Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G',k') in polynomial time with the guarantee that G' has at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number vc(G) since fvs(G) <= vc(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G',X',k') such that |V(G')| <= 2k and |V(G')| <= O(|X'|^3). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP is in coNP/poly and the polynomial hierarchy collapses to the third level.Comment: Published in "Theory of Computing Systems" as an Open Access publicatio

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Kernelization is a concept that enables the formal mathematical analysis of data reduction through the framework of parameterized complexity. Intensive research into the Vertex Cover problem has shown that there is a preprocessing algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G\u27,k\u27) in polynomial time with the guarantee that G\u27 has at most 2k\u27 vertices (and thus O((k\u27)^2) edges) with k\u27 <= k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number VC(G) since fvs(G) <= VC(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G\u27,X\u27,k\u27) such that k\u27 <= k, |X\u27| <= |X| and most importantly |V(G\u27)| <= 2k and |V(G\u27)| in O(|X\u27|^3). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have polynomial kernel when parameterized by fvs(G) unless the polynomial hierarchy collapses to the third level (PH=Sigma_3^p). Our work is one of the first examples of research in kernelization using a non-standard parameter, and shows that this approach can yield interesting computational insights. To obtain our results we make extensive use of the combinatorial structure of independent sets in forests

Structural Parameterizations of Feedback Vertex Set

A feedback vertex set in an undirected graph is a subset of vertices whose removal results in an acyclic graph. It is well-known that the problem of finding a minimum sized (or k sized in case of decision version of) feedback vertex set (FVS) is polynomial time solvable in (sub)-cubic graphs, in pseudo-forests (graphs where each component has at most one cycle) and mock-forests (graph where each vertex is part of at most one cycle). In general graphs, it is known that the problem is NP-Complete, and has an O*((3.619)^k) fixed-parameter algorithm and an O(k^2) kernel where k, the solution size is the parameter. We consider the parameterized and kernelization complexity of feedback vertex set where the parameter is the size of some structure of the input. In particular, we show that * FVS is fixed-parameter tractable, but is unlikely to have polynomial sized kernel when parameterized by the number of vertices of the graph whose degree is at least 4. This answers a question asked in an earlier paper. * When parameterized by k, the number of vertices, whose deletion results in a pseudo-forest, we give an O(k^6) vertices kernel improving from the previously known O(k^{10}) bound. * When parameterized by the number k of vertices, whose deletion results in a mock-d-forest, we give a kernel consisting of O(k^{3d+3}) vertices and prove a lower bound of Omega(k^{d+2}) vertices (under complexity theoretic assumptions). Mock-d-forest for a constant d is a mock-forest where each component has at most d cycles

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

Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most $k$ of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a \BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed $k$. It is known that this implies a polynomial-time compression algorithm that turns OCT instances into equivalent instances of size at most \BigOh(4^k), a so-called kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in $k$, has turned into one of the main open questions in the study of kernelization. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in $k$. For OCT, the matroid is built to allow us to simulate the computation of the iterative compression step of the algorithm of Reed, Smith, and Vetta, applied (for only one round) to an approximate odd cycle transversal which it is aiming to shrink to size $k$. The process is randomized with one-sided error exponentially small in $k$, where the result can contain false positives but no false negatives, and the size guarantee is cubic in the size of the approximate solution. Combined with an \BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a reduction of the instance to size \BigOh(k^{4.5}), implying a randomized polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape

Simultaneous Feedback Edge Set: A Parameterized Perspective

In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G) -> 2^[alpha]and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Here, G_i = (V (G), {e in E(G) | i in col(e)}) and [alpha] = {1,...,alpha}. In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is same as the input of Sim-FVS and the objective is to check whether there is an edge subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Unlike the vertex variant of the problem, when alpha = 1, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for alpha = 3 Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic-graphs. The same reduction shows that the problem does not admit an algorithm of running time O(2^o(k) n^O(1)) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time O(2^((omega k alpha) + (alpha log k)) n^O(1)), where omega is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when alpha = 2. We also give a kernel for Sim-FES with (k alpha)^O(alpha) vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph G, an integer q and, a coloring function col : E(G) -> 2^[alpha] . The question is whether there is a edge subset F of cardinality at least q in G such that for all i in [alpha], G[F_i] is acyclic. Here, F_i = {e in F | i in col(e)}. We give an FPT algorithm for Maximum Simultaneous Acyclic Subgraph running in time O(2^(omega q alpha) n^O(1) ). All our algorithms are based on parameterized version of the Matroid Parity problem