Approximating Submodular Functions Everywhere

Goemans, Michel X.; Harvey, Nicholas J. A.; Iwata, Satoru; Mirrokni, Vahab

doi:10.1137/1.9781611973068.59

Cited by 97 publications

(181 citation statements)

References 30 publications

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“…Our hardness results also hold for a fixed number of players k (approaching 1 − for k fixed but large), and they are independent of a particular oracle model. When the number of players is not fixed, the problem is less understood, and the best known approximation is O(m 1/2+ ) [18,7]. The proof is obtained by a straightforward adaption of the proof for Welfare Maximization, as we explain in Section 4.4.…”

Section: Other Resultsmentioning

confidence: 99%

Communication Complexity of Combinatorial Auctions with Submodular Valuations

Dobzinski¹,

Vondrák²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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We prove the first communication complexity lower bound for constant-factor approximation of the submodular welfare problem. More precisely, we show that a (1− 1 2e + )-approximation ( 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential communication. We also show NP-hardness of (1− 1 2e + )-approximation in a computational model where each valuation is given explicitly by a table of constant size. Both results rule out better than (1 − 1 2e )-approximations in every oracle model with a separate oracle for each player, such as the demand oracle model.Our main tool is a new construction of monotone submodular functions that we call multi-peak submodular functions. Roughly speaking, given a family of sets F, we construct a monotone submodular function f with a high value f (S) for every set S ∈ F (a "peak"), and a low value on every set that does not intersect significantly any set in F.We also study two other related problems: maxmin allocation (for which we also get hardness of (1− 1 2e + )-approximation, in both models), and combinatorial public projects (for which we prove hardness of ( 3 4 + )-approximation in the communication model, and hardness of (1 − 1 e + )-approximation in the computational model, using constant size valuations).

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Section: Other Resultsmentioning

confidence: 99%

Communication Complexity of Combinatorial Auctions with Submodular Valuations

Dobzinski¹,

Vondrák²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…It is of clear interest to learn the submodular functions directly from data. See, e.g., [13,77,197] for several approaches.…”

Section: Graph-based Structured Sparsitymentioning

confidence: 99%

Learning with Submodular Functions: A Convex Optimization Perspective

Bach

2013

FNT in Machine Learning

212

226

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“…The approximability between other classes of valuations has been extensively studied [9,17,14,3], yet very little is known about gross substitutes.…”

Section: 32mentioning

confidence: 99%

Gross substitutability: An algorithmic survey

Leme

2017

Games and Economic Behavior

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Abstract. The concept of gross substitute valuations was introduced by Kelso and Crawford as a sufficient conditions for the existence of Walrasian equilibria in economies with indivisible goods. The proof is algorithmic in nature: gross substitutes is exactly the condition that enables a natural price adjustment procedure -known as Walrasian tatônnement -to converge to equilibrium.The same concept was also introduced independently in other communities with different names: M -concave functions (Murota and Shioura), Matroidal and Well-Layered maps (Dress and Terhalle) and valuated matroids (Dress and Wenzel). Here we survey various definitions of gross substitutability and show their equivalence. We focus on algorithmic aspects of the various definitions. In particular, we highlight that gross substitutes are the exact class of valuations for which demand oracles can be computed via an ascending greedy algorithm. It also corresponds to a natural discrete analogue of concave functions: local maximizers correspond to global maximizers.Finally, we discuss algorithms for the welfare problem (computing an optimal allocation of a set of items when agents have gross substitute valuations) as well as the related problem of computing Walrasian prices. We discuss approximation schemes based on the tatônnement procedure, linear programming approaches and purely combinatorial strongly-polynomial time algorithms.

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Approximating Submodular Functions Everywhere

Cited by 97 publications

References 30 publications

Communication Complexity of Combinatorial Auctions with Submodular Valuations

Communication Complexity of Combinatorial Auctions with Submodular Valuations

Learning with Submodular Functions: A Convex Optimization Perspective

Gross substitutability: An algorithmic survey

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