A Parameterized Perspective on $P_2$-Packings

TL;DR

This paper investigates $P_2$-packings in graphs from a parameterized perspective, providing extremal bounds, a kernelization algorithm, and an improved exponential-time algorithm for finding packings of size $k$.

Contribution

It introduces a new extremal analysis for extending $P_2$-packings, develops a kernelization method with size bound $7k$, and presents an improved algorithm with runtime $ ilde{O}(2.482^{3k})$.

Findings

Vertices reused in packing extensions are at least 2.5 times the current packing size.

Kernelization reduces problem size to at most 7 times the parameter $k$.

New algorithm solves $P_2$-packing problem faster than previous methods.

Abstract

}We study (vertex-disjoint) $P_2$-packings in graphs under a parameterized perspective. Starting from a maximal $P_2$-packing $\p$ of size $j$ we use extremal arguments for determining how many vertices of $\p$ appear in some $P_2$-packing of size $(j+1)$. We basically can 'reuse' $2.5j$ vertices. We also present a kernelization algorithm that gives a kernel of size bounded by $7k$. With these two results we build an algorithm which constructs a $P_2$-packing of size $k$ in time $\Oh^*(2.482^{3k})$.