Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
Rounding of polytopes in the real number model of computation
Mathematics of Operations Research
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
A combinatorial strongly polynomial algorithm for minimizing submodular functions
Journal of the ACM (JACM)
Lectures on Discrete Geometry
Discrete Convex Analysis: Monographs on Discrete Mathematics and Applications 10
Discrete Convex Analysis: Monographs on Discrete Mathematics and Applications 10
An improved approximation algorithm for combinatorial auctions with submodular bidders
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Computation of Minimum-Volume Covering Ellipsoids
Operations Research
An approximation algorithm for max-min fair allocation of indivisible goods
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Maximizing Non-Monotone Submodular Functions
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Optimal approximation for the submodular welfare problem in the value oracle model
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Submodular Approximation: Sampling-based Algorithms and Lower Bounds
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A simple combinatorial algorithm for submodular function minimization
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Inapproximability results for combinatorial auctions with submodular utility functions
WINE'05 Proceedings of the First international conference on Internet and Network Economics
Non-monotone submodular maximization under matroid and knapsack constraints
Proceedings of the forty-first annual ACM symposium on Theory of computing
Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints
SIAM Journal on Discrete Mathematics
Approximability of combinatorial problems with multi-agent submodular cost functions
ACM SIGecom Exchanges
An impossibility result for truthful combinatorial auctions with submodular valuations
Proceedings of the forty-third annual ACM symposium on Theory of computing
Proceedings of the forty-third annual ACM symposium on Theory of computing
Privately releasing conjunctions and the statistical query barrier
Proceedings of the forty-third annual ACM symposium on Theory of computing
Submodular cost allocation problem and applications
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Maximizing Non-monotone Submodular Functions
SIAM Journal on Computing
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Ranking with submodular valuations
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Submodular Approximation: Sampling-based Algorithms and Lower Bounds
SIAM Journal on Computing
ECML PKDD'12 Proceedings of the 2012 European conference on Machine Learning and Knowledge Discovery in Databases - Volume Part II
Annals of Mathematics and Artificial Intelligence
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Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function f such that, for every set S, f(S) approximates f(S) within a factor α(n), where α(n) = √n+1 for rank functions of matroids and α(n), = O(√n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids. Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω(√n/log n), even for rank functions of a matroid.