In parliamentary elections, parties compete for a limited, typically fixed number of seats. Most parliaments are assembled using apportionment methods that distribute the seats based on the parties' vote counts. Common apportionment methods include divisor sequence methods (like D'Hondt or Sainte-Laguë), the largest-remainder method, and first-past-the-post. In many countries, an electoral threshold is implemented to prevent very small parties from entering the parliament. Further, several countries have apportionment systems that incorporate multiple districts. We study how computationally hard it is to change the election outcome (i.e., to increase or limit the influence of a distinguished party) by convincing a limited number of voters to change their vote. We refer to these bribery-style attacks as \emph{strategic campaigns} and study the corresponding problems in terms of their computational (both classical and parameterized) complexity. We also run extensive experiments on real-world election data and study the effectiveness of optimal campaigns, in particular as opposed to using heuristic bribing strategies and with respect to the influence of the threshold and the influence of the number of districts. For apportionment elections with threshold, finally, we propose -- as an alternative to the standard top-choice mode -- the second-chance mode where voters of parties below the threshold receive a second chance to vote for another party, and we establish computational complexity results also in this setting.

Weighted voting games are a popular class of coalitional games that are widely used to model real-life situations of decision-making. They can be applied, for instance, to analyze legislative processes in parliaments or voting in corporate structures. Various ways of tampering with these games have been studied, among them merging or splitting players, fiddling with the quota, and controlling weighted voting games by adding or deleting players. While the complexity of control by adding players to such games so as to change or maintain a given player’s power has been recently settled, the complexity of control by deleting players from such games (with the same goals) remained open. We show that when the players’ power is measured by the probabilistic Penrose–Banzhaf index, some of these problems are complete for NPPP—the class of problems solvable by NP machines equipped with a PP (“probabilistic polynomial time”) oracle. Our results optimally improve the currently known lower bounds of hardness for much smaller complexity classes, thus providing protection against SAT-solving techniques in practical applications.

Apportionment methods allocate a fixed number of seats in a parliament to parties based on their vote counts. In many countries, parliamentary elections are organized by first holding separate elections in several districts and then putting the single results together. We call such elections multidistrict apportionment elections. Moreover, many countries have an additional general electoral threshold, i.e., a minimum number of votes a party must receive to win any seats in a parliament. For such methods, we study the complexity of bribery problems where an external agent seeks to increase the number of seats for a distinguished party by bribing voters within a given budget. Specifically, we investigate how adding either a general electoral threshold, or district limits for the budget, or both influences the complexity of bribery. District limits—which the external agent must adhere to while still respecting the overall budget—denote a maximum budget for each district, each to be spent for changing the votes only in that district. We show that adding a general electoral threshold, with or without district limits, makes the problems NP-complete for the largest-remainder method and all divisor sequence methods (including the prominent D’Hondt and Sainte-Laguë apportionment methods). We also study parameterized complexity and domain restrictions of these problems.

Over the past few years, the concept of quantum routers and their usefulness in quantum communication networks (e.g., in the BB84 protocol [5]) has been popularized in quantum information theory. While quite some work has been done on the theoretical implementation of quantum routers using multiplexing, most of it is constrained to the bipartite case [2, 9]. We extend this setup to quantum routers used in multipartite conference key agreement protocols so as to distribute a secret key among N parties. We formalize the general quantum entanglement matching problem, and show it's NP-completeness. We then study special cases for which we found efficient algorithms. Finally, we consider the weighted case where the weights represent the qubit ages.

We survey the notion of control in various areas of computational social choice (COMSOC) such as voting, fair allocation, cooperative game theory, matching under preferences, and group identification. In all these scenarios, control can be exerted, for instance, by adding or deleting agents with the goal of influencing the outcome. We conclude by briefly covering control in some other COMSOC areas including participatory budgeting, judgment aggregation, and opinion diffusion.

