Faster scaling algorithms for general graph matching problems

An algorithm for minimum-cost matching on a general graph with integral edge costs is presented. The algorithm runs in time close to the fastest known bound for maximum-cardinality matching. Specifically, let n, m, and N denote the number of vertices, number of edges, and largest magnitude of a cost, respectively. The best known time bound for maximum-cardinal ity matching M 0( Am). The new algorithm for minimum-cost matching has time bound 0( in a ( m, n )Iog n m log ( nN)). A slight modification of the new algorithm finds a maximum-cardinality matching in 0( fire) time. Other applications of the new algorlthm are given, mchrding an efficient implementa- tion of Christofides' traveling salesman approximation algorithm and efficient solutions to update problems that require the linear programming duals for matching.

[1]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[2]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[3]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

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

[5]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[6]  Harold N. Gabow,et al.  An Efficient Implementation of Edmonds' Algorithm for Maximum Matching on Graphs , 1976, JACM.

[7]  W. Cunningham,et al.  A primal algorithm for optimum matching , 1978 .

[8]  Robert E. Tarjan,et al.  Applications of Path Compression on Balanced Trees , 1979, JACM.

[9]  G. M. Weber.  Sensitivity analysis of optimal matchings , 1981, Networks.

[10]  Silvio Micali,et al.  Priority queues with variable priority and an O(EV log V) algorithm for finding a maximal weighted matching in general graphs , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[11]  Ulrich Derigs,et al.  An analysis of alternative strategies for implementing matching algorithms , 1983, Networks.

[12]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[13]  Harold N. Gabow,et al.  A scaling algorithm for weighted matching on general graphs , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[14]  Éva Tardos,et al.  A strongly polynomial minimum cost circulation algorithm , 1985, Comb..

[15]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for a Special Case of Disjoint Set Union , 1985, J. Comput. Syst. Sci..

[16]  Harold N. Gabow.  Scaling Algorithms for Network Problems , 1985, J. Comput. Syst. Sci..

[17]  Dimitri P. Bertsekas,et al.  Distributed Asynchronous Relaxation Methods for Linear Network Flow Problems , 1987 .

[18]  Robert E. Tarjan,et al.  Algorithms for Two Bottleneck Optimization Problems , 1988, J. Algorithms.

[19]  Pravin M. Vaidya.  Geometry helps in matching , 1988, STOC '88.

[20]  Zvi Galil,et al.  Efficient implementation of graph algorithms using contraction , 1984, JACM.

[21]  Pravin M. Vaidya,et al.  Geometry helps in matching , 1989, STOC '88.

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

[23]  Andrew V. Goldberg,et al.  Finding Minimum-Cost Circulations by Successive Approximation , 1990, Math. Oper. Res..

[24]  Harold N. Gabow,et al.  Data structures for weighted matching and nearest common ancestors with linking , 1990, SODA '90.

[25]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.