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Efficient algorithms for maximum weight matchings in general graphs with small edge weights
 in: Proceedings 23rd ACMSIAM Symposium on Discrete Algorithms (SODA
"... Let G = (V, E) be a graph with positive integral edge weights. Our problem is to find a matching of maximum weight in G. We present a simple iterative algorithm for this problem that uses a maximum cardinality matching algorithm as a subroutine. Using the current fastest maximum cardinality matching ..."
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Let G = (V, E) be a graph with positive integral edge weights. Our problem is to find a matching of maximum weight in G. We present a simple iterative algorithm for this problem that uses a maximum cardinality matching algorithm as a subroutine. Using the current fastest maximum cardinality matching algorithms, we solve the maximum weight matching problem in O(W √ nm logn(n 2 /m)) time, or in O(W n ω) time with high probability, where n = V , m = E, W is the largest edge weight, and ω < 2.376 is the exponent of matrix multiplication. In relatively dense graphs, our algorithm performs better than all existing algorithms with W = o(log 1.5 n). Our technique hinges on exploiting Edmonds ’ matching polytope and its dual. 1
A Simple Reduction from Maximum Weight Matching to Maximum Cardinality Matching ✩
"... Let mcm(m, n) and mwm(m, n, N) be the complexities of computing a maximum cardinality matching and a maximum weight matching, and let mcmbi, mwmbi be their counterparts for bipartite graphs, where m, n, and N are the edge count, vertex count, and maximum integer edge weight. Kao, Lam, Sung, and Ting ..."
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Let mcm(m, n) and mwm(m, n, N) be the complexities of computing a maximum cardinality matching and a maximum weight matching, and let mcmbi, mwmbi be their counterparts for bipartite graphs, where m, n, and N are the edge count, vertex count, and maximum integer edge weight. Kao, Lam, Sung, and Ting [1] gave a general reduction showing mwmbi(m, n, N) = O(N ·mcmbi(m, n)) and Huang and Kavitha [2] recently proved the analogous result for general graphs, that mwm(m, n, N) = O(N · mcm(m, n)). We show that Gabow’s mwmbi and mwm algorithms from 1983 and 1985 [3, 4] can be modified to replicate the results of Kao et al. and Huang and Kavitha, but with dramatically simpler proofs. We also show that our reduction leads to new bounds on the complexity of mwm on sparse graph classes, e.g., (bipartite) planar graphs, bounded genus graphs, and Hminorfree graphs. Keywords: Graph algorithms, maximum matching
Imperfect Matchings in Bipartite Graphs
"... assignment problem; imperfect matching; minimumcostmatching; unbalanced bipartite graph; weightscaling algorithm Call a bipartite graph G = (X; Y;E) balanced when X  = Y . Given a balanced bipartite graph G with edge costs, the assignment problem asks for a perfect matching in G of minimum to ..."
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assignment problem; imperfect matching; minimumcostmatching; unbalanced bipartite graph; weightscaling algorithm Call a bipartite graph G = (X; Y;E) balanced when X  = Y . Given a balanced bipartite graph G with edge costs, the assignment problem asks for a perfect matching in G of minimum total cost. The Hungarian Method can solve assignment problems in time O(mn+n 2 log n), where n: = X  = Y  and m: = E. If the edge weights are integers bounded in magnitude by C> 1, then algorithms using weight scaling, such as that of Gabow and Tarjan, can lower the time to ( log(nC)). There are important applications in which G is unbalanced, with X  ≠ Y , and we require a mincost matching in G of size r: = min(X, Y ) or, more generally, of some specified size s ≤ r. The Hungarian Method extends easily to find such a matching in time O(ms+s 2 log r), but weightscaling algorithms do not extend so easily. We introduce new machinery that allows us to find such a matching in time ( log(nC)) via weight scaling. Our results also provide insight into the design space of efficient weightscaling matching algorithms. These ideas are presented in greater depth in HPL201240 [17].
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science A WeightScaling Algorithm for MinCost Imperfect Matchings in Bipartite Graphs
"... Abstract — Call a bipartite graph G =(X, Y; E) balanced when X  = Y . Given a balanced bipartite graph G with edge costs, the assignment problem asks for a perfect matching in G of minimum total cost. The Hungarian Method can solve assignment problems in time O(mn+n 2 log n), where n:= X  = ..."
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Abstract — Call a bipartite graph G =(X, Y; E) balanced when X  = Y . Given a balanced bipartite graph G with edge costs, the assignment problem asks for a perfect matching in G of minimum total cost. The Hungarian Method can solve assignment problems in time O(mn+n 2 log n), where n:= X  = Y  and m: = E. If the edge weights are integers bounded in magnitude by C>1, then algorithms using weight scaling, such as that of Gabow and Tarjan, can lower the time to O(m √ n log(nC)). There are important applications in which G is unbalanced, with X  = Y , and we require a mincost matching of size r: = min(X, Y ) or, more generally, of some specified size s ≤ r. The Hungarian Method extends easily to find such a matching in time O(ms + s 2 log r), but weightscaling algorithms do not extend so easily. We introduce new machinery to find such a matching in time O(m √ s log(sC)) via weight scaling. Our results provide some insight into the design space of efficient weightscaling matching algorithms. 1.
Fair Matchings and Related Problems ∗
"... Let G = (A ∪ B, E) be a bipartite graph, where every vertex ranks its neighbors in an order of preference (with ties allowed) and let r be the worst rank used. A matching M is fair in G if it has maximum cardinality, subject to this, M matches the minimum number of vertices to rank r neighbors, subj ..."
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Let G = (A ∪ B, E) be a bipartite graph, where every vertex ranks its neighbors in an order of preference (with ties allowed) and let r be the worst rank used. A matching M is fair in G if it has maximum cardinality, subject to this, M matches the minimum number of vertices to rank r neighbors, subject to that, M matches the minimum number of vertices to rank (r −1) neighbors, and so on. We show an efficient combinatorial algorithm based on LP duality to compute a fair matching in G. We also show a scaling based algorithm for the fair bmatching problem. Our two algorithms can be extended to solve other profilebased matching problems. In designing our combinatorial algorithm, we show how to solve a generalized version of the minimum weighted vertex cover problem in bipartite graphs, using a singlesource shortest paths computation—this can be of independent interest.