Treedepth Parameterized by Vertex Cover Number

To solve hard graph problems from the parameterized perspective, structural parameters have commonly been used. In particular, vertex cover number is frequently used in this context. In this paper, we study the problem of computing the treedepth of a given graph G. We show that there are an O(tau(G)^3) vertex kernel and an O(4^{tau(G)}*tau(G)*n) time fixed-parameter algorithm for this problem, where tau(G) is the size of a minimum vertex cover of G and n is the number of vertices of G

Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters

We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: - Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d+1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. - We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games. - We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ? S[t] ? A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems

Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters

We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d + 1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games. We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ⊆ S[t] ⊆ A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems

Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space

A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of G\"oz\"upek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure

Uniform kernelization complexity of hitting forbidden minors

The F-MINOR-FREE DELETION problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether κ vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. At FOCS 2012, Fomin et al. showed that the special case when F contains at least one planar graph has a kernel of size f (F) · κg(F) for some functions f and g. They left open whether this PLANAR F-MINOR-FREE DELETION problem has kernels whose size is uniformly polynomial, of the form f (F) · κc for some universal constant c. We prove that some PLANAR F-MINOR-FREE DELETION problems do not have uniformly polynomial kernels (unless NP ⊆ coNP/poly), not even when parameterized by the vertex cover number. On the positive side, we consider the problem of determining whether κ vertices can be removed to obtain a graph of treedepth at most η. We prove that this problem admits uniformly polynomial kernels with O(κ6) vertices for every fixed η.</p

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. In this paper we perform a fine-grained analysis of the complexity of q-COLORING with respect to a hierarchy of structural parameters. We show that unless the Exponential Time Hypothesis fails, there is no constant θ such that q-COLORING parameterized by the size k of a vertex cover can be solved in O ∗(θ k) time for all fixed q. We prove that there are O ∗((q−ɛ) k) time algorithms where k is the vertex deletion distance to several graph classes for which q-COLORING is known to be solvable in polynomial time, including all graph classes F whose (q+1)-colorable members have bounded treedepth. In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless the Strong Exponential Time Hypothesis fails.</p