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Multistage Matchings

Bampis, Evripidis; Escoffier, Bruno; Lampis, Michael; Paschos, Vangelis (2018), Multistage Matchings, in Eppstein, David, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018), Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, p. 7:1--7:13. 10.4230/LIPIcs.SWAT.2018.7

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LIPIcs-SWAT-2018-7.pdf (421.7Kb)
Type
Communication / Conférence
Date
2018
Conference title
16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)
Conference date
2018-06
Conference city
Malmo
Conference country
Sweden
Book title
16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)
Book author
Eppstein, David
Publisher
Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik
ISBN
978-3-95977-068-2
Pages
7:1--7:13
Publication identifier
10.4230/LIPIcs.SWAT.2018.7
Metadata
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Author(s)
Bampis, Evripidis

Escoffier, Bruno
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Lampis, Michael cc
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Paschos, Vangelis
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Abstract (EN)
We consider a multistage version of the Perfect Matching problem which models the scenario where the costs of edges change over time and we seek to obtain a solution that achieves low total cost, while minimizing the number of changes from one instance to the next. Formally, we are given a sequence of edge-weighted graphs on the same set of vertices V, and are asked to produce a perfect matching in each instance so that the total edge cost plus the transition cost (the cost of exchanging edges), is minimized. This model was introduced by Gupta et al. (ICALP 2014), who posed as an open problem its approximability for bipartite instances. We completely resolve this question by showing that Minimum Multistage Perfect Matching (Min-MPM) does not admit an n^{1-epsilon}-approximation, even on bipartite instances with only two time steps. Motivated by this negative result, we go on to consider two variations of the problem. In Metric Minimum Multistage Perfect Matching problem (Metric-Min-MPM) we are promised that edge weights in each time step satisfy the triangle inequality. We show that this problem admits a 3-approximation when the number of time steps is 2 or 3. On the other hand, we show that even the metric case is APX-hard already for 2 time steps. We then consider the complementary maximization version of the problem, Maximum Multistage Perfect Matching problem (Max-MPM), where we seek to maximize the total profit of all selected edges plus the total number of non-exchanged edges. We show that Max-MPM is also APX-hard, but admits a constant factor approximation algorithm for any number of time steps.
Subjects / Keywords
Matching; Over time Optimization; Multistage Optimization

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