A simple combinatorial algorithm for submodular function minimization

This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(nEO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function evaluation. The algorithm can be improved to run in O((nEO+n) log nM) time. The strongly polynomial version of this faster algorithm runs in O((nEO + n) log n) time for real valued general submodular functions. These are comparable to the best known running time bounds for submodular function minimization. The algorithm can also be implemented in strongly polynomial time using only additions, subtractions, comparisons, and the oracle calls for function evaluation. This is the first fully combinatorial submodular function minimization algorithm that does not rely on the scaling method.

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

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

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

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

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

[6]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[7]  藤重 悟.  Submodular functions and optimization , 1991 .

[8]  Michel X. Goemans,et al.  Minimizing submodular functions over families of sets , 1995, Comb..

[9]  Maurice Queyranne,et al.  Minimizing symmetric submodular functions , 1998, Math. Program..

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

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

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

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

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

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

[16]  Kiyohito Nagano,et al.  A strongly polynomial algorithm for line search in submodular polyhedra , 2007, Discret. Optim..

[17]  Kiyohito Nagano,et al.  On Convex Minimization over Base Polytopes , 2007, IPCO.

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

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