Elsevier

Discrete Optimization

Volume 8, Issue 3, August 2011, Pages 459-477
Discrete Optimization

On the complexity of submodular function minimisation on diamonds

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Abstract

Let (L;,) be a finite lattice and let n be a positive integer. A function f:LnR is said to be submodular if f(ab)+f(ab)f(a)+f(b) for all a,bLn. In this article we study submodular functions when L is a diamond. Given oracle access to f we are interested in finding xLn such that f(x)=minyLnf(y) as efficiently as possible. We establish

  • a min–max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and

  • a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular f:LnZ and an integer m such that minxLnf(x)=m, there is a proof of this fact which can be verified in time polynomial in n and maxtLnlog|f(t)|; and

  • a pseudopolynomial-time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular f:LnZ one can find mintLnf(t) in time bounded by a polynomial in n and maxtLn|f(t)|.

Keywords

Combinatorial optimization
Submodular function minimization
Lattices
Diamonds

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