Parameterized Approximation Algorithms for Weighted Vertex Cover

Abstract

A vertex cover of a graph is a set of vertices of the graph such that every edge has at least one endpoint in it. In this work, we study Weighted Vertex Cover with solution size as a parameter. Formally, in the $(k,W)$-Vertex Cover problem, given a graph G, an integer k, a positive rational W, and a weight function $w:V(G) \rightarrow \mathbb {Q}^+$, the question is whether G has a vertex cover of size at most k of weight at most W, with k being the parameter. An (a, b)-bi-criteria approximation algorithm for $(k,W)$-Vertex Cover either produces a vertex cover S such that $|S| \le a k$ and $w(S) \le b W$, or decides that there is no vertex cover of size at most k of weight at most W. We obtain the following results.

• A simple (2, 2)-bi-criteria approximation algorithm for $(k,W)$-Vertex Cover in polynomial time by modifying the standard LP-rounding algorithm.

• A simple exact parameterized algorithm for $(k,W)$-Vertex Cover running in $\mathcal {O}^*(1.4656^k)$ time (Here, the $\mathcal {O}^*$ notation hides factors polynomial in n.).

• A $(1+\epsilon ,2)$-approximation algorithm for $(k,W)$-Vertex Cover running in $\mathcal {O}^*(1.4656^{(1-\epsilon )k})$ time.

• A (1.5, 1.5)-approximation algorithm for $(k,W)$-Vertex Cover running in $\mathcal {O}^*(1.414^k)$ time.

• A $(2-\delta ,2-\delta )$-approximation algorithm for $(k,W)$-Vertex Cover running in $\mathcal {O}^*\left( \sum _{i=\frac{\delta k(1-2 \delta )}{1+2 \delta }}^{\frac{\delta k (1-2\delta )}{2 \delta }} {{\delta k+i} \atopwithdelims (){\delta k-\frac{2i \delta }{1-2 \delta }}}\right)$ time for any $\delta < 0.5$. For example, for (1.75, 1.75) and (1.9, 1.9)-approximation algorithms, we get running times of $\mathcal {O}^*(1.272^k)$ and $\mathcal {O}^*(1.151^k)$ respectively.

Our algorithms (expectedly) do not improve upon the running times of the existing algorithms for the unweighted version of Vertex Cover. When compared to algorithms for the weighted version, our algorithms are the first ones to the best of our knowledge which work with arbitrary weights, and they perform well when the solution size is much smaller than the total weight of the desired solution.

S. Mandal: Supported by Council of Scientific and Industrial Research, India.

Ashutosh Rai: Supported by Science and Engineering Research Board (SERB) Grant SRG/2021/002412.

S. Saurabh: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416), and Swarnajayanti Fellowship (No. DST/SJF/MSA01/2017-18).