We study fundamental connections between coalition formation games and clustering, illustrating the cross-disciplinary relevance of these concepts. We focus on graphical hedonic games where agents' preferences are compactly represented by a friendship graph and an enemy graph. In the context of clustering, friendship relations naturally align with data point similarities, whereas enmity corresponds to dissimilarities. We consider two stability notions based on single-agent deviations: local popularity and local stability. Exploring these concepts from an algorithmic viewpoint, we design efficient mechanisms for finding locally stable or locally popular partitions. Besides gaining theoretical insight into the computational complexity of these problems, we perform simulations that demonstrate how our algorithms can be successfully applied in clustering and community detection. Our findings highlight the interplay between coalition formation games and data-driven clustering techniques, offering fresh perspectives and applications in both areas.

Graph-restricted weighted voting games generalize weighted voting games, a well-studied class of succinct simple games, by embedding them into a communication structure: a graph whose vertices are the players some of which are connected by edges. In such games, only sufficiently connected coalitions are taken into consideration for calculating the players' power indices. Focusing on the probabilistic Penrose-Banzhaf index (which Dubey and Shapley proposed in 1979 as an alternative to the normalized Penrose-Banzhaf index) and the Shapley-Shubik index, we study control of these games by an agent who can add edges to or delete edges from the given graph. We determine upper and lower bounds on how much such control actions can change a distinguished player's power and we study the computational complexity of the related problems.

Weighted voting games are a well-known and useful class of succinctly representable simple games that have many real-world applications, e.g., to model collective decision-making in legislative bodies or shareholder voting. Among the structural control types being analyzing, one is control by adding players to weighted voting games, so as to either change or to maintain a player’s power in the sense of the (probabilistic) Penrose–Banzhaf power index or the Shapley–Shubik power index. For the problems related to this control, the best known lower bound is PP-hardness, where PP is “probabilistic polynomial time,” and the best known upper bound is the class NP, i.e., the class NP with a PP oracle. We optimally raise this lower bound by showing NPPP-hardness of all these problems for the Penrose–Banzhaf and the Shapley–Shubik indices, thus establishing completeness for them in that class. Our proof technique may turn out to be useful for solving other open problems related to weighted voting games with such a complexity gap as well.

We study the social welfare maximization problem in the high-multiplicity setting where agents and/or items are available in multiple types, provided that the numbers of types are small. We focus on the egalitarian and Nash social welfare maximization problems, and show that they are NP-hard even when the number of item types is a constant. Furthermore, we present two polynomial-time approximation schemes (PTAS), one for egalitarian social welfare with two item types, and one for Nash social welfare with any constant number of agent types. The first PTAS can be applied to the unrelated machine scheduling problem, thus partially solving an open question raised by Jansen and Maack in 2019. The second PTAS significantly improves upon the existing PTAS for identical agents.

Abstract We consider average- and min-based altruistic hedonic games and study the problem of verifying popular and strictly popular coalition structures. While strict popularity verification has been shown to be coNP-complete in min-based altruistic hedonic games, this problem has been open for equal- and altruistic-treatment average-based altruistic hedonic games. We solve these two open cases of strict popularity verification and then provide the first complexity results for popularity verification in (average- and min-based) altruistic hedonic games, where we cover all three degrees of altruism.

Abstract Frei et al. (J. Comput. Syst. Sci. 123, 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for $$\varvec{\Theta _{2}^{\textrm{P}}}$$ Θ 2 P , the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters $$\varvec{\alpha }$$ α (the independence number), $$\varvec{\beta }$$ β (the vertex cover number), $$\varvec{\omega }$$ ω (the clique number), and $$\varvec{\chi }$$ χ (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.

