Abstract
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. [5] 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 \(\frac{2}{3}\)-relaxed Walrasian equilibrium proving a conjecture posed in [5]. Motivated by practical considerations, we introduce another concept of approximate Walrasian equilibrium called weak Walrasian equilibrium. We show it is strongly NP-complete to determine the existence of δ-weak Walrasian equilibrium, for any 0<δ≤ 1.
In search of positive results, we restrict our attention to the tollbooth problem [15], 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 hard for general graphs.
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Chen, N., Rudra, A. (2005). Walrasian Equilibrium: Hardness, Approximations and Tractable Instances. In: Deng, X., Ye, Y. (eds) Internet and Network Economics. WINE 2005. Lecture Notes in Computer Science, vol 3828. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11600930_15
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DOI: https://doi.org/10.1007/11600930_15
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