TY - GEN
T1 - Improved bounds for online preemptive matching
AU - Epstein, Leah
AU - Levin, Asaf
AU - Segev, Danny
AU - Weimann, Oren
PY - 2013
Y1 - 2013
N2 - When designing a preemptive online algorithm for the maximum matching problem, we wish to maintain a valid matching M while edges of the underlying graph are presented one after the other. When presented with an edge e, the algorithm should decide whether to augment the matching M by adding e (in which case e may be removed later on) or to keep M in its current form without adding e (in which case e is lost for good). The objective is to eventually hold a matching M with maximum weight. The main contribution of this paper is to establish new lower and upper bounds on the competitive ratio achievable by preemptive online algorithms: We provide a lower bound of 1 + ln 2 ≈ 1.693 on the competitive ratio of any randomized algorithm for the maximum cardinality matching problem, thus improving on the currently best known bound of e/(e - 1) ≈ 1.581 due to Karp, Vazirani, and Vazirani [STOC'90]. We devise a randomized algorithm that achieves an expected competitive ratio of 5.356 for maximum weight matching. This finding demonstrates the power of randomization in this context, showing how to beat the tight bound of 3+2p2 ≈ 5.828 for deterministic algorithms, obtained by combining the 5.828 upper bound of McGregor [APPROX'05] and the recent 5.828 lower bound of Varadaraja [ICALP'11].
AB - When designing a preemptive online algorithm for the maximum matching problem, we wish to maintain a valid matching M while edges of the underlying graph are presented one after the other. When presented with an edge e, the algorithm should decide whether to augment the matching M by adding e (in which case e may be removed later on) or to keep M in its current form without adding e (in which case e is lost for good). The objective is to eventually hold a matching M with maximum weight. The main contribution of this paper is to establish new lower and upper bounds on the competitive ratio achievable by preemptive online algorithms: We provide a lower bound of 1 + ln 2 ≈ 1.693 on the competitive ratio of any randomized algorithm for the maximum cardinality matching problem, thus improving on the currently best known bound of e/(e - 1) ≈ 1.581 due to Karp, Vazirani, and Vazirani [STOC'90]. We devise a randomized algorithm that achieves an expected competitive ratio of 5.356 for maximum weight matching. This finding demonstrates the power of randomization in this context, showing how to beat the tight bound of 3+2p2 ≈ 5.828 for deterministic algorithms, obtained by combining the 5.828 upper bound of McGregor [APPROX'05] and the recent 5.828 lower bound of Varadaraja [ICALP'11].
KW - Lower bound
KW - Matching
KW - Online algorithms
UR - http://www.scopus.com/inward/record.url?scp=84892584446&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.STACS.2013.389
DO - 10.4230/LIPIcs.STACS.2013.389
M3 - Conference contribution
AN - SCOPUS:84892584446
SN - 9783939897507
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 389
EP - 399
BT - 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013
T2 - 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013
Y2 - 27 February 2013 through 2 March 2013
ER -