Graph-restricted weighted voting games generalize weighted voting games, a well-studied class of succinct simple games, by embedding them into a communication structure: a graph whose vertices are the players some of which are connected by edges. In such games, only connected coalitions are taken into consideration for calculating the players’ power indices. We focus on the probabilistic Penrose–Banzhaf index [5] and the Shapley–Shubik index [18] and study the computational complexity of manipulating these games by an external agent who can add edges to or delete edges from the graph. For the problems modeling such scenarios, we raise some of the lower bounds obtained by Kaczmarek and Rothe [9] from NP- or DP-hardness to PP-hardness, where PP is probabilistic polynomial time. We also solve one of their open problems by showing that it is a coNP-hard problem to maintain the Shapley–Shubik index of a given player in a graph-restricted weighted voting game when edges are deleted.

Fair allocation of indivisible goods is a central topic in many AI applications. Unfortunately, the corresponding problems are known to be NP-hard for many fairness concepts, so unless P = NP, exact polynomial-time algorithms cannot exist for them. In practical applications, however, it would be highly desirable to find exact solutions as quickly as possible. This motivates the study of algorithms that—even though they only run in exponential time—are as fast as possible and exactly solve such problems. We present known complexity results for them and give a survey of important techniques for designing such algorithms, mainly focusing on four common fairness notions: max-min fairness, maximin share, maximizing Nash social welfare, and envy-freeness. We also highlight the most challenging open problems for future work.

Hedonic games are coalition formation games in which players have preferences over the coalitions they can join. For a long time, all models of representing hedonic games were based upon selfish players only. Among the known ways of representing hedonic games compactly, we focus on friend-oriented hedonic games and propose a novel model for them that takes into account not only the players’ own preferences but also their friends’ preferences. Depending on the order in which players look at their own or their friends’ preferences, we distinguish three degrees of altruism: selfish-first, equal-treatment, and altruistic-treatment preferences. We study both the axiomatic properties of these games and the computational complexity of problems related to various common stability concepts.

Graph parameters such as the clique number and the chromatic number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs in terms of their computational complexity. We show for various central graph parameters that deciding the stability of a given graph is complete for Θ2p, a well-known complexity class in the second level of the polynomial hierarchy.

Game theory is typically used to model the interaction among (software) agents in multiagent systems and, therefore, is a key topic at leading AI conferences. Game-theoretic models, however, are often based on the assumption that agents are perfectly rational and narrowly selfish and are interested only in maximizing their own gains, no matter what the costs to the other agents are. This summary paper presents various ways of introducing certain notions of altruism into existing game-theoretic models in both noncooperative and cooperative games, in the hope that simulating altruistic behavior in AI systems will make AI better suit real-world applications—and thus may make the real world a better place.

Abstract We investigate the computational complexity of electoral control in elections. Electoral control describes the scenario where the election chair seeks to alter the outcome of the election by structural changes such as adding, deleting, or replacing either candidates or voters. Such control actions have been studied in the literature for a lot of prominent voting rules. We complement those results by solving several open cases for Copeland $$^{\alpha }$$ α , maximin,k-veto, plurality with runoff, veto with runoff, Condorcet, fallback, range voting, and normalized range voting.

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.

Expressing incomplete knowledge in abstract argumentation frameworks (AFs) through incomplete AFs has recently received noticeable attention. However, algorithmic aspects of deciding acceptance in incomplete AFs are still under-developed. We address this current shortcoming by developing algorithms for NP-hard and coNP-hard variants of acceptance problems over incomplete AFs via harnessing Boolean satisfiability (SAT) solvers. Focusing on nonempty conflict-free or admissible sets and on stable extensions, we also provide new complexity results for a refined variant of skeptical acceptance in incomplete AFs, ranging from polynomial-time computability to hardness for the second level of the polynomial hierarchy. Furthermore, central to the proposed SAT-based counterexample-guided abstraction refinement approach for the second-level problem variants, we establish conditions for redundant atomic changes to incomplete AFs from the perspective of preserving extensions. We show empirically that the resulting SAT-based approach for incomplete AFs scales at least as well as existing SAT-based approaches to deciding acceptance in AFs.

