In participatory budgeting the stakeholders collectively decide which projects from a set of proposed projects should be implemented. This decision underlies both time and monetary constraints. In reality it is often impossible to figure out the exact cost of each project in advance, it is only known after a project is finished. To reduce risk, one can implement projects one after the other to be able to react to higher costs of a previous project. However, this will increase execution time drastically. We generalize existing frameworks to capture this setting, study desirable properties of algorithms for this problem, and show that some desirable properties are incompatible. Then we present and analyze algorithms that trade-off desirable properties.

Abstract argumentation frameworks (AFs), originally proposed by Dung, constitute a central formal model for the study of computational aspects of argumentation in AI. Credulous and skeptical acceptance of arguments in a given AF are well-studied problems both in terms of theoretical analysis—especially computational complexity—and the development of practical decision procedures for the problems. However, AFs make the assumption that all attacks between arguments are certain (i.e., present attacks are known to exist, and missing attacks are known to not exist), which can in various settings be a restrictive assumption. A generalization of AFs to incomplete AFs was recently proposed as a formalism that allows the representation of both uncertain attacks and uncertain arguments in AFs. In this article, we explore the impact of allowing for modeling such uncertainties in AFs on the computational complexity of natural generalizations of acceptance problems to incomplete AFs under various central AF semantics. Complementing the complexity-theoretic analysis, we also develop the first practical decision procedures for all of the NP-hard variants of acceptance in incomplete AFs. In terms of complexity analysis, we establish a full complexity landscape, showing that depending on the variant of acceptance and property/semantics, the complexity of acceptance in incomplete AFs ranges from polynomial-time decidable to completeness for Σ3p. In terms of algorithms, we show through an extensive empirical evaluation that an implementation of the proposed decision procedures, based on boolean satisfiability (SAT) solving, is effective in deciding variants of acceptance under uncertainties. We also establish conditions for what type of atomic changes are guaranteed to be redundant from the perspective of preserving extensions of completions of incomplete AFs, and show that the results allow for considerably improving the empirical efficiency of the proposed SAT-based counterexample-guided abstraction refinement algorithms for acceptance in incomplete AFs for problem variants with complexity beyond NP.

The bribery problem in elections asks whether an external agent can make some distinguished candidate win or prevent her from winning, by bribing some of the voters. This problem was studied with respect to the weighted swap distance between two votes by Elkind et al. (2009). We generalize this definition by introducing a bound on the distance between the original and the bribed votes. The distance measures we consider include a restriction of the weighted swap distance and variants of the footrule distance, which capture some realworld models of influence an external agent may have on the voters. We study constructive and destructive variants of distance bribery for scoring rules and obtain polynomial-time algorithms as well as NP-hardness results. For the case of element-weighted swap and element-weighted footrule distances, we give a complete dichotomy result for the class of pure scoring rules.

In an election, votes are often given as ordered lists over candidates. A common way of determining the winner is then to apply some scoring system, where each position is associated with a specific score. This setting is also transferable to other situations, such as sports tournaments. The design of such systems, i.e., the choice of the score values, may have a crucial influence on the outcome. We study the computational complexity of two related decision problems. In addition, we provide a case study of data from Formula 1 using ILP formulations. Our results show that under some mild conditions there are cases where the actual scoring system has no influence, whereas in other cases very small changes may lead to a different winner. This may be seen as a measure of robustness of the winning candidate.

Abstract argumentation frameworks are a well-established formalism to model nonmonotonic reasoning processes. However, the standard model cannot express incomplete or conflicting knowledge about the state of a given argumentation. Previously, argumentation frameworks were extended to allow uncertainty regarding the set of attacks or the set of arguments. We combine both models into a model of general incompleteness, complement previous results on the complexity of the verification problem in incomplete argumentation frameworks, and provide a full complexity map covering all three models and all classical semantics. Our main result shows that the complexity of verifying the preferred semantics rises from coNP- to Sigma^p_2-completeness when allowing uncertainty about either attacks or arguments, or both.

We propose natural generalizations of the credulous and skeptical acceptance problems in abstract argumentation for incomplete argumentation frameworks [3]. This continues earlier work on a similar generalization of the verification problem. We provide a full analysis of the computational complexity of the generalized problems for all original semantics, showing that, in almost all cases, acceptance problems for incomplete argumentation frameworks are significantly harder than the respective problems for argumentation frameworks without uncertainty. All our hardness results for the classes NP, coNP, Πp2, and Σp2

In committee elections it is often assumed that voters only (dis)approve of each candidate or that they rank all candidates, as it is common for single-winner elections. We suggest an intermediate approach, where the voters rank the candidates into a fixed number of groups. This allows more diverse votes than approval votes, but leaves more freedom than in a linear order. A committee is then elected by applying the minisum or minimax approach to minimize the voters' dissatisfaction. We study the axiomatic properties of these committee election rules as well as the complexity of winner determination and show fixed-parameter tractability for our minimax rules.

We define a family of rules for dividing m indivisible goods among agents, parameterized by a scoring vector and a social welfare aggregation function. We assume that agents' preferences over sets of goods are additive, but that the input is ordinal: each agent simply ranks single goods. Similarly to (positional) scoring rules in voting, a scoring vector s = (s1,...,sm) consists of m nonincreasing nonnegative weights, where siis the score of a good assigned to an agent who ranks it in position i. The global score of an allocation for an agent is the sum of the scores of the goods assigned to her. The social welfare of an allocation is the aggregation of the scores of all agents, for some aggregation function ★ such as, typically, + or min. The rule associated with s and ★ maps a profile to (one of) the allocation(s) maximizing social welfare. After defining this family of rules, and focusing on some key examples, we investigate some of the social-choice-theoretic properties of this family of rules, such as various kinds of monotonicity, separability, envy-freeness, and Pareto efficiency.

The POSSIBLE WINNER problem asks, given an election where the voters' preferences over the candidates are specified only partially, whether a designated candidate can be made win. Betzler and Dorn [1] proved a result that is only one step away from a full dichotomy of this problem for the important class of pure scoring rules in the case of unweighted voters and an unbounded number of candidates: POSSIBLE WINNER is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule with vector (2,1,…,1,0), but is solvable in polynomial time for plurality and veto. We take the final step to a full dichotomy by showing that POSSIBLE WINNER is NP-complete also for the scoring rule with vector (2,1,…,1,0).

Holzer and Holzer [10] proved that the Tantrix™ rotation puzzle problem is NP-complete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the counting version and the unique version of this problem. We prove that the satisfiability problem parsimoniously reduces to the Tantrix™ rotation puzzle problem. In particular, this reduction preserves the uniqueness of the solution, which implies that the unique Tantrix™ rotation puzzle problem is as hard as the unique satisfiability problem, and so is DP-complete under polynomial-time randomized reductions, where DP is the second level of the boolean hierarchy over NP.