Inapproximability of H-Transversal/Packing

Given an undirected graph G=(V,E) and a fixed pattern graph H with k vertices, we consider the H-Transversal and H-Packing problems. The former asks to find the smallest subset S of vertices such that the subgraph induced by V - S does not have H as a subgraph, and the latter asks to find the maximum number of pairwise disjoint k-subsets S1, ..., Sm such that the subgraph induced by each Si has H as a subgraph. We prove that if H is 2-connected, H-Transversal and H-Packing are almost as hard to approximate as general k-Hypergraph Vertex Cover and k-Set Packing, so it is NP-hard to approximate them within a factor of Omega(k) and Omega(k / polylog(k)) respectively. We also show that there is a 1-connected H where H-Transversal admits an O(log k)-approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of H-Transversal where H is a (family of) cycles, we mention the implication of our result to the related Feedback Vertex Set problem, and give a different hardness proof for directed graphs

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)

On minimum tt-claw deletion in split graphs

For $t\geq 3$, $K_{1, t}$ is called $t$-claw. In minimum $t$-claw deletion problem (\texttt{Min-$t$-Claw-Del}), given a graph $G=(V, E)$, it is required to find a vertex set $S$ of minimum size such that $G[V\setminus S]$ is $t$-claw free. In a split graph, the vertex set is partitioned into two sets such that one forms a clique and the other forms an independent set. Every $t$-claw in a split graph has a center vertex in the clique partition. This observation motivates us to consider the minimum one-sided bipartite $t$-claw deletion problem (\texttt{Min-$t$-OSBCD}). Given a bipartite graph $G=(A \cup B, E)$, in \texttt{Min-$t$-OSBCD} it is asked to find a vertex set $S$ of minimum size such that $G[V \setminus S]$ has no $t$-claw with the center vertex in $A$. A primal-dual algorithm approximates \texttt{Min-$t$-OSBCD} within a factor of $t$. We prove that it is \UGC-hard to approximate with a factor better than $t$. We also prove it is approximable within a factor of 2 for dense bipartite graphs. By using these results on \texttt{Min-$t$-OSBCD}, we prove that \texttt{Min-$t$-Claw-Del} is \UGC-hard to approximate within a factor better than $t$, for split graphs. We also consider their complementary maximization problems and prove that they are \APX-complete.Comment: 11 pages and 1 figur

Kernels for Deletion to Classes of Acyclic Digraphs

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D - S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] considered the kernelization complexity of DFVS with an additional restriction on F, namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k^3) to O(k^2) and of Pumpkin Vertex Deletion Set from O(k^18) to O(k^3). We also prove that the former kernel size is tight under certain complexity theoretic assumptions

Towards a Polynomial Kernel for Directed Feedback Vertex Set

In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O\u27Sullivan and Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus

Towards a Polynomial Kernel for Directed Feedback Vertex Set

In the DIRECTED FEEDBACK VERTEX SET (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity of DFVS parameterized by a natural larger parameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions