Allocating indivisible items among a set of agents is a frequently studied discrete optimization problem. In the setting considered in this work, the agents' preferences over the items are assumed to be identical. We consider a very recent measure for the overall quality of an allocation which does not rely on numerical valuations of the items. Instead, it captures the agents' opinion by a directed acyclic preference graph with vertices representing items. An arc (a,b) in such a graph means that the agents prefer item a over item b. For a given allocation of items the dissatisfaction of an agent is defined as the number of items which the agent does not receive and for which no more preferred item is given to the agent. Our goal is to find an efficient allocation of the items to the agents such that the total dissatisfaction over all agents is minimized. We explore the dichotomy between NP-hard and polynomially solvable instances, depending on properties of the underlying preference graph. While the problem is NP-hard already for three agents even on very restricted graph classes, it is polynomially solvable for two agents on general preference graphs. For an arbitrary number of agents, we derive polynomial-time algorithms for relevant restrictions of the underlying undirected graph. These are trees and, among the graphs of treewidth two, series-parallel graphs and cactus graphs.

Computational social choice (COMSOC) studies principled ways to aggregate conflicting individual preferences into collective decisions. In this paper, we call for an increased effort towards Computational Social Choice: Research Development (COMSOC-R D), a problem-driven research agenda that explicitly aims to design, implement, and test collective decision-making systems in the real world. We articulate the defining features of COMSOC-R D, argue for its value, and discuss various roadblocks and possible solutions.

In this paper we use rank information to construct a measure of preference satisfaction by collective decision rules and to compare the Borda rule with the plurality rule with respect to the inequality their outcome induces. While the construction of our measure of preference satisfaction is formally equivalent to Borda utilities, it turns out that, with respect to equality, the Borda rule is not necessarily superior to plurality rule for the case of more than three alternatives.

We consider the task of allocating indivisible items to agents, when the agents' preferences over the items are identical. The preferences are captured by means of a directed acyclic graph, with vertices representing items and an edge (a,b), meaning that each of the agents prefers item a over item b. The dissatisfaction of an agent is measured by the number of items that the agent does not receive and for which it also does not receive any more preferred item. The aim is to allocate the items to the agents in a fair way, i.e., to minimize the maximum dissatisfaction among the agents. We study the status of computational complexity of that problem and establish the following dichotomy: the problem is NP-hard for the case of at least three agents, even on fairly restricted graphs, but polynomially solvable for two agents. We also provide several polynomial-time results with respect to different underlying graph structures, such as graphs of width at most two and tree-like structures such as stars and matchings. These findings are complemented with fixed parameter tractability results related to path modules and independent set modules. Techniques employed in the paper include bottleneck assignment problem, greedy algorithm, dynamic programming, maximum network flow, and integer linear programming.

We show that Not-All-Equal 3-Sat remains NP-complete when restricted to instances that simultaneously satisfy the following properties: (i) The clauses are given as the disjoint union of k partitions, for any fixed $k \geq 4$, of the variable set into subsets of size 3, and (ii) each pair of distinct clauses shares at most one variable. Property (i) implies that each variable appears in exactly $k$ clauses and each clause consists of exactly 3 unnegated variables. Therewith, we improve upon our earlier result (Darmann and D\"ocker, 2020). Complementing the hardness result for at least $4$ partitions, we show that for $k\leq 3$ the corresponding decision problem is in P. In particular, for $k\in \{1,2\}$, all instances that satisfy Property (i) are nae-satisfiable. By the well-known correspondence between Not-All-Equal 3-Sat} and hypergraph coloring, we obtain the following corollary of our results: For $k\geq 4$, Bicolorability is NP-complete for linear 3-uniform $k$-regular hypergraphs even if the edges are given as a decomposition into $k$ perfect matchings; with the same restrictions, for $k \leq 3$ Bicolorability is in P, and for $k \in \{1,2\}$ all such hypergraphs are bicolorable. Finally, we deduce from a construction in the work by Pilz (Pilz, 2019) that every instance of Positive Planar Not-All-Equal Sat with at least three distinct variables per clause is nae-satisfiable. Hence, when restricted to instances with a planar incidence graph, each of the above variants of Not-All-Equal 3-Sat turns into a trivial decision problem.

Based on data collected in connection with the 2019 parliamentary election in the Austrian region of Styria, we analyze (the use of) different voting rules. Following previous empirical studies in the literature, we use the results of an empirical survey to show that the choice of a voting rule will impact the outcome of an election (if revealed preferences are sincere), at least in certain parts of the social ranking. In addition, we observe a certain desire for voting rules using more fine-grained preference information. In that context, we investigate the degree of consistency in the voters’ declaration of preferences, something of relevance when different voting rules, that require different levels of information, are used. Finally, we discuss the occurrence of strategic behavior that can be observed in the data.

In a hedonic diversity game (HDG) there are two types of agents (red and blue agents) that need to form disjoint coalitions, i.e., subgroups of agents. Each agent’s preferences over the coalitions depend on the relative number of agents of the same type in her coalition. In the special case of a dichotomous hedonic diversity game (DHDG) each agent distinguishes between approved and disapproved fractions only. We aim at outcomes that are stable against agents’ deviations, and at outcomes that maximize social welfare. In particular, we show that the strict core of a DHDG may be empty even in instances with only three agents, while each HDG with two agents has a non-empty strict core. We also provide several computational complexity results for DHDGs with respect to the number of fractions approved per agent. For instance, we prove that deciding whether a DHDG has a non-empty strict core is -complete even when each agent approves of at most three fractions. In addition, we show that deciding whether a DHDG admits a Nash stable outcome is -complete even in restricted settings with only two approved fractions per agent—therewith, improving a result in the literature. For the task of maximizing social welfare, we apply approval scores and Borda scores from voting theory. For DHDGs and approval scores, we draw the sharp separation line between polynomially solvable and -complete cases with respect to the fixed number of approved fractions per agent. We complement these findings with an -completeness result for HDGs under Borda scores.

Several real-world situations can be represented in terms of agents that have preferences over activities in which they may participate. Often, the agents can take part in at most one activity (for instance, since these take place simultaneously), and there are additional constraints on the number of agents that can participate in an activity. In such a setting, we consider the task of assigning agents to activities in a reasonable way. We introduce the simplified group activity selection problem providing a general yet simple model for a broad variety of settings, and start investigating its special case where upper and lower bounds of the groups have to be taken into account. We apply different solution concepts such as envy-freeness and core stability to our setting and provide a computational complexity study for the problem of finding such solutions.

A hedonic diversity game (HDG) is a coalition formation problem, where the set of agents is partitioned into two types of agents (say red and blue agents), and each agent has preferences over the relative number (fraction) of agents of her own type in her coalition. In a dichotomous hedonic diversity game (DHDG) each agent partitions the set of possible fractions into a set of approved and a set of disapproved fractions. The solution concepts for these games considered in the literature so far are concerned with stability notions such as core and Nash stability. We add to the existing literature by providing NP-completeness results for the decision problems whether a DHDG admits (i) a Nash stable outcome and (ii) a strictly core stable outcome respectively, in restricted settings with only two (and three, respectively) approved fractions per agent. In addition, applying approval and Borda scores from voting theory we aim at outcomes that maximize social welfare (i.e., the sum of scores) in (dichotomous) hedonic diversity games. In that context we provide an NP-completeness result for HDGs under the use of Borda scores. For DHDGs with approval scores, we draw the sharp separation line between polynomially solvable and NP-complete cases with respect to the number of approved fractions per agent.

