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On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

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Published:01 September 2021Publication History
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Abstract

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F-Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k, F-Contraction asks whether there exists XE(G) such that G/XF and |X| ≤ k. Here, G/X is the graph obtained from G by contracting edges in X. We obtain the following results for the F-Contraction problem:

Clique Contraction is known to be FPT. However, unless NP⊆ coNP/poly, it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme (PSAKS). That is, it admits a (1 + ε)-approximate kernel with O(kf(ε)) vertices for every ε > 0.

Split Contraction is known to be W[1]-Hard. We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)-FPT-approximation algorithm for Split Contraction). Furthermore, we show that, assuming Gap-ETH, there is no (5/4-δ)-FPT-approximation algorithm for Split Contraction. Here, ε, δ > 0 are fixed constants.

Chordal Contraction is known to be W[2]-Hard. We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPTW[1], there is no F(k)-FPT-approximation algorithm for Chordal Contraction. Here, F(k) is an arbitrary function depending on k alone.

We say that an algorithm is an h(k)-FPT-approximation algorithm for the F-Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists XE(G) satisfying G/XF and |X| ≤ k, it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 13, Issue 4
        December 2021
        198 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3481683
        Issue’s Table of Contents

        Copyright © 2021 Association for Computing Machinery.

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 September 2021
        • Accepted: 1 June 2021
        • Revised: 1 April 2021
        • Received: 1 July 2020
        Published in toct Volume 13, Issue 4

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