Induced Matching below Guarantees: Average Paves the Way for Fixed-Parameter Tractability

TL;DR

This paper investigates the parameterized complexity of the Induced Matching problem, introducing a novel average-based parameterization that enables a fixed-parameter tractable algorithm using Gallai-Edmonds decomposition.

Contribution

It presents the first FPT algorithm for Induced Matching parameterized by the average of maximum matching and independent set sizes minus the target size.

Findings

The problem is unlikely FPT when parameterized by maximum matching or independent set size minus the target.

A new branching algorithm solves Induced Matching in exponential time based on the average of maximum matching and independent set sizes.

Gallai-Edmonds decomposition is effectively used to structure the algorithm.

Abstract

In this work, we study the Induced Matching problem: Given an undirected graph $G$ and an integer $\ell$, is there an induced matching $M$ of size at least $\ell$? An edge subset $M$ is an induced matching in $G$ if $M$ is a matching such that there is no edge between two distinct edges of $M$. Our work looks into the parameterized complexity of Induced Matching with respect to "below guarantee" parameterizations. We consider the parameterization $u - \ell$ for an upper bound $u$ on the size of any induced matching. For instance, any induced matching is of size at most $n / 2$ where $n$ is the number of vertices, which gives us a parameter $n / 2 - \ell$. In fact, there is a straightforward $9^{n/2 - \ell} \cdot n^{O(1)}$-time algorithm for Induced Matching [Moser and Thilikos, J. Discrete Algorithms]. Motivated by this, we ask: Is Induced Matching FPT for a parameter smaller than $n / 2 - \ell$? In search for such parameters, we consider $MM(G) - \ell$ and $IS(G) - \ell$, where $MM(G)$ is the maximum matching size and $IS(G)$ is the maximum independent set size of $G$. We find that Induced Matching is presumably not FPT when parameterized by $MM(G) - \ell$ or $IS(G) - \ell$. In contrast to these intractability results, we find that taking the average of the two helps -- our main result is a branching algorithm that solves Induced Matching in $49^{(MM(G) + IS(G))/ 2 - \ell} \cdot n^{O(1)}$ time. Our algorithm makes use of the Gallai-Edmonds decomposition to find a structure to branch on.