TY - CHAP
T1 - Online Submodular Maximization: Beating 1/2 Made Simple
AU - Buchbinder, Niv
AU - Feldman, Moran
AU - Filmus, Yuval
AU - Garg, Mohit
N1 - Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
PY - 2019
Y1 - 2019
N2 - The problem of Submodular Welfare Maximization (SWM) captures an important subclass of combinatorial auctions and has been studied extensively from both computational and economic perspectives. In particular, it has been studied in a natural online setting in which items arrive one-by-one and should be allocated irrevocably upon arrival. In this setting, it is well known that the greedy algorithm achieves a competitive ratio of, and recently Kapralov et al. [22] showed that this ratio is optimal for the problem. Surprisingly, despite this impossibility result, Korula et al. [25] were able to show that the same algorithm is 0.5052-competitive when the items arrive in a uniformly random order, but unfortunately, their proof is very long and involved. In this work, we present an (arguably) much simpler analysis that provides a slightly better guarantee of 0.5096-competitiveness for the greedy algorithm in the random-arrival model. Moreover, this analysis applies also to a generalization of online SWM in which the sets defining a (simple) partition matroid arrive online in a uniformly random order, and we would like to maximize a monotone submodular function subject to this matroid. Furthermore, for this more general problem, we prove an upper bound of 0.576 on the competitive ratio of the greedy algorithm, ruling out the possibility that the competitiveness of this natural algorithm matches the optimal offline approximation ratio of.
AB - The problem of Submodular Welfare Maximization (SWM) captures an important subclass of combinatorial auctions and has been studied extensively from both computational and economic perspectives. In particular, it has been studied in a natural online setting in which items arrive one-by-one and should be allocated irrevocably upon arrival. In this setting, it is well known that the greedy algorithm achieves a competitive ratio of, and recently Kapralov et al. [22] showed that this ratio is optimal for the problem. Surprisingly, despite this impossibility result, Korula et al. [25] were able to show that the same algorithm is 0.5052-competitive when the items arrive in a uniformly random order, but unfortunately, their proof is very long and involved. In this work, we present an (arguably) much simpler analysis that provides a slightly better guarantee of 0.5096-competitiveness for the greedy algorithm in the random-arrival model. Moreover, this analysis applies also to a generalization of online SWM in which the sets defining a (simple) partition matroid arrive online in a uniformly random order, and we would like to maximize a monotone submodular function subject to this matroid. Furthermore, for this more general problem, we prove an upper bound of 0.576 on the competitive ratio of the greedy algorithm, ruling out the possibility that the competitiveness of this natural algorithm matches the optimal offline approximation ratio of.
KW - Submodular optimization
KW - Online auctions
KW - Greedy algorithms
UR - http://www.scopus.com/inward/record.url?scp=85065872111&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-17953-3_8
DO - 10.1007/978-3-030-17953-3_8
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SN - 978-3-030-17952-6
SN - 9783030179526
VL - 11480
T3 - Lecture Notes in Computer Science
SP - 101
EP - 114
BT - INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION, IPCO 2019
A2 - Lodi, Andrea
A2 - Nagarajan, Viswanath
T2 - 20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019
Y2 - 22 May 2019 through 24 May 2019
ER -