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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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Removal lemmas and approximate homomorphisms
Abstract We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma , states that for each $$\epsilon>0$$ there exists M such that every triangle-free graph G has an $$\epsilon$$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $$\epsilon$$ -approximate homomorphism is a map $$V(G) \to V(F)$$ where all but at most $$\epsilon \left\lvert{V(G)}\right\rvert^2$$ edges of G are mapped to edges of F ). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $$\epsilon^{-1}$$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.
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- PAR ID:
- 10407727
- Date Published:
- Journal Name:
- Combinatorics, Probability and Computing
- Volume:
- 31
- Issue:
- 4
- ISSN:
- 0963-5483
- Page Range / eLocation ID:
- 721 to 736
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0963548321000572
