Boolean hedonic games are a class of cooperative games involving multiple agents in which agents aim to form coalitions based on individual agents’ preferences. In this work, we provide complexity results and exact algorithms for the task of forming Nash stable coalitions under different membership rights in the dichotomous setting where agents specify preferences for which coalitions they are happy/unhappy to join. The membership rights specify veto rights for coalitions, allowing a coalition to forbid an individual agent from moving (exiting the current coalition or entering another coalition) even if the agent themself would become more happy to move. We establish that various problem variants and their refinements in this setting are often situated on the second level of the polynomial hierarchy, complete for Σp2. Building on the complexity results, we develop Boolean satisfiability (SAT) based counterexample-guided abstraction refinement algorithms for the Σp2 problem variants and empirically evaluate a first-of-kind implementation of the approaches.

Apportionment is the task of assigning resources to entities with different entitlements in a fair manner, and specifically a manner that is as proportional as possible. The best-known application is the assignment of parliamentary seats to political parties based on their share in the popular vote. Here we enrich the standard model of apportionment by associating each seat with a weight representing the (objective) value of that seat. A seat’s weight reflects the fact that different seats might come with different roles, such as chair or treasurer. We define several apportionment methods and natural fairness requirements for this new setting, and we study the extent to which our methods satisfy these requirements. Our findings show that full fairness is harder to achieve than in the standard apportionment setting. Yet, for several natural relaxations of those requirements we can achieve stronger results than in the more expressive model of fair division with entitlements, where the values of objects are subjective.

The study of limits of strategic behavior in collective decision making is a central topic in computational social choice. Focusing on judgment aggregation, we provide complexity results and algorithms for manipulation and bribery under various aggregation rules. Specifically, we show that manipulation and bribery are complete for the second level of the Polynomial Hierarchy and detail aggregation-rule-specific strong refinements for effective counterexample-guided abstraction refinement algorithms based on iterative calls to a maximum satisfiability solver for both manipulation and bribery. We provide an open-source implementation of the approach and empirically evaluate its performance on standard PrefLib datasets, showing that the strong refinement strategies developed in this work enable scaling up to solving more instances.

One problem to solve in the context of information fusion, decision-making, and other artificial intelligence challenges is to compute justified beliefs based on evidence. In real-life examples, this evidence may be inconsistent, incomplete, or uncertain, making the problem of evidence fusion highly non-trivial. In this paper, we propose a new model for measuring degrees of beliefs based on possibly inconsistent, incomplete, and uncertain evidence, by combining tools from Dempster-Shafer Theory and Topological Models of Evidence. Our belief model is more general than the aforementioned approaches in two important ways: (1) it can reproduce them when appropriate constraints are imposed, and, more notably, (2) it is flexible enough to compute beliefs according to various standards that represent agents' evidential demands. The latter novelty allows the users of our model to employ it to compute an agent's (possibly) distinct degrees of belief, based on the same evidence, in situations when, e.g, the agent prioritizes avoiding false negatives and when it prioritizes avoiding false positives. Finally, we show that computing degree of belief with this model is #P-complete in general.

Abstract Egalitarian considerations play a central role in many areas of social choice theory. Applications of egalitarian principles range from ensuring everyone gets an equal share of a cake when deciding how to divide it, to guaranteeing balance with respect to gender or ethnicity in committee elections. Yet, the egalitarian approach has received little attention in judgment aggregation—a powerful framework for aggregating logically interconnected issues. We make the first steps towards filling that gap. We introduce axioms capturing two classical interpretations of egalitarianism in judgment aggregation and situate these within the context of existing axioms in the pertinent framework of belief merging. We then explore the relationship between these axioms and several notions of strategyproofness from social choice theory at large. Finally, a novel egalitarian judgment aggregation rule stems from our analysis; we present complexity results concerning both outcome determination and strategic manipulation for that rule.

Over the last 20 years, human interaction with robot swarms has been investigated as a means to mitigate problems associated with the control and coordination of such swarms by either human teleoperation or completely autonomous swarms. Ongoing research seeks to characterize those situations in which such interaction is both viable and preferable. In this article, we contribute to this effort by giving the first computational complexity analyses of problems associated with algorithm, environmental influence, and leader selection methods for the control of swarms performing distributed construction tasks. These analyses are done relative to a simple model in which swarms of deterministic finite-state robots operate in a synchronous error-free manner in 2D grid-based environments. We show that all three of our problems are polynomial-time intractable in general and remain intractable under a number of plausible restrictions (both individually and in many combinations) on robot controllers, environments, target structures, and sequences of swarm control commands. We also give the first restrictions relative to which these problems are tractable, as well as discussions of the implications of our results for both the design and deployment of swarm control assistance software tools and the human control of swarms.

