Maximizing General Set Functions by Submodular Decomposition

TL;DR

This paper introduces a branch and bound approach for maximizing arbitrary set functions by decomposing them into submodular and cut functions, enabling efficient optimization and approximation strategies.

Contribution

It proposes a novel decomposition method for set function maximization, reducing it to submodular problems and analyzing classes that guarantee convergence to global maxima.

Findings

Decomposition into submodular and cut functions facilitates optimization.

Polynomial-time solutions for certain subproblems.

Approximation algorithms for NP-hard subproblems.

Abstract

We present a branch and bound method for maximizing an arbitrary set function h mapping 2^V to R. By decomposing h as f-g, where f is a submodular function and g is the cut function of a (simple, undirected) graph G with vertex set V, our original problem is reduced to a sequence of submodular maximization problems. We characterize a class of submodular functions, which when maximized in the subproblems, lead the algorithm to converge to a global maximizer of f-g. Two "natural" members of this class are analyzed; the first yields polynomially-solvable subproblems, the second, which requires less branching, yields NP-hard subproblems but is amenable to a polynomial-time approximation algorithm. These results are extended to problems where the solution is constrained to be a member of a subset system. Structural properties of the maximizer of f-g are also proved.