Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels

We study two fundamental problems related to finding subgraphs: (1) given graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2) given graphs G, H, and an integer t, PACKING asks if G contains t vertex-disjoint subgraphs isomorphic to H. For every graph class F, let F-Subgraph Test and F-Packing be the special cases of the two problems where H is restricted to be in F. Our goal is to study which classes F make the two problems tractable in one of the following senses: - (randomized) polynomial-time solvable, - admits a polynomial (many-one) kernel (that is, has a polynomial-time preprocessing procedure that creates an equivalent instance whose size is polynomially bounded by the size of the solution), or - admits a polynomial Turing kernel (that is, has an adaptive polynomial-time procedure that reduces the problem to a polynomial number of instances, each of which has size bounded polynomially by the size of the solution). To obtain a more robust setting, we restrict our attention to hereditary classes F. It is known that if every component of every graph in F has at most two vertices, then F-Packing is polynomial-time solvable, and NP-hard otherwise. We identify a simple combinatorial property (every component of every graph in F either has bounded size or is a bipartite graph with one of the sides having bounded size) such that if a hereditary class F has this property, then F-Packing admits a polynomial kernel, and has no polynomial (many-one) kernel otherwise, unless the polynomial hierarchy collapses. Furthermore, if F does not have this property, then F-Packing is either WK[1]-hard, W[1]-hard, or Long Path-hard, giving evidence that it does not admit polynomial Turing kernels either. For F-Subgraph Test, we show that if every graph of a hereditary class F satisfies the property that it is possible to delete a bounded number of vertices such that every remaining component has size at most two, then F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard otherwise. We introduce a combinatorial property called (a, b, c, d)-splittability and show that if every graph in a hereditary class F has this property, then F-Subgraph Test admits a polynomial Turing kernel and it is WK[1]-hard, W[1]-hard, or Long Path-hard otherwise. We do not give a complete characterization of the cases when F-Subgraph Test admits polynomial many-one kernels, but show examples that this question is much more fragile than the characterization for Turing kernels

Crossing Paths with Hans Bodlaender:A Personal View on Cross-Composition for Sparsification Lower Bounds

On the occasion of Hans Bodlaender’s 60th birthday, I give a personal account of our history and work together on the technique of cross-composition for kernelization lower bounds. I present several simple new proofs for polynomial kernelization lower bounds using cross-composition, interlaced with personal anecdotes about my time as Hans’ PhD student at Utrecht University. Concretely, I will prove that Vertex Cover, Feedback Vertex Set, and the H-Factor problem for every graph H that has a connected component of at least three vertices, do not admit kernels of (formula presented) bits when parameterized by the number of vertices n for any (formula presented), unless (formula presented). These lower bounds are obtained by elementary gadget constructions, in particular avoiding the use of the Packing Lemma by Dell and van Melkebeek.</p

Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

Turing kernelization for finding long paths and cycles in restricted graph classes

\u3cp\u3eThe k-PATH problem asks whether a given undirected graph has a (simple) path of length k. We prove that k-PATH has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree, claw-free graphs, or to K \u3csub\u3e3,t\u3c/sub\u3e-minor-free graphs. This means that there is an algorithm that, given a k-PATH instance (G,k) belonging to one of these graph classes, computes its answer in polynomial time when given access to an oracle that solves k-PATH instances of size polynomial in k in a single step. Our techniques also apply to k-CYCLE, which asks for a cycle of length at least k. \u3c/p\u3

Constrained bipartite vertex cover:The easy kernel is essentially tight

\u3cp\u3eThe CONSTRAINED BIPARTITE VERTEX COVER problem asks, for a bipartite graph G with partite sets A and B, and integers k\u3csub\u3eA\u3c/sub\u3e and k\u3csub\u3eB\u3c/sub\u3e, whether there is a vertex cover for G containing at most k\u3csub\u3eA\u3c/sub\u3e vertices from A and k\u3csub\u3eB\u3c/sub\u3e vertices from B. The problem has an easy kernel with 2k\u3csub\u3ea\u3c/sub\u3e · k\u3csub\u3eb\u3c/sub\u3e edges and 4k\u3csub\u3eA\u3c/sub\u3e · k\u3csub\u3eb\u3c/sub\u3e vertices, based on the fact that every vertex in A of degree more than k\u3csub\u3eB\u3c/sub\u3e has to be included in the solution, together with every vertex in B of degree more than k\u3csub\u3eA\u3c/sub\u3e. We show that the number of vertices and edges in this kernel are asymptotically essentially optimal in terms of the product k\u3csub\u3eA\u3c/sub\u3e· k\u3csub\u3eB\u3c/sub\u3e. We prove that if there is a polynomial-time algorithm that reduces any instance (G,A,B, k\u3csub\u3eA\u3c/sub\u3e, k\u3csub\u3eB\u3c/sub\u3e) of CONSTRAINED BIPARTITE VERTEX COVER to an equivalent instance (G', A', B', k'\u3csub\u3eA\u3c/sub\u3e, k'\u3csub\u3eB\u3c/sub\u3e) such that k'\u3csub\u3eA\u3c/sub\u3e ∈ (k\u3csub\u3eA\u3c/sub\u3e) O\u3csup\u3e1\u3c/sup\u3e(k\u3csub\u3eB\u3c/sub\u3e), k'\u3csub\u3eB\u3c/sub\u3e ∈ (k\u3csub\u3eB\u3c/sub\u3e)\u3csup\u3eO(1)\u3c/sup\u3e, and |V(G')| ∈ O((k\u3csub\u3eB\u3c/sub\u3e · kB)\u3csup\u3e1 -ε\u3c/sup\u3e), for some ε &gt; 0, then NP ⊆ coNP/poly and the polynomial-time hierarchy collapses. Using a different construction, we prove that if there is a polynomial-time algorithm that reduces any n-vertex instance into an equivalent instance (of a possibly different problem) that can be encoded in O(n\u3csup\u3e2-ε\u3c/sup\u3e) bits, then NP ⊆ coNP/poly.\u3c/p\u3

