NDLI: A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kratsch, Stefan
Copyright Year	2018
Abstract	In the vertex cover problem we are given a graph $G=(V,E)$ and an integer $k$ and have to determine whether there is a set $X\subseteq V$ of size at most $k$ such that each edge in $E$ has at least one endpoint in $X$. The problem can be easily solved in time $\mathcal{O}^(2^k)$, making it fixed-parameter tractable (FPT) with respect to $k$. While the fastest known algorithm takes only time $\mathcal{O}^(1.2738^k)$, much stronger improvements have been obtained by studying parameters that are smaller than $k$. Apart from treewidth-related results, the arguably best algorithm for vertex cover runs in time $\mathcal{O}^(2.3146^p)$, where $p=k-LP(G)$ is only the excess of the solution size $k$ over the best fractional vertex cover [Lokshtanov et al. ACM Trans. Algorithms, 11 (2014), pp. 15:1--15:31]. Since $p\leq k$ but $k$ cannot be bounded in terms of $p$ alone, this strictly increases the range of tractable instances. Recently, [Garg and Philip, Proceedings of the Symposium on Discrete Algorithms, 2016, pp. 1152--1166] greatly contributed to understanding the parameterized complexity of the vertex cover problem. They prove that $2LP(G)-MM(G)$ is a lower bound for the vertex cover size of $G$, where $MM(G)$ is the size of a largest matching of $G$, and proceed to study parameter $\ell=k-(2LP(G)-MM(G))$. They give an algorithm of running time $\mathcal{O}^(3^\ell)$, proving that vertex cover is FPT in $\ell$. It can be easily observed that $\ell\leq p$ whereas $p$ cannot be bounded in terms of $\ell$ alone. We complement the work of Garg and Philip by proving that vertex cover admits a randomized polynomial kernelization in terms of $\ell$, i.e., an efficient preprocessing to size polynomial in $\ell$. This improves over parameter $p=k-LP(G)$ for which this was previously known [Kratsch and Wahlström, Proceedings of the IEEE Symposium on Foundations of Computer Science, 2012, pp. 450--459].
Starting Page	1806
Ending Page	1839
Page Count	34
File Format	PDF
ISSN	08954801
DOI	10.1137/16M1104585
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	3
Volume Number	32
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-07-26
Access Restriction	Subscribed
Subject Keyword	Nonnumerical algorithms vertex cover parameterized complexity polynomial kernelization Randomized algorithms Analysis of algorithms
Content Type	Text
Resource Type	Article
Subject	Mathematics

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