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The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp [26] shows that maximum bipartite matching can be solved in O(m√n) time on a graph with n vertices and m edges. For the case of very dense graphs, a different approach based on fast matrix multiplication was subsequently developed [27, 39], that achieves a running time of O(n2.371). For the next several decades, these results represented the fastest known algorithms for the problem until in 2013, a ground-breaking work of Madry [36] gave a significantly faster algorithm for sparse graphs. Subsequently, a sequence of works developed increasingly faster algorithms for solving maximum bipartite matching, and more generally directed maximum flow, culminating in a spectacular recent breakthrough [9] that gives an m1+o(1) time algorithm for maximum bipartite matching (and more generally, for min cost flows). These more recent developments collectively represented a departure from earlier combinatorial approaches: they all utilized continuous techniques based on interior-point methods for solving linear programs. This raises a natural question: are continuous techniques essential to obtaining fast algorithms for the bipartite matching problem? Our work makes progress on this question by presenting a new, purely combinatorial algorithm for bipartite matching, that, on moderately dense graphs outperforms both Hopcroft- Karp and the fast matrix multiplication based algorithms. Similar to the classical algorithms for bipartite matching, our approach is based on iteratively augmenting a current matching using augmenting paths in the (directed) residual flow network. A common method for designing fast algorithms for directed flow problems is via the multiplicative weights update (MWU) framework, that effectively reduces the flow problem to decremental single-source shortest paths (SSSP) in directed graphs. Our main observation is that a slight modification of this reduction results in a special case of SSSP that appears significantly easier than general decremental directed SSSP. Our main technical contribution is an efficient algorithm for this special case of SSSP, that outperforms the current state of the art algorithms for general decremental SSSP with adaptive adversary, leading to a deterministic algorithm for bipartite matching, whose running time is Õ(m1/3n5/3). This new algorithm thus starts to outperform the Hopcroft-Karp algorithm in graphs with m = ω(n7/4), and it also outperforms the fast matrix multiplication based algorithms on dense graphs. We believe that this framework for obtaining faster combinatorial algorithms for bipartite matching by exploiting the special properties of the resulting decremental SSSP instances is one of the main conceptual contributions of our work that we hope paves the way for even faster combinatorial algorithms for bipartite matching. Finally, using a standard reduction from the maximum vertex-capacitated s-t flow problem in directed graphs to maximum bipartite matching, we also obtain an O(m1/3n5/3) time deterministic algorithm for maximum vertex-capacitated s-t flow when all vertex capacities are identical.
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(Optimal) Online Bipartite Matching with Degree Information
We propose a model for online graph problems where algorithms are given access to an oracle that predicts (e.g., based on modeling assumptions or on past data) the degrees of nodes in the graph. Within this model, we study the classic problem of online bipartite matching, and a natural greedy matching algorithm called MinPredictedDegree, which uses predictions of the degrees of offline nodes. For the bipartite version of a stochastic graph model due to Chung, Lu, and Vu where the expected values of the offline degrees are known and used as predictions, we show that MinPredictedDegree stochastically dominates any other online algorithm, i.e., it is optimal for graphs drawn from this model. Since the “symmetric” version of the model, where all online nodes are identical, is a special case of the well-studied “known i.i.d. model”, it follows that the competitive ratio of MinPredictedDegree on such inputs is at least 0.7299. For the special case of graphs with power law degree distributions, we show that MinPredictedDegree frequently produces matchings almost as large as the true maximum matching on such graphs. We complement these results with an extensive empirical evaluation showing that MinPredictedDegree compares favorably to state-of-the-art online algorithms for online matching.
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- Award ID(s):
- 2022448
- PAR ID:
- 10430250
- Date Published:
- Journal Name:
- Conference on Neural Information Processing Systems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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