A Modified Decomposition Algorithm for Maximum Weight Bipartite Matching and Its Experimental Evaluation

Let G be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of G in O(|V |W /k(|V |, W /N))-time by employing an algorithm designed by Feder and Motwani as a subroutine, where |V | and N denote the number of nodes and the maximum edge weight of G, respectively and k(x, y) = log x/ log(x 2 /y). The parameter W is smaller than the total edge weight W, essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in any decomposition step of the algorithm. In best case W = O(|E|) where |E| be the number of edges of G and in worst case W = W, that is, |E| ≤ W ≤ W. In addition, we talk about a scaling property of the algorithm and research a better bound of the parameter W. An experimental evaluation on randomly generated data shows that the proposed improvement is significant in general.

[1]  Robert E. Tarjan,et al.  Faster Scaling Algorithms for Network Problems , 1989, SIAM J. Comput..

[2]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[3]  Harold N. Gabow,et al.  Scaling algorithms for network problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[4]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[5]  Moshe Lewenstein,et al.  Approximate Parameterized Matching , 2004, ESA.

[6]  Ran Duan,et al.  Approximating Maximum Weight Matching in Near-Linear Time , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[7]  Kurt Mehlhorn,et al.  Can A Maximum Flow be Computed on o(nm) Time? , 1990, ICALP.

[8]  H. Kuhn The Hungarian method for the assignment problem , 1955 .

[9]  M. Iri A NEW METHOD OF SOLVING TRANSPORTATION· NETWORK PROBLEMS , 1960 .

[10]  R. Jackson Inequalities , 2007, Algebra for Parents.

[11]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[12]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[13]  Piotr Sankowski,et al.  Maximum weight bipartite matching in matrix multiplication time , 2009, Theor. Comput. Sci..

[14]  Rajeev Motwani,et al.  Clique partitions, graph compression and speeding-up algorithms , 1991, STOC '91.

[15]  J. Munkres ALGORITHMS FOR THE ASSIGNMENT AND TRANSIORTATION tROBLEMS* , 1957 .

[16]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[17]  Kurt Mehlhorn,et al.  Computing a Maximum Cardinality Matching in a Bipartite Graph in Time O(^1.5 sqrt m/log n) , 1991, Inf. Process. Lett..

[18]  Ming-Yang Kao,et al.  A Decomposition Theorem for Maximum Weight Bipartite Matchings with Applications to Evolutionary Trees , 1999, ESA.

[19]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[20]  L FredmanMichael,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987 .

[21]  Kalpesh Kapoor,et al.  Fine-Tuning Decomposition Theorem for Maximum Weight Bipartite Matching , 2014, TAMC.