Combinatorial auctions with restricted complements

Abstract

Complements between goods -where one good takes on added value in the presence of another -have been a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymied theoretical progress in the area. For example, there are no known positive results for combinatorial auctions in which bidder valuations are multi-parameter and non-complement-free, other than the relatively weak results known for general valuations.To make inroads on the problem of combinatorial auction design in the presence of complements, we propose a model for valuations with complements that is parameterized by the "size" of the complements. The model permits a succinct representation, a variety of computationally efficient queries, and non-trivial welfare-maximization algorithms and mechanisms. Specifically, a hypergraph-r valuation v for a good set M is represented by a hypergraph H = (M, E), where every (hyper-)edge e ∈ E contains at most r vertices and has a nonnegative weight w e . Each good j ∈ M also has a nonnegative weight w j . The value v(S) for a subset S ⊆ M of goods is defined as the sum of the weights of the goods and edges entirely contained in S.We design the following polynomial-time approximation algorithms and truthful mechanisms for welfare maximization with bidders with hypergraph valuations.2. We give a polynomial-time, r-approximation algorithm for welfare maximization with hypergraph-r valuations. Our algorithm randomly rounds a compact linear programming relaxation of the problem.3. We design a different approximation algorithm and use it to give a polynomial-time, truthful-in-expectation mechanism that has an approximation factor of O(log r m).

Why prices need algorithms

“…The value of a bundle S is v(S) = w(G(S)), where G(S) is the subgraph induced by the vertices in S, and w(G(S)) is the total weight of the subgraph's vertices and edges. The fact that the demand problem given item prices is solvable in polynomial time was observed by [Abraham et al 2012] (Proposition 5.1): The valuation v defined by the positive graph is supermodular, and hence so is the consumer's utility function after subtracting item prices from valuation v; maximizing supermodular functions can be done in polynomial time. On the other hand, the allocation problem is NP-hard by a reduction of [Conitzer et al 2005] (Theorem 6) from the problem of exact cover by 3-sets.…”

“…In particular, it requires communication of a price system that is exponential in the number of items m. (2) Approximation: Algorithmic aspects of approximate welfare-maximization have been extensively studied, especially for the "complement-free" valuation hierarchy [Blumrosen and Nisan 2007]. Recent research expands upon this by studying valuation classes with "limited" complements and good approximation guarantees [Abraham et al 2012;Feige et al 2014]. 1 (3) Representation and elicitation: Both succinctness ("compactness") of valuation classes and their learnability have been studied (see, e.g., [Boutilier et al 2004;Zinkevich et al 2003]), and for classes of non-succinct valuations, simple sketches have been pursued ( [Cohavi and Dobzinski 2014] and references within).…”

Optimal Mechanisms for Combinatorial Auctions and Combinatorial Public Projects via Convex Rounding

“…Each bidder i has a private valuation v i (S) for each subset S of the items. 1 The welfare of an allocation S 1 , . .…”

“…We consider mechanism design optimization problems of the form in (1). In such problems, there are n players, where each player i has a valuation function v i : S → R. We are concerned with welfare maximization problems, where the objective is w(x) = n i=1 v i (x).…”

“…In particular, an attractive potential application is to combinatorial auctions with valuations exhibiting limited complementarity, as defined in [1] and [27]. More generally, a characterization of the family of problems and linear programming relaxations which admit optimal (or near optimal) convex rounding algorithms would open the door to understanding the power and limitations of our technique.…”