Maximum Minimal Feedback Vertex Set: A Parameterized Perspective

In this paper we study a maximization version of the classical Feedback Vertex Set (FVS) problem, namely, the Max Min FVS problem, in the realm of parameterized complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal feedback vertex set of size at least $k$. We obtain following results for Max Min FVS. 1) We first design a fixed parameter tractable (FPT) algorithm for Max Min FVS running in time $10^kn^{\mathcal{O}(1)}$. 2) Next, we consider the problem parameterized by the vertex cover number of the input graph (denoted by $\mathsf{vc}(G)$), and design an algorithm with running time $2^{\mathcal{O}(\mathsf{vc}(G)\log \mathsf{vc}(G))}n^{\mathcal{O}(1)}$. We complement this result by showing that the problem parameterized by $\mathsf{vc}(G)$ does not admit a polynomial compression unless coNP $\subseteq$ NP/poly. 3) Finally, we give an FPT-approximation scheme (fpt-AS) parameterized by $\mathsf{vc}(G)$. That is, we design an algorithm that for every $\epsilon >0$, runs in time $2^{\mathcal{O}\left(\frac{\mathsf{vc}(G)}{\epsilon}\right)} n^{\mathcal{O}(1)}$ and returns a minimal feedback vertex set of size at least $(1-\epsilon){\sf opt}$

An Efficient Local Search for the Minimum Independent Dominating Set Problem

In the present paper, we propose an efficient local search for the minimum independent dominating set problem. We consider a local search that uses k-swap as the neighborhood operation. Given a feasible solution S, it is the operation of obtaining another feasible solution by dropping exactly k vertices from S and then by adding any number of vertices to it. We show that, when k=2, (resp., k=3 and a given solution is minimal with respect to 2-swap), we can find an improved solution in the neighborhood or conclude that no such solution exists in O(n Delta) (resp., O(n Delta^3)) time, where n denotes the number of vertices and Delta denotes the maximum degree. We develop a metaheuristic algorithm that repeats the proposed local search and the plateau search iteratively, where the plateau search examines solutions of the same size as the current solution that are obtainable by exchanging a solution vertex and a non-solution vertex. The algorithm is so effective that, among 80 DIMACS graphs, it updates the best-known solution size for five graphs and performs as well as existing methods for the remaining graphs

Parameterized Algorithms for Maximum Cut with Connectivity Constraints

We study two variants of Maximum Cut, which we call Connected Maximum Cut and Maximum Minimal Cut, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas Maximum Cut on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size

Minimum Stable Cut and Treewidth

A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. Finding a stable cut is a prototypical PLS-complete problem that has been studied in the context of local search and of algorithmic game theory. In this paper we study Min Stable Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time (?? W)^{O(tw)}n^{O(1)}, where tw is the treewidth, ? the maximum degree, and W the maximum weight. On the other hand, bounding ? is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both tw and ? and obtain an FPT algorithm running in time 2^{O(?tw)}(n+log W)^{O(1)}. Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in (nW)^{o(pw)} or 2^{o(?pw)}(n+log W)^{O(1)}, then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of (1+?). Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time ?^{O(tw)}n^{O(1)}. We show that this is also probably essentially optimal: an algorithm running in n^{o(pw)} would contradict the ETH

(In)approximability of Maximum Minimal FVS

We study the approximability of the NP-complete \textsc{Maximum Minimal Feedback Vertex Set} problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: \textsc{Maximum Minimal Vertex Cover}, for which the best achievable approximation ratio is $\sqrt{n}$, and \textsc{Upper Dominating Set}, which does not admit any $n^{1-\epsilon}$ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for \textsc{Max Min FVS} with a ratio of $O(n^{2/3})$, as well as a matching hardness of approximation bound of $n^{2/3-\epsilon}$, improving the previous known hardness of $n^{1/2-\epsilon}$. The approximation algorithm also gives a cubic kernel when parameterized by the solution size. Along the way, we also obtain an $O(\Delta)$-approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from $\Delta\ge 9$ to $\Delta\ge 6$. Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio $r$, produces an $r$-approximate solution in time $n^{O(n/r^{3/2})}$. This time-approximation trade-off is essentially tight: we show that under the ETH, for any ratio $r$ and $\epsilon>0$, no algorithm can $r$-approximate this problem in time $n^{O((n/r^{3/2})^{1-\epsilon})}$, hence we precisely characterize the approximability of the problem for the whole spectrum between polynomial and sub-exponential time, up to an arbitrarily small constant in the second exponent.Comment: 31 pages, 2 figures, ISAAC 2020, Preprint submitted to Journal of Computer and System Science

Upper clique transversals in graphs

A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the study of the "upper" variant of this parameter, the upper clique transversal number, defined as the maximum size of a minimal clique transversal. We investigate this parameter from the algorithmic and complexity points of view, with a focus on various graph classes. We show that the corresponding decision problem is NP-complete in the classes of chordal graphs, chordal bipartite graphs, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs and proper interval graphs.Comment: Full version of a WG 2023 pape

A New Framework for Kernelization Lower Bounds: The Case of Maximum Minimal Vertex Cover

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex-optimization problems, which we call {lop-kernels}. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erd?s-Hajnal property in order to find an appropriate decomposition. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP ? coNP / poly. This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC