3 research outputs found
A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions
This paper presents the first combinatorial polynomial-time algorithm for
minimizing submodular set functions, answering an open question posed in 1981
by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme
that uses a flow in the complete directed graph on the underlying set with each
arc capacity equal to the scaled parameter. The resulting algorithm runs in
time bounded by a polynomial in the size of the underlying set and the largest
length of the function value. The paper also presents a strongly
polynomial-time version that runs in time bounded by a polynomial in the size
of the underlying set independent of the function value.Comment: 17 page
Minimizing Submodular Functions on Diamonds via Generalized Fractional Matroid Matchings
In this paper we show the first polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is solved by polynomial number of solutions of subproblems, each being a generalization of the weighted fractional matroid matching problem. We give a combinatorial polynomial-time algorithm for this optimization problem by extending the result by Gijswijt and Pap [D.~Gijswijt and G.~Pap, An algorithm for weighted fractional matroid matching, J.\ Combin.\ Theory, Ser.~B 103 (2013), 509--520]