On submodular function minimization
Combinatorica
Generalized polymatroids and submodular flows
Mathematical Programming: Series A and B
Valuated Matroid Intersection II: Algorithms
SIAM Journal on Discrete Mathematics
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Minimization of an M-convex function
Discrete Applied Mathematics
Mathematical Programming: Series A and B
M-Convex Function on Generalized Polymatroid
Mathematics of Operations Research
Regular Article: Extension of M-Convexity and L-Convexity to Polyhedral Convex Functions
Advances in Applied Mathematics
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SWAT '96 Proceedings of the 5th Scandinavian Workshop on Algorithm Theory
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A Note on Kelso and Crawford's Gross Substitutes Condition
Mathematics of Operations Research
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EC '04 Proceedings of the 5th ACM conference on Electronic commerce
Combinatorial Auctions
Maximizing Non-Monotone Submodular Functions
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract)
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Maximizing submodular set functions subject to multiple linear constraints
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
On Maximizing Welfare When Utility Functions Are Subadditive
SIAM Journal on Computing
Approximation schemes for multi-budgeted independence systems
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints
SIAM Journal on Discrete Mathematics
Budgeted matching and budgeted matroid intersection via the gasoline puzzle
Mathematical Programming: Series A and B
A note on maximizing a submodular set function subject to a knapsack constraint
Operations Research Letters
Maximizing a Monotone Submodular Function Subject to a Matroid Constraint
SIAM Journal on Computing
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We consider the maximization of a gross substitutes utility function under budget constraints. This problem naturally arises in applications such as exchange economies in mathematical economics and combinatorial auctions in (algorithmic) game theory. We show that this problem admits a polynomial-time approximation scheme (PTAS). More generally, we present a PTAS for maximizing a discrete concave function called an M-concave function under budget constraints. Our PTAS is based on rounding an optimal solution of a continuous relaxation problem, which is shown to be solvable in polynomial time by the ellipsoid method. We also consider the maximization of the sum of two M-concave functions under a single budget constraint. This problem is a generalization of the budgeted max-weight matroid intersection problem to the one with a nonlinear objective function. We show that this problem also admits a PTAS.