In the context of social choice theory, we develop a tableau-based calculus for reasoning about voting rules. This calculus can be used to obtain structured explanations for why a given set of axioms justifies a given election outcome for a given profile of voter preferences. We then show how to operationalise this calculus, using a combination of SAT solving and answer set programming, to arrive at a flexible framework for presenting human-readable justifications to users.

Dempster’s rule of combination allows us to combine various independent pieces of evidence that each have a certain degree of uncertainty. This provides a useful way for dealing with uncertain evidence, but the rule is computationally intractable. In this paper, we analyze the complexity of this rule for differently structured bodies of evidence and we consider a known algorithm by Shafer and Logan to compute this rule efficiently over a hierarchical set of evidence. We show that one can check in polynomial time whether an arbitrary set of evidence has a hierarchical shape, enabling the use of Shafer and Logan’s algorithm. Moreover, we consider two different approaches to deal with non-hierarchical sets of evidence: (i) considering hierarchical subsets and (ii) taking advantage of internal hierarchical structures in the overall set. For the former case, we conclude that getting different hierarchies from an arbitrary set of pieces of evidence corresponds to the VERTEX COVER problem and we present algorithms for obtaining these hierarchies based on this correspondence. For the latter case, we present a fixed-parameter tractable algorithm which computes the belief function of any piece of evidence included in the set.

In a collective decision-making process, having the possibility to provide non-expert agents with a justification for why a target outcome is a good compromise given their individual preferences, is an appealing idea. Such questions have recently been addressed in the computational social choice community at large---whether it was to explain the outcomes of a specific rule in voting theory or to seek transparency and accountability in multi-criteria decision making. Ultimately, the development of real-life applications based on these notions depends on their practical feasibility and on the scalability of the approach taken. In this paper, we provide computational complexity results that address the problem of finding and verifying justifications for collective decisions. In particular, we focus on the recent development of a general notion of justification for outcomes in voting theory. Such a justification consists of a step-by-step explanation, grounded in a normative basis, showing how the selection of the target outcome follows from the normative principles considered. We consider a language in which normative principles can be encoded---either as an explicit list of instances of the principles (by means of quantifier-free sentences), or in a succinct fashion (using quantifiers). We then analyse the computational complexity of identifying and checking justifications. For the case where the normative principles are given in the form of a list of instances, verifying the correctness of a justification is DP-complete and deciding on the existence of such a justification is complete for Sigma 2 P. For the case where the normative principles are given succinctly, deciding whether a justification is correct is in NEXP wedge coNEXP, and NEXP-hard, and deciding whether a justification exists is in EXP with access to an NP oracle and is NEXP-hard.

We introduce an end-to-end model for participatory budgeting grounded in social choice theory. Our model accounts for the interplay between the two stages commonly encountered in real-life partici- patory budgeting. In the first stage participants pro- pose projects to be shortlisted, while in the second stage they vote on which of the shortlisted projects should be funded. Prior work of a formal nature has focused on analysing the second stage only. We in- troduce several shortlisting rules for the first stage and analyse them in both normative and algorith- mic terms. Our main focus is on the incentives of participants to engage in strategic behaviour during the first stage, in which they need to reason about how their proposals will impact the range of strate- gies available to everyone in the second stage.

Abstract We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

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.

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—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.

We provide a comprehensive analysis of the computational complexity of the outcome determination problem for the most important aggregation rules proposed in the literature on logic-based judgment aggregation. Judgment aggregation is a powerful and flexible framework for studying problems of collective decision making that has attracted interest in a range of disciplines, including Legal Theory, Philosophy, Economics, Political Science, and Artificial Intelligence. The problem of computing the outcome for a given list of individual judgments to be aggregated into a single collective judgment is the most fundamental algorithmic challenge arising in this context. Our analysis applies to several different variants of the basic framework of judgment aggregation that have been discussed in the literature, as well as to a new framework that encompasses all existing such frameworks in terms of expressive power and representational succinctness.

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.

Judgment aggregation (JA) studies how to aggregate truth valuations on logically related issues. Computing the outcome of aggregation procedures is notoriously computationally hard, which is the likely reason that no implementation of them exists as of yet. However, even hard problems sometimes need to be solved. The worst-case computational complexity of answer set programming (ASP) matches that of most problems in judgment aggregation. We take advantage of this and propose a natural and modular encoding of various judgment aggregation procedures and related problems in JA into ASP. With these encodings, we achieve two results: (1) paving the way towards constructing a wide range of new benchmark instances (from JA) for answer set solving algorithms; and (2) providing an automated tool for researchers in the area of judgment aggregation.