One of the key topics of computational social choice is electoral control, which models certain ways of how an election chair can seek to influence the outcome of elections via structural changes such as adding, deleting, or partitioning either candidates or voters. Faliszewski and Rothe [13] have surveyed the rich literature on control, giving an overview of previous results on the complexity of the associated problems for the most important voting rules. Among those, only a few results were known for two quite prominent voting rules: Borda Count and maximin voting (a.k.a. the Simpson–Kramer rule). Neveling and Rothe [26, 25] recently settled the remaining open cases for Borda. In this paper, we solve all remaining open cases for the complexity of control in maximin elections all of which concern control by partition of either candidates or voters.

This work focuses on generalizing the existence problems for extensions in abstract argumentation to incomplete argumentation frameworks. In this extended model, incomplete or conflicting knowledge about the state of the arguments and attacks are allowed. We propose possible and necessary variations of the existence and nonemptiness problems, originally defined for (complete) argumentation frameworks, to extend these problems to incomplete argumentation frameworks. While the computational complexity of existence problems is already known for the standard model, we provide a full analysis of the complexity for incomplete argumentation frameworks using the most prominent semantics, namely, the conflict-free, admissible, complete, grounded, preferred, and stable semantics. We show that the complexity rises from NP-completeness to ∏p2-completeness for most "necessary" problem variants when uncertainty is allowed.

In fair division of indivisible goods, finding an allocation that satisfies fairness and efficiency simultaneously is highly desired but computationally hard. We solve this problem approximately in polynomial time by modeling it as a bi-criteria optimization problem that can be solved efficiently by determining an approximate Pareto set of bounded size. We focus on two criteria: max-min fairness and utilitarian efficiency, and study this problem for the setting when there are only a few item types or a few agent types. We show in both cases that one can construct an approximate Pareto set in time polynomial in the input size, either by designing a dynamic programming scheme, or a linear-programming algorithm. Our techniques strengthen known methods and can be potentially applied to other notions of fairness and efficiency as well.

Nguyen et al. [2016] introduced altruistic hedonic games in which agents’ utilities depend not only on their own preferences but also on those of their friends in the same coalition. We propose to extend their model to coalition formation games in general, considering also the friends in other coalitions. Comparing the two models, we argue that excluding some friends from the altruistic behavior of an agent is a major disadvantage that comes with the restriction to hedonic games. After introducing our model, we additionally study some common stability notions and provide a computational analysis of the associated verification and existence problems.

We study a generalized version of the load balancing problem on unrelated machines with cost constraints: Given a set of m machines (of certain types) and a set of n jobs, each job j processed on machine i requires p_{i,j} time units and incurs a cost c_{i,j}, and the goal is to find a schedule of jobs to machines, which is defined as an ordered partition of n jobs into m disjoint subsets, in such a way that some objective function of the vector of the completion times of the machines is optimized, subject to the constraint that the total costs by the schedule must be within a given budget B. Motivated by recent results from the literature, our focus is on the case when the number of machine types is a fixed constant and we develop a bi-criteria approximation scheme for the studied problem. Our result generalizes several known results for certain special cases, such as the case with identical machines, or the case with a constant number of machines with cost constraints. Building on the elegant technique recently proposed by Jansen and Maack [K. Jansen and M. Maack, 2019], we construct a more general approach that can be used to derive approximation schemes to a wider class of load balancing problems with constraints.

We propose a new representation setting for hedonic games, where each agent partitions the set of other agents into friends, enemies, and neutral agents, with friends and enemies being ranked. Under the assumption that preferences are monotonic (respectively, antimonotonic) with respect to the addition of friends (respectively, enemies), we propose a bipolar extension of the responsive extension principle, and use this principle to derive the (partial) preferences of agents over coalitions. Then, for a number of solution concepts, we characterize partitions that necessarily or possibly satisfy them, and we study the related problems in terms of their complexity.

