Abstract. We introduce the notion of combinatorial vote, where a group of agents (or voters) is supposed to express preferences and come to a common decision concerning a set of non-independent variables to assign. We study two key issues pertaining to combinatorial vote, namely preference representation and the automated choice of an optimal decision. For each of these issues, we briefly review the state of the art, we try to define the main problems to be solved and identify their computational complexity.

Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of partial orders and a candidate c, two important questions arise: first, is c guaranteed to win, and second, is it still possible for c to win? These are the necessary winner and possible winner problems, respectively. We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We prove that for Copeland, maximin, Bucklin, and ranked pairs, the possible winner problem is NP-complete; also, we give a sufficient condition on scoring rules for the possible winner problem to be NP-complete (Borda satisfies this condition). We also prove that for Copeland and ranked pairs, the necessary winner problem is coNP-complete. All the hardness results hold even when the number of undetermined pairs in each vote is no more than a constant. We also present polynomial-time algorithms for the necessary winner problem for scoring rules, maximin, and Bucklin.

In group decision making, often the agents need to decide on multiple attributes at the same time, so that there are exponentially many alternatives. In this case, it is unrealistic to ask agents to communicate a full ranking of all the alternatives. To address this, earlier work has proposed decomposing such voting processes by using local voting rules on the individual attributes. Unfortunately, the existing methods work only with rather severe domain restrictions, as they require the voters’ preferences to extend acyclic CP-nets compatible with a common order on the attributes. We first show that this requirement is very restrictive, by proving that the number of linear orders extending an acyclic CP-net is exponentially smaller than the number of all linear orders. Then, we introduce a very general methodology that allows us to aggregate preferences when voters express CP-nets that can be cyclic. There does not need to be any common structure among the submitted CP-nets. Our methodology generalizes the earlier, more restrictive methodology. We study whether properties of the local rules transfer to the global rule, and vice versa. We also address how to compute the winning alternatives.

In this paper we introduce and study a logic of desires. The semantics of our logic is defined by means of two ordering relations representing preference and normality as in Boutilier’s logic QDT. However, the desires are interpreted in a different way: “in context A, I desire B ” is interpreted as “the best among the most normal A ∧ B worlds are preferred to the most normal A ∧ ¬B worlds”. We study the formal properties of these desires, illustrate their expressive power on several classes of examples and position them with respect to previous work in qualitative decision theory. 1