Maximum 0-1 timed matching on temporal graphs

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 0-1 timed matching for temporal graphs, and investigate the problem of finding a maximum 0-1 timed matching for different classes of temporal graphs. We assume that only the edge set of the temporal graph changes with time. Thus, a temporal graph can be represented by associating each edge with one or more non-overlapping discrete time intervals for which that edge exists. 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 vertices 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 vertex is bounded by a constant k ≥ 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 1 n 1 − ϵ -factor approximation algorithm for the problem for any ϵ > 0 even on a rooted temporal tree with n vertices unless NP = ZPP. We then present a 5 2 N ∗ + 3 -factor approximation algorithm for the problem for general temporal graphs where 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 temporal graphs with degree of each vertex bounded by a constant.