Single-minded Auctions Research Articles

Walrasian Equilibrium: Hardness, Approximations and Tractable Instances

We study the complexity issues for Walrasian equilibrium in a special case of combinatorial auction, called single-minded auction, in which every participant is interested in only one subset of commodities. Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004) showed that it is NP-hard to decide the existence of a Walrasian equilibrium for a single-minded auction and proposed a notion of approximate Walrasian equilibrium called relaxed Walrasian equilibrium. We show that every single-minded auction has a relaxed Walrasian equilibrium that satisfies at least two-thirds of the participants, proving a conjecture posed in Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004). Motivated by practical considerations, we introduce another concept of approximate Walrasian equilibrium called weak Walrasian equilibrium. We show NP-completeness and hardness of approximation results for weak Walrasian equilibria. In search of positive results, we restrict our attention to the tollbooth problem (Guruswami et al. in Proceedings of the Symposium on Discrete Algorithms (SODA), pp. 1164–1173, 2005), where every participant is interested in a single path in some underlying graph. We give a polynomial time algorithm to determine the existence of a Walrasian equilibrium and compute one (if it exists), when the graph is a tree. However, the problem is still NP-hard for general graphs.

Approximation of Walrasian equilibrium in single-minded auctions

We consider a social optimization model of pricing scheme in single-minded auctions, in cases where Walrasian equilibrium does not exist. We are interested in the maximization of the ratio, R, of happy bidders over all agents, in a feasible allocation-pricing scheme. We show NP-hardness of the optimization problem, establish lower and upper bounds of R, as well as develop greedy algorithms to approximate the optimal value of R.

On complexity of single-minded auction

We consider complexity issues for a special type of combinatorial auctions, the single-minded auction, where every agent is interested in only one subset of the commodities. First, we present a matching bound on the communication complexity for the single-minded auction under a general communication model. Next, we prove that it is NP-hard to decide whether Walrasian equilibrium exists in a single-minded auction. Finally, we establish a polynomial size duality theorem for the existence of Walrasian equilibrium for the single-minded auction.