Abstract. We consider the two-sided stable matching setting in which there may be uncertainty about the agents' preferences due to limited information or communication. We consider three models of uncertainty:(1) lottery model -in which for each agent, there is a probability distribution over linear preferences, (2) compact indifference model -for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model -there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant.

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The assignment problem is one of the most wellstudied settings in social choice, matching, and discrete allocation. We consider this problem with the additional feature that agents' preferences involve uncertainty. The setting with uncertainty leads to a number of interesting questions including the following ones. How to compute an assignment with the highest probability of being Pareto optimal? What is the complexity of computing the probability that a given assignment is Pareto optimal? Does there exist an assignment that is Pareto optimal with probability one? We consider these problems under two natural uncertainty models: (1) the lottery model in which each agent has an independent probability distribution over linear orders and (2) the joint probability model that involves a joint probability distribution over preference profiles. For both of these models, we present a number of algorithmic and complexity results highlighting the difference and similarities in the complexity of the two models.

Today's propositional satisfiability (SAT) solvers are extremely powerful and can be used as an efficient back-end for solving NP-complete problems. However, many fundamental problems in knowledge representation and reasoning are located at the second level of the Polynomial Hierarchy or even higher, and hence polynomial-time transformations to SAT are not possible, unless the hierarchy collapses. Recent research shows that in certain cases one can break through these complexity barriers by fixed-parameter tractable (fpt) reductions which exploit structural aspects of problem instances in terms of problem parameters.In this paper we develop a general theoretical framework that supports the classification of parameterized problems on whether they admit such an fpt-reduction to SAT or not. This framework is based on several new parameterized complexity classes. As a running example, we use the framework to classify the complexity of the consistency problem for disjunctive answer set programming, with respect to various natural parameters. We underpin the robustness of our theory by providing a characterization of the new complexity classes in terms of weighted QBF satisfiability, alternating Turing machines, and first-order model checking. In addition, we provide a compendium of parameterized problems that are complete for the new complexity classes, including problems related to Knowledge Representation and Reasoning, Logic, and Combinatorics. * This paper contains results that have appeared in shortened and preliminary form in the proceedings of SAT 2014 [40], the proceedings of KR 2014 [41], and the proceedings of SOFSEM 2015 [42].

The assignment problem is one of the most well-studied settings in multi-agent resource allocation. Aziz, de Haan, and Rastegari (2017) considered this problem with the additional feature that agents’ preferences involve uncertainty. In particular, they considered two uncertainty models neither of which is necessarily compact. In this paper, we focus on three uncertain preferences models whose size is polynomial in the number of agents and items. We consider several interesting computational questions with regard to Pareto optimal assignments. We also present some general characterization and algorithmic results that apply to large classes of uncertainty models.

We study a two-sided matching problem where the agents have independent pairwise preferences on their possible partners and these preferences may be uncertain. In this case, the certainly preferred part of an agent's preferences may admit a cycle and there may not even exist a matching that is stable with non-zero probability. We focus on the computational problems of checking the existence of possibly and certainly stable matchings, i.e., matchings whose probability of being stable is positive or one, respectively. We show that finding a possibly stable matching is NP-hard, even if only one side can have cyclic preferences. On the other hand we show that the problem of finding a certainly stable matching is polynomial-time solvable if only one side can have cyclic preferences and the other side has transitive preferences, but that this problem becomes NP-hard when both sides can have cyclic preferences. The latter complexity result also implies the hardness of finding a kernel in a special class of directed graphs. CCS Concepts•Theory of computation ! Design and analysis of algorithms; •Computing methodologies ! Multi-agent systems; •Applied computing ! Economics;

Cognitive science is itself a cognitive activity. Yet, computational cognitive science tools are seldom used to study (limits of) cognitive scientists’ thinking. Here, we do so using computational-level modeling and complexity analysis. We present an idealized formal model of a core inference problem faced by cognitive scientists: Given observations of a system’s behaviors, infer cognitive processes that could plausibly produce the behavior. We consider variants of this problem at different levels of explanation and prove that at each level, the inference problem is intractable, or even uncomputable. We discuss the implications for cognitive science.

We introduce a new approach for designing rules for participatory budgeting, the problem of deciding on the use of public funds based directly on the views expressed by the citizens concerned. The core idea is to embed instances of the participatory budgeting problem into judgment aggregation, a powerful general-purpose framework for modelling collective decision making. Taking advantage of the possibilities offered by judgment aggregation, we enrich the familiar setting of participatory budgeting with additional constraints, namely dependencies between projects and quotas regarding different types of projects. We analyse the rules obtained both in algorithmic and in axiomatic terms.