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Register a new userBipartite Matching in Nearly-linear Time on Moderately Dense Graphs
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Jan van den Brand
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We present an $tilde O(m+n^{1.5})$-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. $m = Omega(n^{1.5})$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $O(msqrt{n})$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $tilde O(n^omega)$-time algorithms [Ibarra-Moran 1981] (where currently $omegaapprox 2.373$). On sparser graphs, i.e. when $m = n^{9/8 + delta}$ for any constant $delta>0$, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an $tilde O(m^{4/3+o(1)})$ 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 $tilde O(n sqrt{m})$ time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art $tilde O(m+n^{1.5})$ time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).
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In this paper we provide an $tilde{O}(nd+d^{3})$ time randomized algorithm for solving linear programs with $d$ variables and $n$ constraints with high probability. To obtain this result we provide a robust, primal-dual $tilde{O}(sqrt{d})$-iteration interior point method inspired by the methods of Lee and Sidford (2014, 2019) and show how to efficiently implement this method using new data-structures based on heavy-hitters, the Johnson-Lindenstrauss lemma, and inverse maintenance. Interestingly, we obtain this running time without using fast matrix multiplication and consequently, barring a major advance in linear system solving, our running time is near optimal for solving dense linear programs among algorithms that do not use fast matrix multiplication.
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