A Fast Parametric Submodular Intersection Algorithm for Strong Map Sequences

This paper presents a fast algorithm to solve the intersection problem for a pair of nondecreasing and nonincreasing strong map sequences of submodular systems. The worst-case time bound is the same as that of the push/relabel algorithm for a single intersection problem. This extends the Gallo-Grigoriadis-Tarjan GGT method for parametric maximum flow problems and reveals an algorithmic significance of the concept of strong maps for submodular systems.

[1]  K. Murota.  Eigensets and power products of a bimatroid , 1990 .

[2]  Joseph P. S. Kung,et al.  Bimatroids and invariants , 1978 .

[3]  Éva Tardos,et al.  Layered Augmenting Path Algorithms , 1986, Math. Oper. Res..

[4]  Satoru Fujishige,et al.  Principal structures of submodular systems , 1980, Discret. Appl. Math..

[5]  András Frank,et al.  Generalized polymatroids and submodular flows , 1988, Math. Program..

[6]  Dan Gusfield,et al.  Computing the Strength of a Graph , 1991, SIAM J. Comput..

[7]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[8]  Henry H. Crapo.  Structure theory for geometric lattices , 1967 .

[9]  S. Fujishige,et al.  New algorithms for the intersection problem of submodular systems , 1992 .

[10]  Donald M. Topkis,et al.  Minimizing a Submodular Function on a Lattice , 1978, Oper. Res..

[11]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[12]  Robert E. Tarjan,et al.  A Fast Parametric Maximum Flow Algorithm and Applications , 1989, SIAM J. Comput..

[13]  William H. Cunningham,et al.  Optimal attack and reinforcement of a network , 1985, JACM.

[14]  S. Fujishige,et al.  A NOTE ON SUBMODULAR FUNCTIONS ON DISTRIBUTIVE LATTICES , 1983 .

[15]  A. Goldberg,et al.  A new approach to the maximum-flow problem , 1988, JACM.

[16]  Satoru Fujishige,et al.  Lexicographically Optimal Base of a Polymatroid with Respect to a Weight Vector , 1980, Math. Oper. Res..

[17]  Eddie Cheng,et al.  A Faster Algorithm for Computing the Strength of a Network , 1994, Inf. Process. Lett..

[18]  Satoru Fujishige,et al.  Submodular functions and optimization , 1991 .

[19]  S. Fujishige.  ALGORITHMS FOR SOLVING THE INDEPENDENT-FLOW PROBLEMS , 1978 .

[20]  Michel Minoux,et al.  Graphs and Algorithms , 1984 .

[21]  S. Thomas McCormick,et al.  Fast algorithms for parametric scheduling come from extensions to parametric maximum flow , 1996, STOC '96.

[22]  Alexander Schrijver,et al.  Matroids and linking systems , 1979, J. Comb. Theory, Ser. B.

[23]  H. Crapo.  Single-element extensions of matroids , 1965 .

[24]  Harold N. Gabow,et al.  Algorithms for graphic polymatroids and parametric s-Sets , 1995, SODA '95.

[25]  D. Higgs,et al.  Strong maps of geometries , 1968 .

[26]  Nimrod Megiddo,et al.  Optimal flows in networks with multiple sources and sinks , 1974, Math. Program..

[27]  Refael Hassin.  Minimum cost flow with set-constraints , 1982, Networks.

[28]  Kazuo Murota,et al.  Combinatorial dynamical system theory: General framework and controllability criteria , 1989, Discret. Appl. Math..

[29]  Werner Dinkelbach.  On Nonlinear Fractional Programming , 1967 .

[30]  András Frank,et al.  Finding feasible vectors of Edmonds-Giles polyhedra , 1984, J. Comb. Theory, Ser. B.