Abstract
The Maximum Carpool Matching problem is a star packing problem in directed graphs. Formally, given a directed graph G = (V, A), a capacity function c: V > N, and a weight function w: A > R^+, a carpool matching is a subset of arcs, M subseteq A, such that every v in V satisfies: (i) d^{in}_M(v) cdot d^{out}_M(v) = 0, (ii) d^{in}_M(v) <= c(v), and (iii) d^{out}_M(v) <= 1. A vertex v for which d^{out}_M(v) = 1 is a passenger, and a vertex for which d^{out}_M(v) = 0 is a driver who has d^{in}_M(v) passengers. In the Maximum Carpool Matching problem the goal is to find a carpool matching M of maximum total weight. The problem arises when designing an online carpool service, such as Zimride, which tries to connect between users based on a similarity function. The problem is known to be NPhard, even in the unweighted and uncapacitated case. The Maximum Group Carpool Matching problem, is an extension of Maximum Carpool Matching where each vertex represents an unsplittable group of passengers. Formally, each vertex u in V has a size s(u) in N, and the constraint d^{in}_M(v) <= c(v) is replaced with sum_{u:(u,v) in M} s(u) <= c(v).
We show that Maximum Carpool Matching can be formulated as an unconstrained submodular maximization problem, thus it admits a 1/2approximation algorithm. We show that the same formulation does not work for Maximum Group Carpool Matching, nevertheless, we present a local search (1/2  epsilon)approximation algorithm for Maximum Group Carpool Matching. For the unweighted variant of both problems when the maximum possible capacity, c_{max}, is bounded by a constant, we provide a local search (1/2 + 1/{2c_{max}}  epsilon)approximation algorithm. We also show that the problem is APXhard, even if the maximum degree and c_{max} are at most 3.
BibTeX  Entry
@InProceedings{kutiel_et_al:LIPIcs:2017:7821,
author = {Gilad Kutiel and Dror Rawitz},
title = {{Local Search Algorithms for Maximum Carpool Matching}},
booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)},
pages = {55:155:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770491},
ISSN = {18688969},
year = {2017},
volume = {87},
editor = {Kirk Pruhs and Christian Sohler},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7821},
URN = {urn:nbn:de:0030drops78210},
doi = {10.4230/LIPIcs.ESA.2017.55},
annote = {Keywords: approximation algorithms, local search, star packing, submodular maximization}
}
Keywords: 

approximation algorithms, local search, star packing, submodular maximization 
Collection: 

25th Annual European Symposium on Algorithms (ESA 2017) 
Issue Date: 

2017 
Date of publication: 

01.09.2017 