Saurabh, Saket ;
Tale, Prafullkumar
On the Parameterized Complexity of Maximum Degree Contraction Problem
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 bruteforce algorithm that checks all possible sets of edges for a solution runs in time n^πͺ(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) β
n^πͺ(1) = 2^πͺ(k log (k+d)) β
n^πͺ(1), and remains NPhard for every constant d β₯ 2 (Acta Informatica (2014)). We present a different FPT algorithm that runs in time 2^πͺ(dk) β
n^πͺ(1). In particular, our algorithm runs in time 2^πͺ(k) β
n^πͺ(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 β coNP/poly.
BibTeX  Entry
@InProceedings{saurabh_et_al:LIPIcs:2020:13329,
author = {Saket Saurabh and Prafullkumar Tale},
title = {{On the Parameterized Complexity of Maximum Degree Contraction Problem}},
booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
pages = {26:126:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771726},
ISSN = {18688969},
year = {2020},
volume = {180},
editor = {Yixin Cao and Marcin Pilipczuk},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13329},
URN = {urn:nbn:de:0030drops133297},
doi = {10.4230/LIPIcs.IPEC.2020.26},
annote = {Keywords: Graph Contraction Problems, FPT Algorithm, Lower Bound, ETH, No Polynomial Kernel}
}
04.12.2020
Keywords: 

Graph Contraction Problems, FPT Algorithm, Lower Bound, ETH, No Polynomial Kernel 
Seminar: 

15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Issue date: 

2020 
Date of publication: 

04.12.2020 