A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Symmetry and Approximability of Submodular Maximization Problems
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Matroid matching: the power of local search
Proceedings of the forty-second ACM symposium on Theory of computing
Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints
SIAM Journal on Discrete Mathematics
Maximizing Non-monotone Submodular Functions
SIAM Journal on Computing
Maximizing a Monotone Submodular Function Subject to a Matroid Constraint
SIAM Journal on Computing
A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
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We devise a method for proving inequalities on submodular functions, with a term rewriting flavor. Our method comprises of the following steps: (1) start with a linear combination X of the values of the function; (2) define a set of simplification rules; (3) conclude that X=Y, where Y is a linear combination of a small number of terms which cannot be simplified further; (4) calculate the coefficients of Y by evaluating X and Y on functions on which the inequality is tight. The crucial third step is non-constructive, since it uses compactness of the dual cone of submodular functions. Its proof uses the classical uncrossing technique with a quadratic potential function. We prove several inequalities using our method, and use them to tightly analyze the performance of two natural (but non-optimal) algorithms for submodular maximization, the random set algorithm and local search.