# Temporal Matching

Abstract : A link stream is a sequence of pairs of the form $(t,\{u,v\})$, where $t\in\mathbb N$ represents a time instant and $u\neq v$. Given an integer $\gamma$, the $\gamma$-edge between vertices $u$ and $v$, starting at time $t$, is the set of temporally consecutive edges defined by $\{(t',\{u,v\}) | t' \in [t,t+\gamma-1]\}$. We introduce the notion of temporal matching of a link stream to be an independent $\gamma$-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 $\gamma>1$. 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 behaviour 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.
Document type :
Journal articles

https://hal.sorbonne-universite.fr/hal-02075865
Contributor : Binh-Minh Bui-Xuan <>
Submitted on : Thursday, March 21, 2019 - 4:25:56 PM
Last modification on : Tuesday, January 14, 2020 - 11:23:55 AM

### Citation

Julien Baste, Binh-Minh Bui-Xuan, Antoine Roux. Temporal Matching. Theoretical Computer Science, Elsevier, In press, ⟨10.1016/j.tcs.2019.03.026⟩. ⟨hal-02075865⟩

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