We consider simplified versions of 3-SAT, the variant of the famous SATISFIABILITY PROBLEM where each clause is made up of exactly three distinct literals formed over pairwise distinct variables. More precisely, the focus of this work is laid on MONOTONE 3-SAT, the restriction of 3-SAT to formulas with monotone clauses, where a clause is monotone if it contains only unnegated variables or only negated variables. We prove several hardness results for MONOTONE 3-SAT with respect to a variety of restrictions imposed on the variable appearances. In particular, we show that for any k >= 2, MONOTONE 3-SAT turns out to be NP-complete even if each variable appears exactly k times unnegated and exactly k times negated. Therewith, for MONOTONE 3-SAT with balanced variable appearances we establish a sharp boundary between NP-complete and polynomial time solvable cases. In addition, we prove that for any k >= 5, MONOTONE 3-SAT is NP-complete even if each variable appears exactly k times unnegated and exactly once negated. Further, we prove that the problem remains NP-complete when restricted to instances in which each variable appears either exactly once unnegated and three times negated or the other way around. Thereby, we improve on a result by Darmann et al. (2018) showing NP-completeness for four appearances per variable. Our stronger result also implies that 3-SAT remains NP-complete even if each variable appears exactly three times unnegated and once negated, therewith complementing a result by Berman et al. (2003). (C) 2020 Elsevier B.V. All rights reserved.

We consider a simplified version of Not-All-Equal 3-Sat, a variation of the famous Satisfiability problem, where each clause is made up of exactly three distinct literals and the question is whether there exists a truth assignment such that for each clause at least one literal is set to true and at least one is set to false. We show that Not-All-Equal 3-Sat remains NP-complete if (1) each variable appears exactly four times, (2) there are no negations in the formula, and (3) the formula is linear, i.e., each pair of distinct clauses shares at most one variable. Therewith, we improve upon two results in the literature.

We consider simplified, monotone versions of Not-All-Equal 3-Sat and 3-Sat, variants of the famous Satisfiability Problem where each clause is made up of exactly three distinct literals. We show that Not-All-Equal 3-Sat remains NP-complete even if (1) each variable appears exactly four times, (2) there are no negations in the formula, and (3) the formula is linear, i.e., each pair of distinct clauses shares at most one variable. Concerning 3-Sat we prove several hardness results for monotone formulas with respect to a variety of restrictions imposed on the variable appearances. Monotone 3-Sat is the restriction of 3-Sat to monotone formulas, i.e. to formulas in which each clause contains only unnegated variables or only negated variables, respectively. In particular, we show that, for any $k\geq 5$, Monotone 3-Sat is NP-complete even if each variable appears exactly $k$ times unnegated and exactly once negated. In addition, we show that Monotone 3-Sat is NP-complete even if each variable appears exactly three times unnegated and three times negated, respectively. In fact, we provide a complete analysis of Monotone 3-Sat with exactly six appearances per variable. Further, we prove that the problem remains NP-complete when restricted to instances in which each variable appears either exactly once unnegated and three times negated or the other way around. Thereby, we improve on a result by Darmann et al. [DDD18] showing NP-completeness for four appearances per variable. Our stronger result also implies that 3-Sat remains NP-complete even if each variable appears exactly three times unnegated and once negated, therewith complementing a result by Berman et al. [BKS03].

We analyze the problem of ranking sets of objects based on a ranking over the single objects. In recent years various papers used the sum of individual scores for the objects, in particular Borda scores, to make such comparisons. The advantage of this approach lies in providing a complete ranking of sets of objects and therefore can be seen as an alternative to other methods based on best and/or worst objects. The paper contributes in two ways. On the one hand, we highlight certain drawbacks that arise when using Borda scores in such comparisons. On the other hand, we provide two characterization results for Borda-sum rankings, one for the restricted setting of sets of equal cardinality and one for the general setting which allows for comparisons of sets of unequal cardinality.

We provide empirical evidence for Tullock's claim (Tullock (1967, 1981)) that many of the problems known from social choice literature do not show up in practice. In particular, even though theoretically the use of different voting rules can lead to very different outcomes, there appears to be rather high consensus among voting rules when applied to real-world preference data. In addition, the famous and widely studied problem of majority cycles seems to be of little significance in practice. In this study, based on data collected in an online-survey in connection with the 2015 parliament election in the Austrian federal state of Styria, we confirm these findings to a high degree. Our analysis is based on an approach using a nonparametric bootstrap and includes various forms of evaluative voting (which has recently received increasing attention).

