Maximum 0-1 Timed Matching on Temporal Graphs

TL;DR

This paper introduces the concept of maximum 0-1 timed matching in temporal graphs, analyzing its computational complexity and proposing algorithms and approximations for different classes of such graphs.

Contribution

It defines the maximum 0-1 timed matching problem, proves NP-Completeness in various cases, and develops efficient algorithms and approximation strategies.

Findings

NP-Complete for rooted temporal trees with multiple intervals per edge

O(n log n) algorithm for rooted temporal trees with single interval edges

No good approximation for multiple interval edges unless NP=ZPP

Abstract

Temporal graphs are graphs where the topology and/or other properties of the graph change with time. They have been used to model applications with temporal information in various domains. Problems on static graphs become more challenging to solve in temporal graphs because of dynamically changing topology, and many recent works have explored graph problems on temporal graphs. In this paper, we define a type of matching called {\em 0-1 timed matching} for temporal graphs, and investigate the problem of finding a {\em maximum 0-1 timed matching} for different classes of temporal graphs. We first prove that the problem is NP-Complete for rooted temporal trees when each edge is associated with two or more time intervals. We then propose an $O(n \log n)$ time algorithm for the problem on a rooted temporal tree with $n$ nodes when each edge is associated with exactly one time interval. The problem is then shown to be NP-Complete also for bipartite temporal graphs even when each edge is associated with a single time interval and degree of each node is bounded by a constant $k \geq 3$. We next investigate approximation algorithms for the problem for temporal graphs where each edge is associated with more than one time intervals. It is first shown that there is no $\frac{1}{n^{1-\epsilon}}$-factor approximation algorithm for the problem for any $\epsilon > 0$ even on a rooted temporal tree with $n$ nodes unless NP = ZPP. We then present a $\frac{5}{2\mathcal{N}^* + 3}$-factor approximation algorithm for the problem for general temporal graphs where $\mathcal{N^*}$ is the average number of edges overlapping in time with each edge in the temporal graph. The same algorithm is also a constant-factor approximation algorithm for degree bounded temporal graphs.