TY - JOUR

T1 - Maximizing symmetric submodular functions

AU - Feldman, Moran

N1 - Funding Information: This work was supported in part by the European Research Council under the ERC Starting Grant 335288-OptApprox and by the Isreal Science Foundation under grant number 1357/16. Publisher Copyright: © 2017 ACM.

PY - 2017/5

Y1 - 2017/5

N2 - 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 : 2N → ℝ+ subject to a down-monotone solvable polytope ℘ ⊆ [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 : 2N → 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.

AB - 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 : 2N → ℝ+ subject to a down-monotone solvable polytope ℘ ⊆ [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 : 2N → 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.

KW - Cardinality constraint

KW - Matroid constraint

KW - Submodular welfare

KW - Symmetric submodular functions

UR - http://www.scopus.com/inward/record.url?scp=85027001294&partnerID=8YFLogxK

U2 - https://doi.org/10.1145/3070685

DO - https://doi.org/10.1145/3070685

M3 - Article

SN - 1549-6325

VL - 13

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 3

M1 - 39

ER -