Maximizing Symmetric Submodular Functions

Abstract

Symmetric submodular functions are an important family of submodular functions capturing many interesting cases including cut functions of graphs and hypergraphs. In this work, we identify submodular maximization problems for which one can get a better approximation for symmetric objectives compared to what is known for general submodular functions

For the problem of maximizing a non-negative symmetric submodular function $f: 2^\mathcal{N} \to \mathbb{R}^{+}$ subject to a down-monotone solvable polytope $\mathcal{P} \subseteq[0, 1]^\mathcal{N}$ we describe an algorithm producing a fractional solution of value at least 0.432 ·f(OPT), where OPT is the optimal integral solution. Our second result is a 0.432-approximation algorithm for the problem max {f(S) : |S| = k} with a non-negative symmetric submodular function $f: 2^\mathcal{N} \to \mathbb{R}^{+}$. Our method also applies non-symmetric functions, in which case it produces $\frac{1}{e} - o(1)$ approximation. Finally, we describe a deterministic linear-time $\frac{1}{2}$-approximation algorithm for unconstrained maximization of a non-negative symmetric submodular function.