TY - JOUR

T1 - Online Bipartite Matching with Amortized O(log2 n) Replacements

AU - Bernstein, Aaron

AU - Holm, Jacob

AU - Rotenberg, Eva

PY - 2019

Y1 - 2019

N2 - In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O (log2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n) lower bound. The previous best strategy known achieved amortized O (√n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O (n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O (log2 n) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O (min{L log2 n, √n log n}) reassignments. We also show that this is close to tight, because Ω (min{L, √n}) reassignments can be necessary.

AB - In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O (log2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n) lower bound. The previous best strategy known achieved amortized O (√n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O (n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O (log2 n) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O (min{L log2 n, √n log n}) reassignments. We also show that this is close to tight, because Ω (min{L, √n}) reassignments can be necessary.

KW - Bipartite graphs

KW - Load balancing

KW - Maximum matching

KW - Online algorithms

KW - Shortest augmenting path

U2 - 10.1145/3344999

DO - 10.1145/3344999

M3 - Journal article

AN - SCOPUS:85074506166

VL - 66

JO - Journal of the ACM

JF - Journal of the ACM

SN - 0004-5411

IS - 5

M1 - 37

ER -