Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Novotná, Jana; Okrasa, Karolina; Pilipczuk, Michał; Rzążewski, Paweł; Leeuwen, Erik Jan van; Walczak, Bartosz

doi:10.1007/s00453-020-00745-z

Cited by 7 publications

(7 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Every string graph that does not contain K , as a subgraph has degeneracy O( log ). This property of string graphs was also used algorithmically [34,35]. Our Theorem 4 is the analogue of Theorem 10 in C t -free graphs.…”

Section: Introductionmentioning

confidence: 95%

“…This property turned to be very useful in the design of algorithms, as for many natural problems, including MAX INDEPENDENT SET, MAX INDUCED MATCHING, or 3-COLORING, each of the two possible outcomes allows us to compute the solution efficiently. Such a win-win approach leads to a subexponential running time for the considered problems [34,35].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Degeneracy of $P_t$-free and $C_{\geq t}$-free graphs with no large complete bipartite subgraphs

Bonamy,

Bousquet,

Pilipczuk

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

A hereditary class of graphs G is χ-bounded if there exists a function f such that every graph G ∈ G satisfies χ(G) f (ω(G)), where χ(G) and ω(G) are the chromatic number and the clique number of G, respectively. As one of the first results about χ-bounded classes, Gyárfás proved in 1985 that if G is Ptfree, i.e., does not contain a t-vertex path as an induced subgraph, then χ(G) (t − 1) ω(G)−1 . In 2017, Chudnovsky, Scott, and Seymour proved that C t -free graphs, i.e., graphs that exclude induced cycles with at least t vertices, are χ-bounded as well, and the obtained bound is again superpolynomial in the clique number. Note that Pt−1-free graphs are in particular C t -free. It remains a major open problem in the area whether for C t -free, or at least Pt-free graphs G, the value of χ(G) can be bounded from above by a polynomial function of ω(G). We consider a relaxation of this problem, where we compare the chromatic number with the size of a largest balanced biclique contained in the graph as a (not necessarily induced) subgraph. We show that for every t there exists a constant c such that for and every C t -free graph which does not contain K , as a subgraph, it holds that χ(G)c .

show abstract

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Degeneracy of $P_t$-free and $C_{\geq t}$-free graphs with no large complete bipartite subgraphs

Bonamy,

Bousquet,

Pilipczuk

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the first case, we identify two more positive cases: we show that if H ∈ {P 3 , C 4 }, then the problem admits a subexponential-time algorithm for every F ∈ S. The algorithm itself uses a win-win strategy: we combine branching on a high-degree vertex with a separator theorem that can be used if the maximum degree is bounded. A similar approach was used for various other problems [20,31], however, the specifics of our problem require a slightly more complicated approach.…”

Section: Our Contributionmentioning

confidence: 99%

List Locally Surjective Homomorphisms in Hereditary Graph Classes

Dvořák¹,

Krawczyk²,

Masařík³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V (G) to V (H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V (H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F -free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H, F ) the problem can be solved in subexponential time.We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F -free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of boundeddegree forests F , that might lead to some tractability results, is the family S consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P3, C4} are the only connected ones that allow for a subexponential-time algorithm in F -free graphs for every F ∈ S (unless the ETH fails).

show abstract

“…The existence of such a subexponential-time algorithm for F -free graphs is excluded under the Exponential Time Hypothesis whenever F is not a subdivided claw forest (see e.g. the discussion in [27]), which shows a qualitative difference between the negative and the potentially positive cases. Also, Chudnovsky et al [10] recently gave a quasi-polynomial-time approximation scheme (QPTAS) for Maximum Independent Set in F -free graphs, for every fixed subdivided claw forest F .…”

Section: Introductionmentioning

confidence: 99%

“…For Maximum Independent Set this property is being edgeless, but for instance the property of being acyclic corresponds to the Maximum Induced Forest problem, which by complementation is equivalent to Feedback Vertex Set. Work in this direction so far focused on properties that imply bounded treewidth [1,17] or, more generally, that imply sparsity [27].…”

Section: Introductionmentioning

confidence: 99%

Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes

Chudnovsky¹,

King²,

Pilipczuk³

et al. 2021

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k = 2 is equivalent (by complementation) to Odd Cycle Transversal.We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved: in {P5, F }-free graphs in polynomial time, whenever F is a threshold graph; in {P5, bull}-free graphs in polynomial time; in P5-free graphs in time n O(ω(G)) ; in {P6, 1-subdivided claw}-free graphs in time n O(ω(G) 3 ) . Here, n is the number of vertices of the input graph G and ω(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P5-free and for {P6, 1-subdivided claw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs.Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P5-free graphs, if we allow loops on H.

show abstract

Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Cited by 7 publications

References 31 publications

Degeneracy of $P_t$-free and $C_{\geq t}$-free graphs with no large complete bipartite subgraphs

Degeneracy of $P_t$-free and $C_{\geq t}$-free graphs with no large complete bipartite subgraphs

List Locally Surjective Homomorphisms in Hereditary Graph Classes

Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes

Contact Info

Product

Resources

About