)-degenerate (depends on expansion). Linear number of edges. Linear number of edges. No large cliques. No large cliques. No large clique-minors. Can contain large clique minors. Closed under taking minors. “Closed” under taking shallow minors. Degeneracy of every minor is d. Degeneracy of r-shallow minors at most 2 · f(r). Techniques from H-topo-minor-free graphs don’t work! (They use large (non-shallow) topological minors.)

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Natural parameters

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The problem

Treewidth-t Deletion Input: A graph G, an integer k. Problem: Is there a set X ⊆ V (G) of size at most k such that tw(G − X) ≤ t?

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The problem

Treewidth-t Deletion Input: A graph G, an integer k. Problem: Is there a set X ⊆ V (G) of size at most k such that tw(G − X) ≤ t? Treewidth-1 Deletion = Feedback Vertex Set.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

The problem

Treewidth-t Deletion Input: A graph G, an integer k. Problem: Is there a set X ⊆ V (G) of size at most k such that tw(G − X) ≤ t? Treewidth-1 Deletion = Feedback Vertex Set. Model problem for previous results.

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The problem

Treewidth-t Deletion Input: A graph G, an integer k. Problem: Is there a set X ⊆ V (G) of size at most k such that tw(G − X) ≤ t? Treewidth-1 Deletion = Feedback Vertex Set. Model problem for previous results. kf(t)-kernel on general graphs.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

The problem

Treewidth-t Deletion Input: A graph G, an integer k. Problem: Is there a set X ⊆ V (G) of size at most k such that tw(G − X) ≤ t? Treewidth-1 Deletion = Feedback Vertex Set. Model problem for previous results. kf(t)-kernel on general graphs. Probably none of size O(f(t) · kc) (c independent of t).

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The problem

Treewidth-t Deletion Input: A graph G, an integer k. Problem: Is there a set X ⊆ V (G) of size at most k such that tw(G − X) ≤ t? Treewidth-1 Deletion = Feedback Vertex Set. Model problem for previous results. kf(t)-kernel on general graphs. Probably none of size O(f(t) · kc) (c independent of t).

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The problem

Treewidth-t Deletion Input: A graph G, an integer k. Problem: Is there a set X ⊆ V (G) of size at most k such that tw(G − X) ≤ t? Treewidth-1 Deletion = Feedback Vertex Set. Model problem for previous results. kf(t)-kernel on general graphs. Probably none of size O(f(t) · kc) (c independent of t). An f(k) kernel on bounded expansion graphs implies f(k) kernel on general graphs.

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A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

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A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

A treewidth-t modulator remains unchanged under edge subdivision.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

A treewidth-t modulator remains unchanged under edge subdivision. Instances of Treewidth-t Deletion closed under subdivision of edges.

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A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

A treewidth-t modulator remains unchanged under edge subdivision. Instances of Treewidth-t Deletion closed under subdivision of edges.

2 Any graph class C can be transformed into a class of bounded

expansion by edge subdivision.

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A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

A treewidth-t modulator remains unchanged under edge subdivision. Instances of Treewidth-t Deletion closed under subdivision of edges.

2 Any graph class C can be transformed into a class of bounded

expansion by edge subdivision.

For each G ∈ C, subdivide each edge |G| times.

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A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

A treewidth-t modulator remains unchanged under edge subdivision. Instances of Treewidth-t Deletion closed under subdivision of edges.

2 Any graph class C can be transformed into a class of bounded

expansion by edge subdivision.

For each G ∈ C, subdivide each edge |G| times.

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

A treewidth-t modulator remains unchanged under edge subdivision. Instances of Treewidth-t Deletion closed under subdivision of edges.

2 Any graph class C can be transformed into a class of bounded

expansion by edge subdivision.

For each G ∈ C, subdivide each edge |G| times.

A kernel on general graphs Reduce (G, k) to ( ˜ G, k) by subdividing every edge |G| times;

  • utput kernel of ( ˜

G, k).

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Brief history Sparse graphs Natural parameters Structural parameters Conclusion

A kernel on general graphs from sparse graphs

1 Treewidth closed under subdivision of edges.

A treewidth-t modulator remains unchanged under edge subdivision. Instances of Treewidth-t Deletion closed under subdivision of edges.

