TY - GEN
T1 - Lossless Online Rounding for Online Bipartite Matching (Despite its Impossibility)
AU - Buchbinder, Niv
AU - Naor, Joseph
AU - Wajc, David
N1 - Publisher Copyright:
Copyright © 2023 by SIAM.
PY - 2023
Y1 - 2023
N2 - For numerous online bipartite matching problems, such as edge-weighted matching and matching under two-sided vertex arrivals, the state-of-the-art fractional algorithms outperform their randomized integral counterparts. This gap is surprising, given that the bipartite fractional matching polytope is integral, and so lossless rounding is possible. This gap was explained by Devanur et al. (SODA'13), who showed that online lossless rounding is impossible. Despite the above, we initiate the study of lossless online rounding for online bipartite matching problems. Our key observation is that while lossless online rounding is impossible in general, randomized algorithms induce fractional algorithms of the same competitive ratio which by definition are losslessly roundable online. This motivates the addition of constraints that decrease the “online integrality gap”, thus allowing for lossless online rounding. We characterize a set of non-convex constraints which allow for such lossless online rounding, and better competitive ratios than yielded by deterministic algorithms. As applications of our lossless online rounding approach, we obtain two results of independent interest: (i) a doubly-exponential improvement, and a sharp threshold for the amount of randomness (or advice) needed to outperform deterministic online (vertex-weighted) bipartite matching algorithms, and (ii) an optimal semi-OCS, matching a recent result of Gao et al. (FOCS'21) answering a question of Fahrbach et al. (FOCS'20).
AB - For numerous online bipartite matching problems, such as edge-weighted matching and matching under two-sided vertex arrivals, the state-of-the-art fractional algorithms outperform their randomized integral counterparts. This gap is surprising, given that the bipartite fractional matching polytope is integral, and so lossless rounding is possible. This gap was explained by Devanur et al. (SODA'13), who showed that online lossless rounding is impossible. Despite the above, we initiate the study of lossless online rounding for online bipartite matching problems. Our key observation is that while lossless online rounding is impossible in general, randomized algorithms induce fractional algorithms of the same competitive ratio which by definition are losslessly roundable online. This motivates the addition of constraints that decrease the “online integrality gap”, thus allowing for lossless online rounding. We characterize a set of non-convex constraints which allow for such lossless online rounding, and better competitive ratios than yielded by deterministic algorithms. As applications of our lossless online rounding approach, we obtain two results of independent interest: (i) a doubly-exponential improvement, and a sharp threshold for the amount of randomness (or advice) needed to outperform deterministic online (vertex-weighted) bipartite matching algorithms, and (ii) an optimal semi-OCS, matching a recent result of Gao et al. (FOCS'21) answering a question of Fahrbach et al. (FOCS'20).
UR - http://www.scopus.com/inward/record.url?scp=85160867122&partnerID=8YFLogxK
U2 - 10.1137/1.9781611977554.ch78
DO - 10.1137/1.9781611977554.ch78
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AN - SCOPUS:85160867122
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 2030
EP - 2068
BT - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
PB - Association for Computing Machinery
T2 - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
Y2 - 22 January 2023 through 25 January 2023
ER -