Abstract Ramsey quantifiers are a natural object of study not only for logic and computer science but also for the formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomial-time computable or NP-hard, and whether we can give a natural characterization of the polynomial-time computable quantifiers. In this paper, we first show that there exist intermediate Ramsey quantifiers and then we prove a dichotomy result for a large and natural class of Ramsey quantifiers, based on a reasonable and widely believed complexity assumption. We show that the polynomial-time computable quantifiers in this class are exactly the constant-log-bounded Ramsey quantifiers.

The assignment problem is one of the most well-studied 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 logic, in knowledge representation and reasoning, and in artificial intelligence are located at the second level of the Polynomial Hierarchy or even higher, and hence for these problems 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 to SAT which exploit structural aspects of problem instances in terms of problem parameters. These reductions are more powerful because their running times can grow superpolynomially in the 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.

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

We investigate the parameterized complexity of computing an outcome of the Kemeny rule in judgment aggregation, providing the first parameterized complexity results for this problem for any judgment aggregation procedure. As parameters, we consider (i) the number of issues, (ii) the maximum size of formulas used to represent issues, (iii) the size of the integrity constraint used to restrict the set of feasible opinions, (iv) the number of individuals, and (v) the maximum Hamming distance between any two individual opinions, as well as all possible combinations of these parameters. We provide parameterized complexity results for two judgment aggregation frameworks: formula-based judgment aggregation and constraint-based judgment aggregation. Whereas the classical complexity of computing an outcome of the Kemeny rule in these two frameworks coincides, the parameterized complexity results differ.

Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k-CNF-Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this paper, we extend the framework of Impagliazzo et al., and identify a larger set of problems that are equivalent to k-CNF-Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem. This research direction can be traced back to Fagin's celebrated theorem stating that NP coincides with the class of problems expressible in existential second order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter. We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat.

Not all NP-complete problems share the same practical hardness with respect to exact computation. Whereas some NP-complete problems are amenable to efficient computational methods, others are yet to show any such sign. It becomes a major challenge to develop a theoretical framework that is more fine-grained than the theory of NP-completeness, and that can explain the distinction between the exact complexities of various NP-complete problems. This distinction is highly relevant for constraint satisfaction problems under natural restrictions, where various shades of hardness can be observed in practice. Acknowledging the NP-hardness of such problems, one has to look beyond polynomial time computation. The theory of subexponential-time complexity provides such a framework, and has been enjoying increasing popularity in complexity theory. An instance of the constraint satisfaction problem with n variables over a domain of d values can be solved by brute-force in dn steps (omitting a polynomial factor). In this paper we study the existence of subexponential-time algorithms, that is, algorithms running in do(n) steps, for various natural restrictions of the constraint satisfaction problem. We consider both the constraint satisfaction problem in which all the constraints are given extensionally as tables, and that in which all the constraints are given intensionally in the form of global constraints. We provide tight characterizations of the subexponential-time complexity of the aforementioned problems with respect to several natural structural parameters, which allows us to draw a detailed landscape of the subexponential-time complexity of the constraint satisfaction problem. Our analysis provides fundamental results indicating whether and when one can significantly improve on the brute-force search approach for solving the constraint satisfaction problem.

The problem of finding small unsatisfiable subsets of a set of constraints is important for various applications in computer science and artificial intelligence. We study the problem of identifying whether a given instance to the constraint satisfaction problem (CSP) has an unsatisfiable subset of size at most k from a parameterized complexity point of view. We show that the problem of finding small unsatisfiable subsets of a CSP instance is harder than the corresponding problem for CNF formulas. Moreover, we show that the problem is not fixed-parameter tractable when restricting the problem to any maximal tractable Boolean constraint language (for which the problem is nontrivial). We show that the problem is hard even when the maximum number of occurrences of any variable is bounded by a constant, a restriction which leads to fixed-parameter tractability for the case of CNF formulas. Finally, we relate the problem of finding small unsatisfiable subsets to the problem of identifying variable assignments that are enforced already by a small number of constraints (backbones), or that are ruled out already by a small number of constraints (anti-backbones).

Planning is a notoriously difficult computational problem of high worst-case complexity. Researchers have been investing significant efforts to develop heuristics or restrictions to make planning practically feasible. Case-based planning is a heuristic approach where one tries to reuse previous experience when solving similar problems in order to avoid some of the planning effort. Plan reuse may offer an interesting alternative to plan generation in some settings. We provide theoretical results that identify situations in which plan reuse is provably tractable. We perform our analysis in the framework of parameterized complexity, which supports a rigorous worst-case complexity analysis that takes structural properties of the input into account in terms of parameters. A central notion of parameterized complexity is fixed-parameter tractability which extends the classical notion of polynomial-time tractability by utilizing the effect of parameters. We draw a detailed map of the parameterized complexity landscape of several variants of problems that arise in the context of case-based planning. In particular, we consider the problem of reusing an existing plan, imposing various restrictions in terms of parameters, such as the number of steps that can be added to the existing plan to turn it into a solution of the planning instance at hand.