Borda Count is one of the earliest and most important voting rules. Going far beyond voting, we summarize recent advances related to Borda in computational social choice and, more generally, in collective decision making. We first present a variety of well known attacks modeling strategic behavior in voting—including manipulation, control, and bribery—and discuss how resistant Borda is to them in terms of computational complexity. We then describe how Borda can be used to maximize social welfare when indivisible goods are to be allocated to agents with ordinal preferences. Finally, we illustrate the use of Borda in forming coalitions of players in a certain type of hedonic game. All these approaches are central to applications in artificial intelligence.

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

A key issue in cooperative game theory is coalitional stability, usually captured by the notion of the core---the set of outcomes that are resistant to group deviations. However, some coalitional games have empty cores, and any outcome in such a game is unstable. We investigate the possibility of stabilizing a coalitional game by using subsidies. We consider scenarios where an external party that is interested in having the players work together offers a supplemental payment to the grand coalition, or, more generally, a particular coalition structure. This payment is conditional on players not deviating from this coalition structure, and may be divided among the players in any way they wish. We define the cost of stability as the minimum external payment that stabilizes the game. We provide tight bounds on the cost of stability, both for games where the coalitional values are nonnegative (profit-sharing games) and for games where the coalitional values are nonpositive (cost-sharing games), under natural assumptions on the characteristic function, such as superadditivity, anonymity, or both. We also investigate the relationship between the cost of stability and several variants of the least core. Finally, we study the computational complexity of problems related to the cost of stability, with a focus on weighted voting games.

Standard (offline) control scenarios in elections (such as adding, deleting, or partitioning either voters or candidates) have been studied for many voting systems, natural and less natural ones, and the related control problems have been classified in terms of their complexity. However, for one of the most important natural voting systems, the Borda Count, only a few such complexity results are known. We reduce the number of missing cases by pinpointing the complexity of three control scenarios for Borda elections, including some that arguably are among the practically most relevant ones. We also study online candidate control, an interesting dynamical, partial-information model due to Hemaspaandra et al. (2012a), who mainly focused on general complexity bounds by constructing artificial voting systems—only recently they succeeded in classifying four problems of online candidate control for one natural voting system: sequential plurality (Hemaspaandra et al. 2016). We settle the complexity of another four natural cases: constructive and destructive online control by deleting and adding candidates in sequential Borda elections.

Control by partition refers to situations where an election chair seeks to influence the outcome of an election by partitioning either the candidates or the voters into two groups, thus creating two first-round subelections that determine who will take part in a final round. The model of partition-of-voters control attacks is remotely related to “gerrymandering” (maliciously resizing election districts). While the complexity of control by partition (and other control actions) has been studied thoroughly for many voting systems, there are no such results known for the important veto and maximin voting systems. We settle the complexity of control by partition for veto in a broad variety of models and for maximin with respect to destructive control by partition of candidates. We also observe that a reduction from the literature [8] showing the parameterized complexity of control by adding candidates to plurality elections, parameterized by the number of voters, is technically flawed by giving a counterexample, and we show how this reduction can be fixed.

Inspired by the study of control scenarios in elections and complementing manipulation and bribery settings in cooperative games with transferable utility, we introduce the notion of structural control in weighted voting games. We model two types of influence, adding players to and deleting players from a game, with goals such as increasing a given player's Shapley-Shubik or probabilistic Penrose-Banzhaf index in relation to the original game. We study the computational complexity of the problems of whether such structural changes can achieve the desired effect.

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.

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Relatedly, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. For the problems of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley--Shubik and the normalized Banzhaf index, merely NP-hardness lower bounds are known, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," a class considered to be by far a larger class than NP. For both power indices, we provide matching upper bounds for beneficial merging and, whenever the new players' weights are given, also for beneficial splitting, thus resolving previous conjectures in the affirmative. Relatedly, we consider the beneficial annexation problem, asking whether a single player can increase her power by taking over other players' weights. It is known that annexation is never disadvantageous for the Shapley--Shubik index, and that beneficial annexation is NP-hard for the normalized Banzhaf index. We show that annexation is never disadvantageous for the probabilistic Banzhaf index either, and for both the Shapley--Shubik index and the probabilistic Banzhaf index we show that it is NP-complete to decide whether annexing another player is advantageous. Moreover, we propose a general framework for merging and splitting that can be applied to different classes and representations of games.

