Subset feedback vertex set is fixed parameter tractable

The classical Feedback Vertex Set problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an instance comes additionally with a set S ? V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00]. The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2^k n^O(1) instances with the size of S bounded by O(k^3), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.Comment: full version of a paper presented at ICALP'1

Ehrhart clutters: Regularity and Max-Flow Min-Cut

If C is a clutter with n vertices and q edges whose clutter matrix has column vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then C is an Ehrhart clutter and in this case we provide sharp bounds on the Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols conjecture on packing problems, we conjecture that if C is both ideal and the clique clutter of a perfect graph, then C has the MFMC property. We prove this conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof of our conjecture when C is a uniform clique clutter of a perfect graph. We close with a generalization of Ehrhart clutters as it relates to total dual integrality.Comment: Electronic Journal of Combinatorics, to appea

Reducing Graph Transversals via Edge Contractions

For a graph parameter ?, the Contraction(?) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which ? has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where ? is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ? according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ?, which in particular imply that Contraction(?) is co-NP-hard even for fixed k = d = 1 when ? is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when ? is the size of a minimum vertex cover, the problem is in XP parameterized by d

Blocking Dominating Sets for H-Free Graphs via Edge Contractions

In this paper, we consider the following problem: given a connected graph G, can we reduce the domination number of G by one by using only one edge contraction? We show that the problem is NP-hard when restricted to {P6, P4 + P2}-free graphs and that it is coNP-hard when restricted to subcubic claw-free graphs and 2P3-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on H-free graphs when H is connected

A branch-and-cut algorithm for the Edge Interdiction Clique Problem

Given a graph G and an interdiction budget k∈N, the Edge Interdiction Clique Problem (EICP) asks to find a subset of at most k edges to remove from G so that the size of the maximum clique, in the interdicted graph, is minimized. The EICP belongs to the family of interdiction problems with the aim of reducing the clique number of the graph. The EICP optimal solutions, called optimal interdiction policies, determine the subset of most vital edges of a graph which are crucial for preserving its clique number. We propose a new set-covering-based Integer Linear Programming (ILP) formulation for the EICP with an exponential number of constraints, called the clique-covering inequalities. We design a new branch-and-cut algorithm which is enhanced by a tailored separation procedure and by an effective heuristic initialization phase. Thanks to the new exact algorithm, we manage to solve the EICP in several sets of instances from the literature. Extensive tests show that the new exact algorithm greatly outperforms the state-of-the-art approaches for the EICP

A large and natural Class of Σ2p\Sigma^p_2- and Σ3p\Sigma^p_3-complete Problems in Bilevel and Robust Optimization

Because $\Sigma^p_2$- and $\Sigma^p_3$-hardness proofs are usually tedious and difficult, not so many complete problems for these classes are known. This is especially true in the areas of min-max regret robust optimization, network interdiction, most vital vertex problems, blocker problems, and two-stage adjustable robust optimization problems. Even though these areas are well-researched for over two decades and one would naturally expect many (if not most) of the problems occurring in these areas to be complete for the above classes, almost no completeness results exist in the literature. We address this lack of knowledge by introducing over 70 new $\Sigma^p_2$-complete and $\Sigma^p_3$-complete problems. We achieve this result by proving a new meta-theorem, which shows $\Sigma^p_2$- and $\Sigma^p_3$-completeness simultaneously for a huge class of problems. The majority of all earlier publications on $\Sigma^p_2$- and $\Sigma^p_3$-completeness in said areas are special cases of our meta-theorem. Our precise result is the following: We introduce a large list of problems for which the meta-theorem is applicable (including clique, vertex cover, knapsack, TSP, facility location and many more). For every problem on this list, we show: The interdiction/minimum cost blocker/most vital nodes problem (with element costs) is $\Sigma^p_2$-complete. The min-max-regret problem with interval uncertainty is $\Sigma^p_2$-complete. The two-stage adjustable robust optimization problem with discrete budgeted uncertainty is $\Sigma^p_3$-complete. In summary, our work reveals the interesting insight that a large amount of NP-complete problems have the property that their min-max versions are 'automatically' $\Sigma^p_2$-complete