Abstract
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 edgeweighted 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 (MinMPM) does not admit an n^{1epsilon}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 (MetricMinMPM) we are promised that edge weights in each time step satisfy the triangle inequality. We show that this problem admits a 3approximation when the number of time steps is 2 or 3. On the other hand, we show that even the metric case is APXhard already for 2 time steps. We then consider the complementary maximization version of the problem, Maximum Multistage Perfect Matching problem (MaxMPM), where we seek to maximize the total profit of all selected edges plus the total number of nonexchanged edges. We show that MaxMPM is also APXhard, but admits a constant factor approximation algorithm for any number of time steps.
BibTeX  Entry
@InProceedings{bampis_et_al:LIPIcs:2018:8833,
author = {Evripidis Bampis and Bruno Escoffier and Michael Lampis and Vangelis Th. Paschos},
title = {{Multistage Matchings}},
booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)},
pages = {7:17:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770682},
ISSN = {18688969},
year = {2018},
volume = {101},
editor = {David Eppstein},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8833},
URN = {urn:nbn:de:0030drops88338},
doi = {10.4230/LIPIcs.SWAT.2018.7},
annote = {Keywords: Perfect Matching, Temporal Optimization, Multistage Optimization}
}
Keywords: 

Perfect Matching, Temporal Optimization, Multistage Optimization 
Collection: 

16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018) 
Issue Date: 

2018 
Date of publication: 

04.06.2018 