Charting the Tractability Frontier of Mixed Multi-Unit Combinatorial Auctions

Mixed multi-unit combinatorial auctions (MMUCAs) are extensions of classical combinatorial auctions (CAs) where bidders trade transformations of goods rather than just sets of goods. Solving MMUCAs, i.e., determining the sequences of bids to be accepted by the auctioneer, is computationally intractable in general. However, differently from CAs, little was known about whether polynomial-time solvable classes of MMUCAs can be singled out based on constraining their characteristics. The paper precisely fills this gap, by depicting a clear picture of the "tractability frontier" for MMUCA instances under both structural and qualitative restrictions, which characterize interactions among bidders and types of bids involved in the various transformations, respectively. By analyzing these restrictions, a sharp frontier is charted based on various dichotomy results. In particular, tractability islands resulting from the investigation generalize on MMUCAs the largest class of tractable CAs emerging from the literature.

[1]  Christophe Lecoutre.  Constraint Networks , 1992 .

[2]  Rina Dechter,et al.  Network-Based Heuristics for Constraint-Satisfaction Problems , 1987, Artif. Intell..

[3]  Georg Gottlob,et al.  On the complexity of combinatorial auctions: structured item graphs and hypertree decomposition , 2007, EC '07.

[4]  Ulrich Endriss,et al.  Bidding Languages and Winner Determination for Mixed Multi-unit Combinatorial Auctions , 2007, EUMAS.

[5]  Kidane Asrat Ghebreamlak.  Winner determination problem in combinatorial auctions , 2002 .

[6]  B. Mohar,et al.  Graph Minors , 2009 .

[7]  Eugene C. Freuder.  A sufficient condition for backtrack-bounded search , 1985, JACM.

[8]  Moshe Babaioff,et al.  On the Approximability of Combinatorial Exchange Problems , 2008, SAGT.

[9]  Georg Gottlob,et al.  Hypertree decompositions and tractable queries , 1998, J. Comput. Syst. Sci..

[10]  Raimund Seidel,et al.  A New Method for Solving Constraint Satisfaction Problems , 1981, IJCAI.

[11]  D. Lehmann,et al.  The Winner Determination Problem , 2003 .

[12]  Toby Walsh,et al.  Binary vs. non-binary constraints , 2002, Artif. Intell..

[13]  Éva Tardos,et al.  Algorithm design , 2005 .

[14]  Phokion G. Kolaitis,et al.  Conjunctive-query containment and constraint satisfaction , 1998, PODS.

[15]  Yoav Shoham,et al.  Combinatorial Auctions , 2005, Encyclopedia of Wireless Networks.

[16]  D. Goldsman.  Operations Research Models and Methods , 2003 .

[17]  Michael P. Wellman,et al.  Decentralized Supply Chain Formation: A Market Protocol and Competitive Equilibrium Analysis , 2003, J. Artif. Intell. Res..

[18]  Marc Gyssens,et al.  Decomposing Constraint Satisfaction Problems Using Database Techniques , 1994, Artif. Intell..

[19]  Meritxell Vinyals,et al.  Mixed multi-unit combinatorial auctions for supply chain management , 2007, SECO.

[20]  Georg Gottlob,et al.  A Comparison of Structural CSP Decomposition Methods , 1999, IJCAI.

[21]  Rina Dechter,et al.  Tree Clustering for Constraint Networks , 1989, Artif. Intell..

[22]  Peter Jeavons,et al.  A Survey of Tractable Constraint Satisfaction Problems , 1997 .

[23]  Anand Rajaraman,et al.  Conjunctive query containment revisited , 1997, Theor. Comput. Sci..

[24]  Ulrich Endriss,et al.  Comparing winner determination algorithms for mixed multi-unit combinatorial auctions , 2008, AAMAS.

[25]  Vincent Conitzer,et al.  Combinatorial Auctions with Structured Item Graphs , 2004, AAAI.

[26]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.