An algorithm applicable to clearing combinatorial exchanges

It is important to approach negotiations in a way that ensures privacy. So far, research has focused on securely solving restricted classes of negotiation techniques, mainly the (M+1)-st-price auctions. Here we show how these results can be adapted to more general problems. This paper extends our previous results on how distributed finite discrete problems can be solved securely. Such problems can model larger classes of negotiation problems, e.g. Combinatorial Exchanges [SF02]. In Finite Discrete Maximization, each tuple in the problem space is associated with an integer value in a predefined interval and we search for a maximizing input. Values from different subproblems are combined additively. We show that unconstrained distributed Finite Discrete Maximization problems can be solved securely using a scheme that we propose for translating shared secret values into shared differential bids. Differential bid vectors are already used in [AS02, Bra02b]. Constrained distributed Finite Discrete Maximization poses additional challenges, due to the loss of additivity of the maximized cost, when infeasibility is marked as the lowest finite value. We found two ways of solving this problem: a) by using an additional multiplication value; and b) by using larger variable domains. While the first alternative enforces a threshold to the privacy level in our current protocol, the second one increases much the complexity of the computation. The proposed algorithms are only (t/3)-private, where t is the number of participants.