A Faster Scaling Algorithm for Minimizing Submodular Functions

Combinatorial strongly polynomial algorithms for minimizing submodular functions have been developed by Iwata, Fleischer, and Fujishige (IFF) and by Schrijver. The IFF algorithm employs a scaling scheme for submodular functions, whereas Schrijver's algorithm exploits distance labeling. This paper combines these two techniques to yield a faster combinatorial algorithm for submodular function minimization. The resulting algorithm improves over the previously best known bound by an almost linear factor in the size of the underlying ground set.

[1]  William H. Cunningham.  On submodular function minimization , 1985, Comb..

[2]  Satoru Iwata,et al.  A capacity scaling algorithm for convex cost submodular flows , 1996, SODA '96.

[3]  William H. Cunningham,et al.  Testing membership in matroid polyhedra , 1984, J. Comb. Theory, Ser. B.

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

[5]  L. Shapley.  Cores of convex games , 1971 .

[6]  László Lovász,et al.  Submodular functions and convexity , 1982, ISMP.

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

[8]  Satoru Iwata,et al.  Improved algorithms for submodular function minimization and submodular flow , 2000, STOC '00.

[9]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[10]  Satoru Iwata,et al.  A faster capacity scaling algorithm for minimum cost submodular flow , 2002, Math. Program..

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

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

[13]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

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

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