The prominent Boolean formula satisfiability problem SAT is known to be [Formula: see text]-complete even for very restricted variants such as 3-SAT, Monotone 3-SAT, or Planar 3-SAT, or instances with bounded variable appearance. We settle the computational complexity status for two variants with bounded variable appearance: We show that Planar Monotone Sat — the variant of Monotone Sat in which the incidence graph is required to be planar — is [Formula: see text]-complete even if each clause consists of at most three distinct literals and each variable appears exactly three times, and that Monotone Sat is [Formula: see text]-complete even if each clause consists of three distinct literals and each variable appears exactly four times in the formula. The latter confirms a conjecture stated in scribe notes [7] of an MIT lecture by Eric Demaine. In addition, we provide hardness results with respect to bounded variable appearances for two variants of Planar Monotone Sat.

We consider the situation in which group activities need to be organized for a set of agents when each agent can take part in at most one activity. The agents’ preferences depend both on the activity and the number of participants in that activity. In particular, the preferences are given by means of strict orders over pairs “(activity, group size)”, including the possibility “do nothing”. Our goal will be to assign agents to activities on basis of their preferences, the minimum requirement being that no agent prefers doing nothing, i.e., not taking part in any activity at all. Taking a social choice perspective, we aim at establishing such an assignment by two approaches. On the one hand, we use k-approval and Borda scores, and we apply the Condorcet criterion on the other hand. We analyze the computational complexity involved in finding a desired assignment, with focus on two natural special cases of agents’ preferences which allow for some positive complexity results.

We consider a setting where one has to organize one or several group activities for a set of agents. Each agent will participate in at most one activity, and her preferences over activities depend on the number of participants in the activity. The goal is to assign agents to activities based on their preferences in a way that is socially optimal and/or stable. We put forward a general model for this setting, which is a natural generalization of anonymous hedonic games. We then focus on a special case of our model where agents' preferences are binary, i.e., each agent classifies all pairs of the form '(activity, group size)' into ones that are acceptable and ones that are not. We formulate several solution concepts for this scenario, and study them from the computational point of view, providing hardness results for the general case as well as efficient algorithms for settings where agents' preferences satisfy certain natural constraints.

We use preference data from the 2015 parliament election in the Austrian federal state of Styria to analyze different voting rules. An exit poll right after the election collected data on ordinal and cardinal preferences from approximately 1000 actual voters. Our analysis is threefold. First, we determine the hypothetical social outcomes under different voting rules; second, we investigate the stability of the outcomes under those rules. Finally, we provide a categorization of different types of parties and analyze the impact of certain voting rules (Plurality Rule, Plurality Run Off, Hare System, Condorcet Method, Approval Voting, Borda Rule, Evaluative Voting, and Majority Judgment) on the performances of parties in those scenarios.

In this work we address a game theoretic variant of the shortest path problem, in which two decision makers (players) move together along the edges of a graph from a given starting vertex to a given destination. The two players take turns in deciding in each vertex which edge to traverse next. The decider in each vertex also has to pay the cost of the chosen edge. We want to determine the path where each player minimizes its costs taking into account that also the other player acts in a selfish and rational way. Such a solution is a subgame perfect equilibrium and can be determined by backward induction in the game tree of the associated finite game in extensive form. We show that the decision problem associated with such a path is PSPACE-complete even for bipartite graphs both for the directed and the undirected version. The latter result is a surprising deviation from the complexity status of the closely related game Geography. On the other hand, we can give polynomial time algorithms for directed acyclic graphs and for cactus graphs even in the undirected case. The latter is based on a decomposition of the graph into components and their resolution by a number of fairly involved dynamic programming arrays. Finally, we give some arguments about closing the gap of the complexity status for graphs of bounded treewidth.