2026-07-22T18:55:37Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13329 2024-03-06T10:51:43Z ddc:004 open_access

On the Parameterized Complexity of Maximum Degree Contraction Problem Saurabh, Saket Tale, Prafullkumar Graph Contraction Problems FPT Algorithm Lower Bound ETH No Polynomial Kernel 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Saket Saurabh and Prafullkumar Tale 2020 Is Part Of LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2020.26 urn:nbn:de:0030-drops-133297 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.26 eng https://creativecommons.org/licenses/by/3.0/legalcode