Symmetric submodular functions are an important family of submodular functions capturing many interesting cases, including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little attention by current research, unlike similar minimization problems that have been widely studied. In this work, we identify a few submodular maximization problems for which one can get a better approximation for symmetric objectives than the state-of-the-art approximation for general submodular functions. We first consider the problem of maximizing a non-negative symmetric submodular function f : 2 N → R + subject to a down-monotone solvable polytope P ⊆ [0, 1] N . For this problem, 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 considers the problem max{ f (S) : |S| = k} for a non-negative symmetric submodular function f : 2 N → R + . For this problem, we give an approximation ratio that depends on the value k/|N | and is always at least 0.432. Our method can also be applied to non-negative non-symmetric submodular functions, in which case it produces 1 /e − o(1) approximation, improving over the best-known result for this problem. For unconstrained maximization of a non-negative symmetric submodular function, we describe a deterministic linear-time 1 /2-approximation algorithm. Finally, we give a [1 − (1 − 1/k) k−1 ]-approximation algorithm for Submodular Welfare with k players having identical non-negative submodular utility functions and show that this is the best possible approximation ratio for the problem.
This gap is somewhat bridged by a result of REF giving a 0.432-approximation algorithm for the maximization of a special class of non-monotone submodular functions, known as symmetric submodular functions, subject to a general matroid constraint.
Maximizing Symmetric Submodular Functions
TALG
Mathematics Computer Science