WÃJÇ

Lossless Online Rounding for Online Bipartite Matching (Despite its Impossibility)

Niv Buchbinder, Joseph (Seffi) Naor, David Wajc

C25SODA 2023

Abstract

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).

Dynamic Matching with Better-than-2 Approximation in Polylogarithmic Update Time

Sayan Bhattacharya, Peter Kiss, Thatchaphol Saranurak, David Wajc

C26SODA 2023 Best paper award.

Abstract

We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than 2. Specifically, we obtain a 1+1/√2+ϵ≈1.707+ϵ approximation in bipartite graphs and a 1.973+ϵ approximation in general graphs. We thus answer in the affirmative the major open question first posed in the influential work of Onak and Rubinfeld (STOC'10) and repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms also work against an adaptive adversary and guarantee worst-case polylog update time, both w.h.p.

Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS'21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.

Improved Online Contention Resolution for Matchings and Applications to the Gig Economy

Niv Buchbinder, Joseph (Seffi) Naor, David Wajc

C25SODA 2023

Abstract

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).

Beating the Folklore Algorithm for Dynamic Matching

Niv Buchbinder, Joseph (Seffi) Naor, David Wajc

C25SODA 2023

Abstract

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).

The Stationary Prophet Inequality Problem

Niv Buchbinder, Joseph (Seffi) Naor, David Wajc

C25SODA 2023

Abstract

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).

Lossless Online Rounding for Online Bipartite Matching (Despite its Impossibility)

Niv Buchbinder, Joseph (Seffi) Naor, David Wajc

C25SODA 2023

Abstract

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).