DeepAI
Log In Sign Up

Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs

09/03/2020
by   Jan van den Brand, et al.
0

We present an Õ(m+n^1.5)-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on m-edge, n-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. m = Ω(n^1.5), our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic O(m√(n))-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and Õ(n^ω)-time algorithms [Ibarra-Moran 1981] (where currently ω≈ 2.373). On sparser graphs, i.e. when m = n^9/8 + δ for any constant δ>0, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an Õ(m^4/3+o(1)) runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand et al.] and our new IPMs. Combining this general machinery yields a simpler Õ(n √(m)) time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art Õ(m+n^1.5) time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).

READ FULL TEXT

page 1

page 2

page 3

page 4

03/25/2019

A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings

We present a weighted approach to compute a maximum cardinality matching...
04/27/2022

Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching

Box-simplex games are a family of bilinear minimax objectives which enca...
07/03/2022

Decremental Matching in General Graphs

We consider the problem of maintaining an approximate maximum integral m...
01/14/2021

Minimum Cost Flows, MDPs, and ℓ_1-Regression in Nearly Linear Time for Dense Instances

In this paper we provide new randomized algorithms with improved runtime...
07/12/2018

A Faster Algorithm for Minimum-Cost Bipartite Matching in Minor-Free Graphs

We give an Õ(n^7/5 (nC))-time algorithm to compute a minimum-cost maximu...
05/04/2021

Deterministic Rounding of Dynamic Fractional Matchings

We present a framework for deterministically rounding a dynamic fraction...
03/12/2021

Computing Zigzag Persistence on Graphs in Near-Linear Time

Graphs model real-world circumstances in many applications where they ma...