Walrasian Equilibrium: Hardness, Approximations and Tractable Instances

  • Ning Chen
  • Atri Rudra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3828)


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.


Polynomial Time Optimal Allocation Polynomial Time Algorithm General Graph Market Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ning Chen
    • 1
  • Atri Rudra
    • 1
  1. 1.Department of Computer Science & EngineeringUniversity of WashingtonSeattleUSA

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