Temporal matching

Abstract

A link stream is a sequence of pairs of the form (t,{u,v}), where t∈N represents a time instant and u≠v. Given an integer γ, the γ-edge between vertices u and v, starting at time t, is the set of temporally consecutive edges defined by {(t′,{u,v})|t′∈〚t,t+γ−1〛}. We introduce the notion of temporal matching of a link stream to be an independent γ-edge set belonging to the link stream. We show that the problem of computing a temporal matching of maximum size is NP-hard as soon as γ>1. We depict a kernelization algorithm parameterized by the solution size for the problem. As a byproduct we also give a 2-approximation algorithm.Both our 2-approximation and kernelization algorithms are implemented and confronted to link streams collected from real world graph data. We observe that finding temporal matchings is a sensitive question when mining our data from such a perspective as: managing peer-working when any pair of peers X and Y are to collaborate over a period of one month, at an average rate of at least two email exchanges every week. We furthermore design a link stream generating process by mimicking the behavior of a random moving group of particles under natural simulation, and confront our algorithms to these generated instances of link streams. All the implementations are open source.