A Submodular Function Minimization Algorithm Based on the Minimum-Norm Base ⁄

We consider an application of the minimum-norm-point algorithm to submodular function minimization. Although combinatorial polynomial algorithms for submodular function minimization (SFM) have recently been obtained, there still remain (open) problems of reducing the complexity of the SFM algorithms and of constructing a practically fast SFM algorithms. We show some possible approach to the problems by means of the minimum-norm-point algorithm. Computational results on submodular function minimization reveal that our algorithm outperforms existing polynomial algorithms for SFM.

[1]  Balder Von Hohenbalken,et al.  A finite algorithm to maximize certain pseudoconcave functions on polytopes , 1975, Math. Program..

[2]  Philip Wolfe,et al.  Finding the nearest point in A polytope , 1976, Math. Program..

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

[4]  S. Fujishige.  Submodular systems and related topics , 1984 .

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

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

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

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

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

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

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

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

[13]  S. Thomas McCormick,et al.  Submodular Function Minimization , 2005 .

[14]  Satoko Moriguchi,et al.  Discrete L-/ M-Convex Function Minimization Based on Continuous Relaxation , 2007 .

[15]  Satoru Iwata,et al.  Submodular function minimization , 2007, Math. Program..

[16]  Satoru Iwata,et al.  A simple combinatorial algorithm for submodular function minimization , 2009, SODA.

[17]  James B. Orlin,et al.  A faster strongly polynomial time algorithm for submodular function minimization , 2007, Math. Program..