A simple combinatorial algorithm for submodular function minimization

Abstract

This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) [O (n superscript 6 E O 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 ((n4EO+n5)log nM) [O ((n superscript 4 EO + n superscript 5) log nM)] time. The strongly polynomial version of this faster algorithm runs in O((n5EO + n6) log n) [O ((n superscript 5 EO + n superscript 6) 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.