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We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm.
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On the cover time of dense graphs
We consider arbitrary graphs $$G$$ with $$n$$ vertices and minimum degree at least $$\delta n$$ where $$\delta>0$$ is constant.\\ (a) If the conductance of $$G$$ is sufficiently large then we obtain an asymptotic expression for the cover time $$C_G$$ of $$G$$ as the solution to an explicit transcendental equation.\\ (b) If the conductance is not large enough to apply (a), but the mixing time of a random walk on $$G$$ is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic estimate via a decomposition into a bounded number of dense subgraphs with high conductance. \\ (c) If $$G$$ fits neither (a) nor (b) then we give a deterministic asymptotic (2+o(1))-approximation of $$C_G$$.
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- Award ID(s):
- 1661063
- PAR ID:
- 10158498
- Date Published:
- Journal Name:
- SIAM journal on discrete mathematics
- Volume:
- 33
- ISSN:
- 0895-4801
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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