Kernelization using structural parameters on sparse graph classes - - PowerPoint PPT Presentation
Kernelization using structural parameters on sparse graph classes Jakub Gajarsk 1 en 1 Jan Obdrlek 1 Petr Hlin Sebastian Ordyniak 1 Felix Reidl 2 Peter Rossmanith 2 Fernando Snchez Villaamil 2 Somnath Sikdar 2 1 Faculty of Informatics
Jakub Gajarský1 Petr Hlinˇ ený1 Jan Obdržálek1 Sebastian Ordyniak1 Felix Reidl2 Peter Rossmanith2 Fernando Sánchez Villaamil2 Somnath Sikdar2
1Faculty of Informatics 2Theoretical Computer Science
Bidimensional Structures: Algorithms, Combinatorics and Logic @Dagstuhl 2013
The story so far Beyond excluded minors The exemplary obstacle: ❚r❡❡✇✐❞t❤✲t✲❉❡❧❡t✐♦♥ Structural parameterization to the rescue Conclusion
kernelization algorithm
Basic technique of kernelization: Devise reduction rules that preserve equivalence of instances; apply exhaustively, prove kernel size.
Algorithmic meta-results: nail down as many problems as possible
Guo and Niedermeier: Linear problem kernels for NP-hard problems on planar graphs
Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh and Thilikos: (Meta) Kernelization
Fomin, Lokshtanov, Saurabh and Thilikos: Bidimensionality and kernels
topological minor
Kim, Langer, Paul, R., Rossmanith, Sau and Sikdar: Linear kernels and single-exponential algorithms via protrusion decompositions
expansion, local bounded expansion and nowhere-dense graphs using structural parameterization
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
Bounded genus (planar) Topological H-minor free General H-minor free Bounded expansion No polynomial kernels No kernels Linear kernel Polynomial kernel Dominating Set Vertex Cover Connected Vertex Cover Longest Path Feedback Vertex Set Treewidth
(probably)
Dichotomy: either easy instance
⇒ Bounded treewidth gives enough structure to make reduction rule work (more on that later)
graphs beyond H-minor-free
Ω(r2 log r) with no r × r-grid ⇒ At least not much hope for linear kernels
For a graph G we denote by G ▽ r the set of its r-shallow minors.
Definition (Grad, Expansion)
For a graph G, the greatest reduced average density is defined as ∇r(G) = max
H∈G ▽r
|E(H)| |V (H)| For a graph class G the expansion of G is defined as ∇r(G) = sup
G∈G
∇r(G) A graph class G has bounded expansion if there exists a function f such that ∇r(G) ≤ f(r) for all r ∈ N.
Excluded minors Bounded expansion
d-degenerate (depening
ex- cluded minor) f(0)-degenerate (depening on ex- pansion) 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 mi- nors Degeneracy of every minor is d Degeneracy of minors depends on its “size”
Techniques from result on H-topological-minor-free graphs stop working because they use large (non-shallow) topological minors.
❚r❡❡✇✐❞t❤✲t ❉❡❧❡t✐♦♥ 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?
⇒ Probably none of size O(f(t)kc) (c independent of t)
Kernel on bounded expansion graphs implies same kernel on general graphs
1 Treewidth closed under subdivision of edges
⇒ Treewidth-modulator closed under subdivision of edges ⇒ Instances of ❚r❡❡✇✐❞t❤✲t ❉❡❧❡t✐♦♥ closed under subdivision of edges
2 Subdividing each edge of a graph |G| yields a graph of
bounded expansion General kernel from sparse kernel: Reduce (G, k) to ( ˜ G, k) by subdividing every edge |G| times,
G, k).
If we want a kernel, we need a parameter that is not closed under edge subdivision
Bounded Genus H-Minor-Free H-Topological- Minor-Free Bounded Expansion Quasi-compact Treewidth-bounding
Bidimensional
+separation property
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)
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
For a graph G with td(G) ≤ d:
Not closed under subdivision!
If X is a treedepth-d-modulator, G − X does not contain long paths
Definition
X ⊆ V (G) is a t-protrusion if
1 |∂(X)| = |N(X) \ X| ≤ t
(small boundary)
2 tw(G[X]) ≤ t
(small treewidth)
smaller
protrusion) helps
Find approximate treedepth-d-modulator Reduce neighbourhood size
( )-components in Reduce size of components with same neighbours in
Theorem
Any graph-theoretic problem that has finite integer index on graphs of constant treedepth∗ admits linear kernels on graphs
constant treedepth.
∗ Structural parameter enables us to relax the FII condition
⇒ Kernels for problems like ❚r❡❡✇✐❞t❤ and ▲♦♥❣❡st P❛t❤
like 3✲❈♦❧♦r❛❜✐❧✐t② and ❍❛♠✐❧t✐♦♥✐❛♥ P❛t❤
The problems. . .
❉♦♠✐♥❛t✐♥❣ ❙❡t, ❈♦♥♥❡❝t❡❞ ❉♦♠✐♥❛t✐♥❣ ❙❡t, r✲❉♦♠✐♥❛t✐♥❣ ❙❡t, ❊❢❢✐❝✐❡♥t ❉♦♠✐♥❛t✐♥❣ ❙❡t, ❈♦♥♥❡❝t❡❞ ❱❡rt❡① ❈♦✈❡r, ❍❛♠✐❧t♦♥✐❛♥ P❛t❤✴❈②❝❧❡, 3✲❈♦❧♦r❛❜✐❧✐t②, ■♥❞❡♣❡♥❞❡♥t ❙❡t, ❋❡❡❞❜❛❝❦ ❱❡rt❡① ❙❡t, ❊❞❣❡ ❉♦♠✐♥❛t✐♥❣ ❙❡t, ■♥❞✉❝❡❞ ▼❛t❝❤✐♥❣, ❈❤♦r❞❛❧ ❱❡rt❡① ❉❡❧❡t✐♦♥, ■♥t❡r✈❛❧ ❱❡rt❡① ❉❡❧❡t✐♦♥, ❖❞❞ ❈②❝❧❡ ❚r❛♥s✈❡rs❛❧, ■♥❞✉❝❡❞ d✲❉❡❣r❡❡ ❙✉❜❣r❛♣❤, ▼✐♥ ▲❡❛❢ ❙♣❛♥♥✐♥❣ ❚r❡❡, ▼❛① ❋✉❧❧ ❉❡❣r❡❡ ❙♣❛♥♥✐♥❣ ❚r❡❡, ▲♦♥❣❡st P❛t❤✴❈②❝❧❡, ❊①❛❝t s, t✲P❛t❤, ❊①❛❝t ❈②❝❧❡, ❚r❡❡✇✐❞t❤, P❛t❤✇✐❞t❤
. . . parameterized by a treedepth-modulator have . . .
Our interpretation:
small treewidth modulator: Quasi-compactness and bidimensionality are tangible properties which guarantee this on the respective graph classes
Open questions:
classes using their natural parameter?