A Faster Combinatorial Algorithm for Maximum Bipartite Matching
CoRR(2023)
摘要
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 (1973) shows that maximum
bipartite matching can be solved in $O(m \sqrt{n})$ time on a graph with $n$
vertices and $m$ edges. For the case of very dense graphs, a fast matrix
multiplication based approach gives a running time of $O(n^{2.371})$. These
results represented the fastest known algorithms for the problem until 2013,
when Madry introduced a new approach based on continuous techniques achieving
much faster runtime in sparse graphs. This line of research has culminated in a
spectacular recent breakthrough due to Chen et al. (2022) that gives an
$m^{1+o(1)}$ time algorithm for maximum bipartite matching (and more generally,
for min cost flows).
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 runs in $\tilde{O}(m^{1/3}n^{5/3})$ time, and
hence outperforms both Hopcroft-Karp and the fast matrix multiplication based
algorithms on moderately dense graphs. Using a standard reduction, we also
obtain an $\tilde{O}(m^{1/3}n^{5/3})$ time deterministic algorithm for maximum
vertex-capacitated $s$-$t$ flow in directed graphs when all vertex capacities
are identical.
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