On the Parameterized Complexity of Maximum Degree Contraction Problem
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^đȘ(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) â
n^đȘ(1) = 2^đȘ(k log (k+d)) â
n^đȘ(1), and remains NP-hard for every constant d â„ 2 (Acta Informatica (2014)). We present a different FPT algorithm that runs in time 2^đȘ(dk) â
n^đȘ(1). In particular, our algorithm runs in time 2^đȘ(k) â
n^đȘ(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 â coNP/poly.
Graph Contraction Problems
FPT Algorithm
Lower Bound
ETH
No Polynomial Kernel
Theory of computation~Fixed parameter tractability
26:1-26:16
Regular Paper
Full version: http://arxiv.org/abs/2009.11793.
We want to thank the anonymous reviewers for their valuable feedback.
Saket
Saurabh
Saket Saurabh
The Institute Of Mathematical Sciences, HBNI, Chennai, India
University of Bergen, Norway
This project has received funding from the European Research Council (ERC) under the European Unionâs Horizon 2020 research and innovation programme (grant agreement No 819416), and Swarnajayanti Fellowship (No DST/SJF/MSA01/2017-18).
Prafullkumar
Tale
Prafullkumar Tale
CISPA - Helmholtz Center for Information Security, SaarbrĂŒcken, Germany
This research is a part of a project that has received funding from the European Research Council (ERC) under the European Unionâs Horizon 2020 research and innovation programme under grant agreement SYSTEMATICGRAPH (No. 725978).
10.4230/LIPIcs.IPEC.2020.26
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Saket Saurabh and Prafullkumar Tale
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