A Monotone Branch-and-Bound Search for Restricted Combinatorial Auctions

Given a system of self-interested agents, each with private information about their preferences, and a set of outcomes, the problem of mechanism design is to select an outcome with desirable properties despite the ability of agents to misreport their preferences. Computational mechanism design (CMD) also insists on computational efficiency, which is a significant concern in domains such as combinatorial auctions (CAs), where the winner determination problem is NP-hard. Although typical instances of NP-hard problems such as combinatorial auctions can be routinely solved through the use of heuristic search such as branch-and-bound search [Sandholm et al. 2005; Andersson et al. 2000], a common theme in CMD is to insist on worst-case polynomial time algorithms, and look for algorithms for which there is theoretical support through worst-case approximation guarantees. In the context of single-minded CAs, where each agent is interested in exactly one bundle, [Lehmann et al. 2002] provide a greedy algorithm and associated payment rule with a √ m welfare guarantee (relative to the optimal welfare), where m is the number of items being allocated, and a matching lower-bound. More recently, [Mu’alem and Nisan 2008] provide an approximation for the special case of known single-minded CAs with guarantee √ m for any fixed > 0, with runtime that is exponential in 1/ . In a known single-minded CA, the bundle is known to the mechanism, and this is a singledimensional mechanism design problem.1 However, if incentives were not a concern, we have more sophisticated algorithms such as BnB search that can efficiently find optimal solutions to the winner determination problem on typical instances. Following a research agenda on heuristic mechanism design [Parkes 2009], we seek to leverage these heuristic algorithms for the purpose of CMD. Branch-and-bound search is a canonical method for solving optimization problems that are formulated as integer programs (IPs). In particular, branch-and-bound (BnB) search has been found to be very effective for computing optimal allocations for combinatorial auctions. Search proceeds in BnB by branching on decisions in regard to whether or not an agent is allocated (“branch”), and looking to prune large parts of