We study the complexity of closure operators, with applications to machine learning and decision theory. In machine learning, closure operators emerge naturally in data classification and clustering. In decision theory, they can model equivalence of choice menus, and therefore situations with a preference for flexibility. Our contribution is to formulate a notion of complexity of closure operators, which translate into the complexity of a classifier in ML, or of a utility function in decision theory.

We introduce the domain of preferences that are single-peaked on a circle, which is a generalization of the well-studied single-peaked domain. This preference restriction is useful, e.g., for scheduling decisions, certain facility location problems, and for one-dimensional decisions in the presence of extremist preferences. We give a fast recognition algorithm of this domain, provide a characterisation by finitely many forbidden subprofiles, and show that many popular single- and multi-winner voting rules are polynomial-time computable on this domain. In particular, we prove that Proportional Approval Voting can be computed in polynomial time for profiles that are single-peaked on a circle. In contrast, Kemeny's rule remains hard to evaluate, and several impossibility results from social choice theory can be proved using only profiles in this domain.

Incomplete preferences are likely to arise in real-world preference aggregation scenarios. This paper deals with determining whether an incomplete preference profile is single-peaked. This is valuable information since many intractable voting problems become tractable given singlepeaked preferences. We prove that the problem of recognizing single-peakedness is NP-complete for incomplete profiles consisting of partial orders. Despite this intractability result, we find several polynomial-time algorithms for reasonably restricted settings. In particular, we give polynomial-time recognition algorithms for weak orders, which can be viewed as preferences with indifference.

We introduce the (j, k)-Kemeny rule - a generalization of Kemeny's voting rule that aggregates j-chotomous weak orders into a k-chotomous weak order. Special cases of (j, k)- Kemeny include approval voting, the mean rule and Borda mean rule, as well as the Borda count and plurality voting. Why, then, is the winner problem computationally tractable for each of these other rules, but intractable for Kemeny? We show that intractability of winner determination for the (j, k)-Kemeny rule first appears at the j 3, k 3 level. The proof rests on a reduction of max cut to a related problem on weighted tournaments, and reveals that computational complexity arises from the cyclic part in the fundamental decomposition of a weighted tournament into cyclic and cocyclic components. Thus the existence of majority cycles - the engine driving both Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem - also serves as a source of computational complexity in social choice.

We introduce the domain of preferences that are single-peaked on a circle, which is a generalization of the well-studied single-peaked domain. This preference restriction is useful, e.g., for scheduling decisions, and for one-dimensional decisions in the presence of extremist preferences. We give a fast recognition algorithm of this domain, provide a characterisation by finitely many forbidden subprofiles, and show that many popular single- and multi-winner voting rules are polynomial-time computable on this domain. In contrast, Kemeny's rule remains hard to evaluate, and several impossibility results from social choice theory can be proved using only profiles that are single-peaked on a circle

We study the complexity of deciding if a given profile of incomplete votes (i.e., a profile of partial orders over a given set of alternatives) can be extended to a single-crossing profile of complete votes (total orders). This problem models settings where we have partial knowledge regarding voters' preferences and we would like to understand whether the given preference profile may be single-crossing. We show that this problem admits a polynomial-time algorithm when the order of votes is fixed and the input profile consists of top orders, but becomes NP-complete if we are allowed to permute the votes and the input profile consists of weak orders or independent-pairs orders. Also, we identify a number of practical special cases of both problems that admit polynomial-time algorithms.

Incomplete preferences are likely to arise in real-world preference aggregation and voting systems. This paper deals with determining whether an incomplete preference profile is single-peaked. This is essential information since many intractable voting problems become tractable for single-peaked profiles. We prove that for incomplete profiles the problem of determining single-peakedness is NP-complete. Despite this computational hardness result, we find four polynomial-time algorithms for reasonably restricted settings.

While decision theory provides an appealing normative framework for representing rich preference structures, eliciting utility or value functions typically incurs a large cost. For many applications involving interactive systems this overhead precludes the use of formal decision-theoretic models of preference. Instead of performing elicitation in a vacuum, it would be useful if we could augment directly elicited preferences with some appropriate default information. In this paper we propose a case-based approach to alleviating the preference elicitation bottleneck. Assuming the existence of a population of users from whom we have elicited complete or incomplete preference structures, we propose eliciting the preferences of a new user interactively and incrementally, using the closest existing preference structures as potential defaults. Since a notion of closeness demands a measure of distance among preference structures, this paper takes the first step of studying various distance measures over fully and partially specified preference structures. We explore the use of Euclidean distance, Spearman's footrule, and define a new measure, the probabilistic distance. We provide computational techniques for all three measures.

We present a method for using standard techniques from satisfiability checking to automatically verify and discover theorems in an area of economic theory known as ranking sets of objects. The key question in this area, which has important applications in social choice theory and decision making under uncertainty, is how to extend an agent's preferences over a number of objects to a preference relation over nonempty sets of such objects. Certain combinations of seemingly natural principles for this kind of preference extension can result in logical inconsistencies, which has led to a number of important impossibility theorems. We first prove a general result that shows that for a wide range of such principles, characterised by their syntactic form when expressed in a many-sorted first-order logic, any impossibility exhibited at a fixed (small) domain size will necessarily extend to the general case. We then show how to formulate candidates for impossibility theorems at a fixed domain size in propositional logic, which in turn enables us to automatically search for (general) impossibility theorems using a SAT solver. When applied to a space of 20 principles for preference extension familiar from the literature, this method yields a total of 84 impossibility theorems, including both known and nontrivial new results.

We consider methods for aggregating preferences that are base d on the resolution of discrete optimization problems. For a review and references see Cook and Kress (1992), and Belkin and Levin (1990), and also David (1988) and Van B lokland-Vogelesang (1991). Some algorithmic aspects can be found in Barth elemy (1989) and Litvak (1982). The preferences are represented by arbitra ry binary relations (possibly weighted) or incomplete paired comparison matrices. The o utcome of an aggregation method is a set of "optimal" rankings (linear or weak ord ers) of the alternatives. Namely, a ranking is said to be optimal if it provides an ex tremum of some chosen objective function that expresses the connectio n (or proximity) between an arbitrary ranking and the original preferences.