On the Parameterized Complexity of \textsc{Maximum Degree Contraction} Problem

TL;DR

This paper investigates the computational complexity of transforming a graph into one with bounded maximum degree via edge contractions, establishing optimal algorithms and complexity bounds, and answering open questions about kernelization.

Contribution

The paper proves the brute-force algorithm is optimal under ETH, introduces a new fixed-parameter tractable algorithm, and shows the problem lacks a polynomial kernel when parameterized by k+d.

Findings

Brute-force algorithm is asymptotically optimal under ETH.

New FPT algorithm with running time 2^{O(dk)} for the problem.

The problem does not admit a polynomial kernel unless NP ⊆ coNP/poly.

Abstract

In the \textsc{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 $n^{\mathcal{O}(k)}$. 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 $(d + k)^{2k} \cdot n^{\mathcal{O}(1)} = 2^{\mathcal{O}(k \log (k+d) )} \cdot n^{\mathcal{O}(1)}$, and remains \NP-hard for every constant $d \ge 2$ (Acta Informatica $(2014)$). We present a different \FPT\ algorithm that runs in time $2^{\mathcal{O}(dk)} \cdot n^{\mathcal{O}(1)}$. In particular, our algorithm runs in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$, 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 $\NP \subseteq \coNP/poly$.