On the Approximability of Combinatorial Exchange Problems

In a combinatorial exchange a set of indivisible products is traded between buyers and sellers whichare interested in bundles (multi-sets of products). Although combinatorial exchanges are a natural andimportant generalization of combinatorial auctions, their approximability has not been studied. We in-vestigate the computational approximability of several social goals and show that the problems of surplusmaximization and volume maximization (subject to positive surplus) are inapproximable even with freedisposal and even if each agent’s bundle is of size at most 3. Similar results based on communicationcomplexity are shown for agents with general valuation functions.In light of the negative results for surplus maximization we consider a complementary goal of socialcost minimization and present tight approximation results for several social cost minimization problems.Considering the more general supply chain problem we prove that social cost minimization remains inap-proximable even with bundles of size 3, yet becomes polynomial time solvable for agents trading bundlesof size 1 or 2. This yields a complete characterization of the approximability of supply chain and combi-natorial exchange problems based on the size of traded bundles.Finally, we point out that economic considerations prevent any social cost approximation when agentshold private information about their valuations that needs to be elicited via some truthful mechanism.