Approval-Based Shortlisting

Shortlisting is the task of reducing a long list of alternatives to a (smaller) set of best or most suitable alternatives from which a final winner will be chosen. Shortlisting is often used in the nomination process of awards or in recommender systems to display featured objects. In this paper, we analyze shortlisting methods that are based on approval data, a common type of preferences. Furthermore, we assume that the size of the shortlist, i.e., the number of best or most suitable alternatives, is not fixed but determined by the shortlisting method. We axiomatically analyze established and new shortlisting methods and complement this analysis with an experimental evaluation based on biased voters and noisy quality estimates. Our results lead to recommendations which shortlisting methods to use, depending on the desired properties

The core of an approval-based PB instance can be empty for nearly all cost-based satisfaction functions and for the share

The core is a strong fairness notion in multiwinner voting and participatory budgeting (PB). It is known that the core can be empty if we consider cardinal utilities, but it is not known whether it is always satisfiable with approval-ballots. In this short note, I show that in approval-based PB the core can be empty for nearly all satisfaction functions that are based on the cost of a project. In particular, I show that the core can be empty for the cost satisfaction function, satisfaction functions based on diminishing marginal returns and the share. However, it remains open whether the core can be empty for the cardinality satisfaction function

Ranking Sets of Objects: The Complexity of Avoiding Impossibility Results

The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating properties the lifted order should have. Well-known impossibility results by Kannai and Peleg and by Barber\`a and Pattanaik tell us that some desirable axioms - namely dominance and (strict) independence - are not jointly satisfiable for any linear order on the objects if all non-empty sets of objects are to be ordered. On the other hand, if not all non-empty sets of objects are to be ordered, the axioms are jointly satisfiable for all linear orders on the objects for some families of sets. Such families are very important for applications as they allow for the use of lifted orders, for example, in combinatorial voting. In this paper, we determine the computational complexity of recognizing such families. We show that it is $\Pi_2^p$-complete to decide for a given family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for all linear orders on the objects if the lifted order needs to be total. Furthermore, we show that the problem remains coNP-complete if the lifted order can be incomplete. Additionally, we show that the complexity of these problem can increase exponentially if the family of sets is not given explicitly but via a succinct domain restriction. Finally, we show that it is NP-complete to decide for family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for at least one linear orders on the objects

AM-modulus and Hausdorff measure of codimension one in metric measure spaces

Let Gamma(E) be the family of all paths which meet a set E in the metric measure space X. The set function E bar right arrow AM (Gamma(E)) defines the AM-modulus measure in X where AM refers to the approximation modulus [22]. We compare AM (Gamma(E)) to the Hausdorff measure coH(1) (E) of codimension one in X and show that coH(1)(E) approximate to AM(Gamma(E)) for Suslin sets E in X. This leads to a new characterization of sets of finite perimeter in X in terms of the AM-modulus. We also study the level sets of BV functions and show that for a.e. t. these sets have finite coH(1)-measure. Most of the results are new also in R-n.Peer reviewe

A version of the Stokes theorem using test curves

We prove that a parametric Lipschitz surface of codimension 1 in a smooth manifold induces a boundary in the sense of currents (roughly speaking, surrounds a "domain" with an eventual multiplicity and together with it forms a pair for the Stokes theorem) if and only if it passes a test in terms of crossing the surface by "almost all" curves. We use the AM-modulus recently introduced in [22] to measure the exceptional family of curves.Peer reviewe