A tight linear time (1/2)-approximation for unconstrained submodular maximization

Niv Buchbinder, Moran Feldman, Joseph Seffi Naor, Roy Schwartz

Research output: Contribution to journalArticlepeer-review


We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f : 2N → R+, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by Unconstrained Submodular Maximization include Max-Cut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige, Mirrokni, and Vondrák [SIAM J. Comput., 40 (2011), pp. 1133-1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem.

Original languageEnglish
Pages (from-to)1384-1402
Number of pages19
JournalSIAM Journal on Computing
Issue number5
StatePublished - 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.


  • Approximation algorithms
  • Linear time
  • Submodular functions


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