Online Bipartite Matching with Amortized $O(\log^2 n)$ Replacements

TL;DR

This paper presents an efficient online bipartite matching algorithm that guarantees amortized O(log^2 n) replacements per insertion, nearly matching the lower bound, and extends to related problems like capacitated assignment and load minimization.

Contribution

It introduces the SAP protocol with a polylogarithmic replacement bound, the first such analysis for any strategy, improving over previous O(√n) bounds, and applies to related optimization problems.

Findings

SAP protocol uses at most O(log^2 n) replacements per insertion.

Achieves near-optimal polylogarithmic bounds for online matching.

Extends to capacitated assignment and load balancing problems.

Abstract

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 $O(\log^2 n)$ 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 \emph{any} replacement strategy, almost matching the $\Omega(\log n)$ lower bound. The previous best strategy known achieved amortized $O(\sqrt{n})$ replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial $O(n)$ bound was known except in special cases. Our analysis immediately implies the same upper bound of $O(\log^2 n)$ 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 $O( \min\{L \log^2 n , \sqrt{n}\log n\})$ reassignments. We also show that this is close to tight because $\Omega(\min\{L, \sqrt{n}\})$ reassignments can be necessary.