A strongly polynomial algorithm for line search in submodular polyhedra

A submodular polyhedron is a polyhedron associated with a submodular function. This paper presents a strongly polynomial time algorithm for line search in submodular polyhedra with the aid of a fully combinatorial algorithm for submodular function minimization. The algorithm is based on the parametric search method proposed by Megiddo.

[1]  David Hartvigsen,et al.  A strongly polynomial time algorithm for a constrained submodular optimization problem , 2001, Discret. Appl. Math..

[2]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[3]  Tomasz Radzik.  Fractional Combinatorial Optimization , 1998 .

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

[5]  Tomasz Radzik,et al.  Parametric Flows, Weighted Means of Cuts, and Fractional Combinatorial Optimization , 1993 .

[6]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[7]  Kazuo Murota,et al.  Discrete convex analysis , 1998, Math. Program..

[8]  Jens Vygen,et al.  A note on Schrijver's submodular function minimization algorithm , 2003, J. Comb. Theory, Ser. B.

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

[10]  William H. Cunningham,et al.  Minimum cuts, modular functions, and matroid polyhedra , 1985, Networks.

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

[12]  Satoru Iwata,et al.  A push-relabel framework for submodular function minimization and applications to parametric optimization , 2003, Discret. Appl. Math..

[13]  Nimrod Megiddo.  Combinatorial Optimization with Rational Objective Functions , 1979, Math. Oper. Res..

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

[15]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[16]  Alexander Schrijver,et al.  A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time , 2000, J. Comb. Theory B.

[17]  R. E. Bixby,et al.  The Partial Order of a Polymatroid Extreme Point , 1985, Math. Oper. Res..

[18]  P. Pardalos.  Complexity in numerical optimization , 1993 .

[19]  David Hartvigsen.  A Submodular Optimization Problem with Side Constraints , 1998, Math. Oper. Res..

[20]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[21]  J. Picard.  Maximal Closure of a Graph and Applications to Combinatorial Problems , 1976 .

[22]  D. Gale.  A theorem on flows in networks , 1957 .

[23]  C. SIAMJ..  A FASTER SCALING ALGORITHM FOR MINIMIZING SUBMODULAR FUNCTIONS∗ , 2001 .

[24]  Satoru Iwata,et al.  A fully combinatorial algorithm for submodular function minimization , 2001, SODA '02.