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On the Parameterized Complexity of Maximum Degree Contraction Problem

Authors Saket Saurabh, Prafullkumar Tale



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Author Details

Saket Saurabh
  • The Institute Of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Prafullkumar Tale
  • CISPA - Helmholtz Center for Information Security, Saarbrücken, Germany

Acknowledgements

We want to thank the anonymous reviewers for their valuable feedback.

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Saket Saurabh and Prafullkumar Tale. On the Parameterized Complexity of Maximum Degree Contraction Problem. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 26:1-26:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.IPEC.2020.26

Abstract

In the Maximum Degree Contraction problem, input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time n^𝒪(k). As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm of Parameterized Complexity and proved, among other things, that it admits an FPT algorithm running in time (d + k)^(2k) ⋅ n^𝒪(1) = 2^𝒪(k log (k+d)) ⋅ n^𝒪(1), and remains NP-hard for every constant d ≥ 2 (Acta Informatica (2014)). We present a different FPT algorithm that runs in time 2^𝒪(dk) ⋅ n^𝒪(1). In particular, our algorithm runs in time 2^𝒪(k) ⋅ n^𝒪(1), for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by k + d. We answer this question in the negative and prove that it does not admit a polynomial compression unless NP ⊆ coNP/poly.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Graph Contraction Problems
  • FPT Algorithm
  • Lower Bound
  • ETH
  • No Polynomial Kernel

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