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On the Parameterized Complexity of Maximum Degree Contraction Problem

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conference contribution
posted on 2023-11-29, 18:15 authored by Saket Saurabh, Prafullkumar Tale
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^????(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.

History

Preferred Citation

Saket Saurabh and Prafullkumar Tale. On the Parameterized Complexity of Maximum Degree Contraction Problem. In: International Symposium on Parameterized and Exact Computation (IPEC). 2020.

Primary Research Area

  • Algorithmic Foundations and Cryptography

Name of Conference

International Symposium on Parameterized and Exact Computation (IPEC)

Legacy Posted Date

2021-05-04

Open Access Type

  • Unknown

BibTeX

@inproceedings{cispa_all_3395, title = "On the Parameterized Complexity of Maximum Degree Contraction Problem", author = "Saurabh, Saket and Tale, Prafullkumar", booktitle="{International Symposium on Parameterized and Exact Computation (IPEC)}", year="2020", }

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