2 Any graph class C can be transformed into a class of bounded

expansion by edge subdivision.

For each G ∈ C, subdivide each edge |G| times.

A kernel on general graphs Reduce (G, k) to ( ˜ G, k) by subdividing every edge |G| times;

  • utput kernel of ( ˜

G, k). For a meta-kernel result, the parameter must not be closed under edge subdivision!

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Structural parameters

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The natural view

Bounded Genus H-Minor-Free H-Topological- Minor-Free Bounded Expansion Quasi-compact Treewidth-bounding

?

Bidimensional

+separation property

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The structural view

Bounded Genus H-Minor-Free H-Topological- Minor-Free Bounded Expansion Treewidth-t Modulator Treewidth-t Modulator Treewidth-t Modulator

(implied by Lemma 3.2) (implied by Lemma 9)

?

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The structural view

Bounded Genus H-Minor-Free H-Topological- Minor-Free Bounded Expansion Treewidth-t Modulator Treewidth-t Modulator Treewidth-t Modulator

(implied by Lemma 3.2) (implied by Lemma 9)

Treedepth-d Modulator

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Treedepth?

For a graph G with td(G) ≤ d: G embeddable in closure of tree (forest) of depth d.

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Treedepth?

For a graph G with td(G) ≤ d: G embeddable in closure of tree (forest) of depth d. Graph does not contain path of length 2d.

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Treedepth?

For a graph G with td(G) ≤ d: G embeddable in closure of tree (forest) of depth d. Graph does not contain path of length 2d. tw(G) ≤ pw(G) ≤ d − 1.

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Treedepth?

For a graph G with td(G) ≤ d: G embeddable in closure of tree (forest) of depth d. Graph does not contain path of length 2d. tw(G) ≤ pw(G) ≤ d − 1.

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Treedepth?

For a graph G with td(G) ≤ d: G embeddable in closure of tree (forest) of depth d. Graph does not contain path of length 2d. tw(G) ≤ pw(G) ≤ d − 1. Not closed under subdivision!

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Treedepth?

For a graph G with td(G) ≤ d: G embeddable in closure of tree (forest) of depth d. Graph does not contain path of length 2d. tw(G) ≤ pw(G) ≤ d − 1. Not closed under subdivision! If X is a treedepth-d-modulator, G − X does not contain long paths.

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Theorem Any graph-theoretic problem that has finite integer index on graphs

  • f constant treedepth admits linear kernels on graphs of bounded

expansion if parameterized by a modulator to constant treedepth. Kernelization possible in linear time. And . . . . . . quadratic kernels on graphs of locally bounded expansion; . . . polynomial kernels on nowhere dense graphs.

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Protrusion anatomy

Definition X ⊆ V (G) is a t-protrusion if

1 |∂(X)| = |N(X) \ X| ≤ t

(small boundary)

2 tw(G[X]) ≤ t

(small treewidth)

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The magic reduction rule

We want to replace a large protrusion by something smaller.

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The magic reduction rule

We want to replace a large protrusion by something smaller. Possible if problem has finite integer index.

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The magic reduction rule

We want to replace a large protrusion by something smaller. Possible if problem has finite integer index. Recursive structure of graphs of small treewidth (i.e. protrusion) helps.

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The magic reduction rule

We want to replace a large protrusion by something smaller. Possible if problem has finite integer index. Recursive structure of graphs of small treewidth (i.e. protrusion) helps. Lots of technicalities omitted . . .

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The magic reduction rule . . .

Our results assume finite integer index on graphs of bounded treedepth. How does one ensure that the graph obtained by replacing protrusions is in the same class?

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The magic reduction rule . . .

We can show that the replacements are always induced subgraphs

  • f the original protrusions.
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The magic reduction rule . . .

We can show that the replacements are always induced subgraphs

  • f the original protrusions.

Graphs of treedepth d are well-quasi-ordered wrt the induced subgraph relation [Nešetřil and Ossona de Mendez, Sparsity]. Every equivalence class of the FII-relation is partitioned into a finite number of posets. The minimal elements of the posets of each equivalence class are its representatives.