Peer reviewing is the key ingredient of evaluating the quality of scientific work. Based on the review scores assigned by the individual reviewers to the submissions, program committees of conferences and journal editors decide which papers to accept for publication and which to reject. However, some reviewers may be more rigorous than others, they may be biased one way or the other, and they often have highly subjective preferences over the papers they review. Moreover, each reviewer usually has only a very local view, as he or she evaluates only a small fraction of the submissions. Despite all these shortcomings, the review scores obtained need to be aggregrated in order to globally rank all submissions and to make the acceptance/rejection decision. A common method is to simply take the average of each submission's review scores, possibly weighted by the reviewers' confidence levels. Unfortunately, the global ranking thus produced often suffers a certain unfairness, as the reviewers' biases and limitations are not taken into account. We propose a method for calibrating the scores of reviewers that are potentially biased and blindfolded by having only partial information. Our method uses a maximum likelihood estimator, which estimates both the bias of each individual reviewer and the unknown "ideal" score of each submission. This yields a quadratic program whose solution transforms the individual review scores into calibrated, globally comparable scores. We argue why our method results in a fairer and more reasonable global ranking than simply taking the average of scores. To show its usefulness, we test our method empirically using real-world data.

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).

The Banzhaf power index is a prominent measure of a player's influence for coalition formation in weighted voting games, an important class of simple coalitional games that are fully expressive but compactly representable. For the normalized Banzhaf index, Aziz and Paterson [1] show that it is NP-hard to decide whether merging any coalition of players is beneficial, and that in unanimity games, merging is always disadvantageous, whereas splitting is always advantageous. We show that for the probabilistic Banzhaf index (which is considered more natural than the normalized Banzhaf index), the merging problem is in P for coalitions of size two, and is NP-hard for coalitions of size at least three. We also prove a corresponding result for the splitting problem. In unanimity games and for the probabilistic Banzhaf index (in strong contrast with the results for the normalized Banzhaf index), we show that splitting is always disadvantageous or neutral, whereas merging is neutral for size-two coalitions, yet advantageous for coalitions of size at least three.

Much work has been devoted, during the past twenty years, to using complexity to protect elections from manipulation and control. Many results have been obtained showing NP-hardness shields, and recently there has been much focus on whether such worst-case hardness protections can be bypassed by frequently correct heuristics or by approximations. This paper takes a very different approach: We argue that when electorates follow the canonical political science model of societal preferences the complexity shield never existed in the first place. In particular, we show that for electorates having single-peaked preferences, many existing NP-hardness results on manipulation and control evaporate.

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.

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate's victory. An election system in which an agent can sometimes affect the result and it can be determined in polynomial time on which inputs the agent can succeed is said to be vulnerable to the given type of control. An election system in which an agent can sometimes affect the result, yet in which it is NP-hard to recognize the inputs on which the agent can succeed, is said to be resistant to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. This paper studies a parameterized version of Copeland voting, denoted by Copeland^\alpha, where the parameter \alpha is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive control scenario, we determine which of resistance or vulnerability holds for Copeland^\alpha for each rational \alpha, 0 \leq \alpha \leq 1. In particular, we prove that Copeland^{0.5}, the system commonly referred to as ``Copeland voting,'' provides full resistance to constructive control, and we prove the same for Copeland^\alpha, for all rational \alpha, 0 < \alpha < 1. Among systems with a polynomial-time winner problem, Copeland voting is the first natural election system proven to have full resistance to constructive control. In addition, we prove that both Copeland^0 and Copeland^1 (interestingly, Copeland^1 is an election system developed by the thirteenth-century mystic Llull) are resistant to all standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational \alpha, 0 \leq \alpha \leq 1, Copeland^\alpha voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^\alpha. We also study Copeland^\alpha elections under more flexible models such as microbribery and extended control, we integrate the potential irrationality of voter preferences into many of our results, and we prove our results in both the unique-winner model and the nonunique-winner model. Our vulnerability results for microbribery are proven via a novel technique involving min-cost network flow.

Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan (1997) initiated the search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard, and that a number of closely related problems are SPP-hard. The present paper studies whether their UP-hardness result itself can be improved to SPP-hardness. We show that their UP-hardness result cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard. We also raise their general lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism.

We establish a condition necessary and sufficient for the existence of one-way permutations: One-way permutations exist if and only if there exist total one-one one-way functions whose range is P-rankable.

Rabi and Sherman presented novel digital signature and unauthenticated secret-key agreement protocols, developed by themselves and by Rivest and Sherman. These protocols use strong, total, commutative (in the case of multiparty secret-key agreement), associative one-way functions as their key building blocks. Although Rabi and Sherman did prove that associative one-way functions exist if P≠NP, they left as an open question whether any natural complexity-theoretic assumption is sufficient to ensure the existence of strong, total, commutative, associative one-way functions. In this paper, we prove that if P≠NP then strong, total, commutative, associative one-way functions exist.

Bodlaender, Thilikos, and Yamazaki (1997) investigate the computational complexity of the problem of whether the Minimum Degree Greedy Algorithm can approximate a maximum independent set of a graph within a constant factor of r, for fixed rational r ⩾ 1. They denote this problem by Sr and prove that for each rational r ⩾ 1, Sr is coNP-hard. They also provide a PNP upper bound of Sr, leaving open the question of whether this gap between the upper and the lower bound of Sr can be closed. For the special case of r = 1, they show that S1 is even DP-hard, again leaving open the question of whether S1 can be shown to be complete for DP or some larger class such as PNP. In this note, we completely solve all the questions left open by Bodlaender et al. Our main result is that for each rational r ⩾ 1, Sr is complete for P∥NP, the class of sets solvable via parallel access to NP.

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL2, yet their join is in EL2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL2 and prove that EL2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL2 lower bounds for certain notions generalizing P-selectivity, which may be regarded as an interesting result in its own right.

In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters' preferences becomes a Condorcet winner—a candidate who beats all other candidates in pairwise majority-rule elections. Bartholdi, Tovey, and Trick provided a lower bound—NP-hardness—on the computational complexity of determining the election winner in Carroll's system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carroll's system is complete for parallel access to NP, that is, it is complete for Theta_ 2 p for which it becomes the most natural complete problem known. It follows that determining the winner in Carroll's elections is not NP-complete unless the polynomial hierarchy collapses.

It is known that for any class $\tweak{\cal C}$ closed under union and intersection, the Boolean closure of ${\cal C}$, the Boolean hierarchy over $\tweak{\cal C}$, and the symmetric difference hierarchy over $\tweak{\cal C}$ all are equal. We prove that these equalities hold for any complexity class closed under intersection; in particular, they thus hold for unambiguous polynomial time (UP). In contrast to the NP case, we prove that the Hausdorff hierarchy and the nested difference hierarchy over UP both fail to capture the Boolean closure of UP in some relativized worlds. Karp and Lipton proved that if nondeterministic polynomial time has sparse Turing-complete sets, then the polynomial hierarchy collapses. We establish the first consequences from the assumption that unambiguous polynomial time has sparse Turing-complete sets: (a) $\up \seq \mbox{Low}_2$, where $\mbox{Low}_2$ is the second level of the low hierarchy, and (b) each level of the unambiguous polynomial hierarchy is contained one level lower in the promise unambiguous polynomial hierarchy than is otherwise known to be the case.