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Register a new userOn the Approximate Compressibility of Connected Vertex Cover
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Added by
Diptapriyo Majumdar
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have efficient preprocessing algorithms, also known as polynomial kernelizations. However, it has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one considered an appropriate notion of approximate kernelization, then this problem parameterized by the solution size does admit an approximate polynomial kernelization. In fact, Lokhtanov et al. were able to obtain a polynomial size approximate kernelization scheme (PSAKS) for Connected Vertex Cover parameterized by the solution size. A PSAKS is essentially a preprocessing algorithm whose error can be made arbitrarily close to 0. In this paper we revisit this problem, and consider parameters that are strictly smaller than the size of the solution and obtain the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph to the class of cographs, the class of bounded treewidth graphs, and the class of all chordal graphs.
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A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this conjecture, and also studied its implied fraction
The CONNECTED VERTEX COVER problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al.[STOC 2017], have shown an $alpha$-approximate kernel for the problem for every $alpha > 1$, in the framework of approximate or lossy kernelization. In this work, we exhibit lossy kernels and FPT algorithms for CONNECTED VERTEX COVER for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size. The parameters we consider are the sizes of a split deletion set, clique deletion set, clique cover, cluster deletion set and chordal deletion set.
We give efficient distributed algorithms for the minimum vertex cover problem in bipartite graphs in the CONGEST model. From KH{o}nigs theorem, it is well known that in bipartite graphs the size of a minimum vertex cover is equal to the size of a maximum matching. We first show that together with an existing $O(nlog n)$-round algorithm for computing a maximum matching, the constructive proof of KH{o}nigs theorem directly leads to a deterministic $O(nlog n)$-round CONGEST algorithm for computing a minimum vertex cover. We then show that by adapting the construction, we can also convert an emph{approximate} maximum matching into an emph{approximate} minimum vertex cover. Given a $(1-delta)$-approximate matching for some $delta>1$, we show that a $(1+O(delta))$-approximate vertex cover can be computed in time $O(D+mathrm{poly}(frac{log n}{delta}))$, where $D$ is the diameter of the graph. When combining with known graph clustering techniques, for any $varepsilonin(0,1]$, this leads to a $mathrm{poly}(frac{log n}{varepsilon})$-time deterministic and also to a slightly faster and simpler randomized $O(frac{log n}{varepsilon^3})$-round CONGEST algorithm for computing a $(1+varepsilon)$-approximate vertex cover in bipartite graphs. For constant $varepsilon$, the randomized time complexity matches the $Omega(log n)$ lower bound for computing a $(1+varepsilon)$-approximate vertex cover in bipartite graphs even in the LOCAL model. Our results are also in contrast to the situation in general graphs, where it is known that computing an optimal vertex cover requires $tilde{Omega}(n^2)$ rounds in the CONGEST model and where it is not even known how to compute any $(2-varepsilon)$-approximation in time $o(n^2)$.
We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless $NP subseteq coNP/poly$, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an $O(2^{O(pk)}n^{O(1)})$-time algorithm for the $p$-Edge-Connected VC and an $O(2^{O(k^2)}n^{O(1)})$-time algorithm for the $p$-Vertex-Connected VC. Finally, we describe a $2(p+1)$-approximation algorithm for the $p$-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning $p$-vertex/edge-connected subgraph of a $p$-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).
We give a characterization result for the integrality gap of the natural linear programming relaxation for the vertex cover problem. We show that integrality gap of the standard linear programming relaxation for any graph G equals $left(2-frac{2}{chi^f(G)}right)$ where $chi^f(G)$ denotes the fractional chromatic number of G.
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