On structural parameterizations of hitting set : hitting paths in graphs using 2-SAT

Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T. We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results

On structural parameterizations of hitting set : hitting paths in graphs using 2-SAT

\u3cp\u3eHitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T. We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 2 \u3csup\u3e5k\u3c/sup\u3e(|G| + |P|) \u3csup\u3eO(1)\u3c/sup\u3ewhether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results. \u3c/p\u3

Approximation and kernelization for chordal vertex deletion

The Chordal Vertex Deletion (ChVD) problem asks to delete a minimum number of vertices from an input graph to obtain a chordal graph. In this paper we develop a polynomial kernel for ChVD under the parameterization by the solution size, as well as poly(opt) approximation algorithm. The first result answers an open problem of Marx from 2006 [WG 2006, LNCS 4271, 37-48]

Fine-grained parameterized complexity analysis of graph coloring problems

The q-Coloring problem asks whether the vertices of a graph can be properly colored with q colors. Lokshtanov et al. [SODA 2011] showed that q-Coloring on graphs with a feedback vertex set of size k cannot be solved in time O ∗ ((q−ε) k ) O∗((q−ε)k) , for any ε&gt;0 ε&gt;0 , unless the Strong Exponential-Time Hypothesis (SETH SETH ) fails. In this paper we perform a fine-grained analysis of the complexity of q-Coloring with respect to a hierarchy of parameters. We show that unless ETH ETH fails, there is no universal constant θ θ such that q-Coloring parameterized by vertex cover can be solved in time O ∗ (θ k ) O∗(θk) for all fixed q. We prove that there are O ∗ ((q−ε) k ) O∗((q−ε)k) time algorithms where k is the vertex deletion distance to several graph classes F F for which q-Coloring is known to be solvable in polynomial time, including all graph classes whose (q+1) (q+1) -colorable members have bounded treedepth. In contrast, we prove that if F F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless SETH SETH fails. This research was partially funded by the Networks programme via the Dutch Ministry of Education, Culture and Science through the Netherlands Organisation for Scientific Research. The research was done while the first author was at CWI, Amsterdam. The second author was supported by NWO Veni grant “Frontiers in Parameterized Preprocessing”

A Turing kernelization dichotomy for structural parameterizations of ℱ -minor-free deletion

\u3cp class= para style= margin:0cm;margin-bottom:.0001pt;background:white \u3eFor a fixed finite family of graphs FF, the FF-Minor-Free Deletion problemtakes as input a graph \u3cem style= box-sizing: border-box \u3eG\u3c/em\u3e and an integer ℓℓ and asks whether there exists a set X⊆V(G)X⊆V(G) of size at most ℓℓ such that G−XG−X is FF-minor-free. For F={K2}F={K2} and F={K3}F={K3} this encodes VertexCover and FeedbackVertex Set respectively. When parameterized by thefeedback vertex number of \u3cem style= box-sizing: border-box \u3eG\u3c/em\u3e these two problems areknown to admit a polynomial kernelization. Such a polynomial kernelization alsoexists for any FF containing a planar graph but no forests.\u3c/p\u3e\u3cp/\u3e\u3cp class= para style= margin: 0cm 0cm 0.0001pt; background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial; /\u3e\u3cp class= para style= margin:0cm;margin-bottom:.0001pt;background:white; box-sizing: border-box;overflow-wrap: break-word;word-break:break-word; font-variant-ligatures: normal;font-variant-caps: normal;orphans: 2;text-align: start;widows: 2;-webkit-text-stroke-width: 0px;text-decoration-style: initial; text-decoration-color: initial;word-spacing:0px \u3eIn this paper we show that FF-Minor-Free Deletion parameterizedby the feedback vertex number is MK[2]MK[2]-hard for F={P3}F={P3}. This rules out the existence of a polynomial kernel ssuming NP⊈coNP/polyNP⊈coNP/poly, and also gives evidence that the problem does not admit a polynomialTuring kernel. Our hardness result generalizes to any FF not containing a P3P3-subgraph-free graph, using as parameter the vertex-deletion distance totreewidth mintw(F)mintw(F), where mintw(F)mintw(F) denotes the minimum treewidth of the graphs in FF. For the other case, where FF contains a P3P3-subgraph-free graph, we present a polynomial Turing kernelization. Ourresults extend to FF-Subgraph-Free Deletion.\u3c/p\u3e\u3cp/\u3e\u3cp class= para style= margin:0cm;margin-bottom:.0001pt;background:white; box-sizing: border-box;overflow-wrap: break-word;word-break:break-word; font-variant-ligatures: normal;font-variant-caps: normal;orphans: 2;text-align: start;widows: 2;-webkit-text-stroke-width: 0px;text-decoration-style: initial; text-decoration-color: initial;word-spacing:0px /\u3e\u3cp class= MsoNormal \u3e \u3c/p\u3