An Improved Kernel and Parameterized Algorithm for Almost Induced Matching

TL;DR

This paper advances algorithms for the Almost Induced Matching problem by providing a smaller kernel and a faster parameterized algorithm, improving efficiency in solving this graph problem.

Contribution

It introduces a 6k-vertex kernel and an improved O*(1.6765^k) algorithm for the problem, surpassing previous bounds.

Findings

Reduced kernel size from 7k to 6k.

Improved running time from O*(1.7485^k) to O*(1.6765^k).

Provides efficient parameterized algorithms for Almost Induced Matching.

Abstract

An induced subgraph is called an induced matching if each vertex is a degree-1 vertex in the subgraph. The \textsc{Almost Induced Matching} problem asks whether we can delete at most $k$ vertices from the input graph such that the remaining graph is an induced matching. This paper studies parameterized algorithms for this problem by taking the size $k$ of the deletion set as the parameter. First, we prove a $6k$-vertex kernel for this problem, improving the previous result of $7k$. Second, we give an $O^*(1.6765^k)$-time and polynomial-space algorithm, improving the previous running-time bound of $O^*(1.7485^k)$.