A Faster Parameterized Algorithm for Temporal Matching

TL;DR

This paper introduces a significantly faster algorithm for finding maximum $\Delta$-temporal matchings in temporal graphs, improving the computational efficiency over previous methods by exponential factors in $\Delta$.

Contribution

The authors develop an improved algorithm with a running time of $\Delta^{O( u)} imes | ext{graph}|$, offering exponential speedup over prior algorithms based on $ u$ and $\Delta$.

Findings

New algorithm reduces runtime from $2^{O(\Delta u)}$ to $\Delta^{O( u)}$

Provides practical efficiency improvements for large temporal graphs

Enhances understanding of parameterized complexity in temporal graph problems

Abstract

A temporal graph is a sequence of graphs (called layers) over the same vertex set -- describing a graph topology which is subject to discrete changes over time. A $\Delta$-temporal matching $M$ is a set of time edges $(e,t)$ (an edge $e$ paired up with a point in time $t$) such that for all distinct time edges $(e,t),(e',t') \in M$ we have that $e$ and $e'$ do not share an endpoint, or the time-labels $t$ and $t'$ are at least $\Delta$ time units apart. Mertzios et al. [STACS '20] provided a $2^{O(\Delta\nu)}\cdot |{\mathcal G}|^{O(1)}$-time algorithm to compute the maximum size of a $\Delta$-temporal matching in a temporal graph $\mathcal G$, where $|\mathcal G|$ denotes the size of $\mathcal G$, and $\nu$ is the $\Delta$-vertex cover number of $\mathcal G$. The $\Delta$-vertex cover number is the minimum number $\nu$ such that the classical vertex cover number of the union of any $\Delta$ consecutive layers of the temporal graph is upper-bounded by $\nu$. We show an improved algorithm to compute a $\Delta$-temporal matching of maximum size with a running time of $\Delta^{O(\nu)}\cdot |\mathcal G|$ and hence provide an exponential speedup in terms of $\Delta$.