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Proof Idea: By a picture

Find approximate treedepth-d-modulator Reduce neighbourhood size

  • f

( )-components in Reduce size of components with same neighbours in

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Using sparseness

Each Y ′

i for 1 ≤ i ≤ ℓ is a protrusion and has constant size

(after protrusion reduction). |Y0| = O(|X|) (follows from degeneracy of 2d-shallow minors). ℓ = O(|Y0|) = O(|X|) (ditto). Hidden constants depend on expansion ∇2d(G) ≤ f(2d).

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The result

Theorem Any graph-theoretic problem that has finite integer index on graphs

  • f constant treedepth admits linear kernels on graphs of bounded

expansion if parameterized by a modulator to constant treedepth. Kernelization possible in linear time.

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The result

Theorem Any graph-theoretic problem that has finite integer index on graphs

  • f constant treedepth admits linear kernels on graphs of bounded

expansion if parameterized by a modulator to constant treedepth. Kernelization possible in linear time. Structural parameter enables us to relax the FII condition.

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The result

Theorem Any graph-theoretic problem that has finite integer index on graphs

  • f constant treedepth admits linear kernels on graphs of bounded

expansion if parameterized by a modulator to constant treedepth. Kernelization possible in linear time. Structural parameter enables us to relax the FII condition. Kernels for problems like Treewidth and Longest Path.

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The result

Theorem Any graph-theoretic problem that has finite integer index on graphs

  • f constant treedepth admits linear kernels on graphs of bounded

expansion if parameterized by a modulator to constant treedepth. Kernelization possible in linear time. Structural parameter enables us to relax the FII condition. Kernels for problems like Treewidth and Longest Path. Structural parameter helps to include decision problems like 3-Colorability and Hamiltionian Path.

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The result

Theorem Any graph-theoretic problem that has finite integer index on graphs

  • f constant treedepth admits linear kernels on graphs of bounded

expansion if parameterized by a modulator to constant treedepth. Kernelization possible in linear time. Structural parameter enables us to relax the FII condition. Kernels for problems like Treewidth and Longest Path. Structural parameter helps to include decision problems like 3-Colorability and Hamiltionian Path. Quadratic kernels on graphs of locally bounded expansion.

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The result

Theorem Any graph-theoretic problem that has finite integer index on graphs

  • f constant treedepth admits linear kernels on graphs of bounded

expansion if parameterized by a modulator to constant treedepth. Kernelization possible in linear time. Structural parameter enables us to relax the FII condition. Kernels for problems like Treewidth and Longest Path. Structural parameter helps to include decision problems like 3-Colorability and Hamiltionian Path. Quadratic kernels on graphs of locally bounded expansion. Polynomial kernels on nowhere dense graphs.

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Consequences

The problems. . .

Dominating Set, Connected Dominating Set, r-Dominating Set, Efficient Dominating Set, Connected Vertex Cover, (Connected) Vertex Cover, Hamiltonian Path/Cycle, 3-Colorability, Independent Set, Feedback Vertex Set, Edge Dominating Set, Induced Matching, Chordal Vertex Deletion, Interval Vertex Deletion, Odd Cycle Transversal, Induced d-Degree Subgraph, Min Leaf Spanning Tree, Max Full Degree Spanning Tree, Longest Path/Cycle, Exact s, t-Path, Exact Cycle, Treewidth, Pathwidth

. . . parameterized by a treedepth-modulator have . . .

. . . linear kernels on graphs of bounded expansion . . . quadratic kernels on graphs of locally bounded expansion . . . polynomial kernels on nowhere-dense graphs

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Conclusion

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Interpretation of meta-theorems

For meta-theorems up until H-topo-minor-free graphs, a small treewidth modulator is crucial: quasi-compactness on bounded genus graphs, and bidimensionality + separability on H-minor-free graphs are tangible properties which guarantee this on these classes. Larger graph classes need stronger (structural) parameters. Treedepth-modulator is a useful parameter (generalization of vertex cover).

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Open questions

Which problems admit kernels on these classes with a natural parameter? Problem categories: closed under subdivision vs. not closed. Weaker parameterization for latter? Linear kernels for graphs with locally bounded treewidth? Lower bounds!

Thanks!