We study the computational complexity of explaining preference data through Boolean attribute models (BAMs), motivated by extensive research involving attribute models and their promise in understanding preference structure and enabling more efficient decision-making processes. In a BAM, each alternative possesses a subset of binary attributes, each voter cares about a subset of attributes, and voters prefer alternatives with more of their desired attributes. In the BAM problem, we are given a preference profile and want to know whether there is a k-attribute model explaining the profile. We establish a complexity dichotomy for the number of attributes k: BAM is linear-time solvable for k≤2 but NP-complete for k≥3. The problem remains hard even when preference orders have length two. On the positive side, BAM becomes fixed-parameter tractable when parameterized by the number of alternatives m. For the special case of two voters, we provide a linear-time algorithm. We also analyze variants where partial information is given: When voter preferences over attributes are known (BAM With Cares) or when alternative attributes are specified (BAM With Has), showing that for most parameters BAM With Cares is more difficult whereas BAM With Has is more tractable except for being NP-hard even for one voter.

We propose a class of cooperative games, called PARTITIONED COMBINATORIAL OPTIMIZATION GAMEs (PCOGs). The input of PCOG consists of a set of agents and a combinatorial structure (typically a graph) with a fixed optimization goal on this structure (e.g., finding a minimum dominating set on a graph) such that the structure is divided among the agents. The value of each coalition of agents is derived from the optimal solution for the part of the structure possessed by the coalition. We study two fundamental questions related to the core: CORE STABILITY VERIFICATION and CORE STABILITY EXISTENCE. We analyze the algorithmic complexity of both questions for four classic graph optimization tasks: minimum vertex cover, minimum dominating set, minimum spanning tree, and maximum matching.

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 investigate multi-organizational scheduling problems, building upon the framework introduced by Pascual et al. in 2009. In this setting, multiple organizations each own a set of identical machines and sequential jobs with distinct processing times. The challenge lies in optimally assigning jobs across organizations’ machines to minimize the overall makespan while ensuring no organization’s performance deteriorates. To formalize this fairness constraint, we introduce individual rationality, a game-theoretic concept that guarantees each organization benefits from participation. Our analysis reveals that finding an individually rational schedule with minimum makespan is ΘP2-hard, placing it in a complexity class strictly harder than both NP and coNP. We further extend the model by considering an alternative objective: minimizing the sum of job completion times, both within individual organizations and across the entire system. The corresponding decision variant proves to be NP-complete. Through comprehensive parameterized complexity analysis of both problems, we provide new insights into these computationally challenging multi-organizational scheduling scenarios.

We investigate congested assignment problems where agents have preferences over both resources and their associated congestion levels. These agents are \emph{averse} towards congestion, i.e., consistently preferring lower congestion for identical resources. Such scenarios are ubiquitous across domains including traffic management and school choice, where fair resource allocation is essential. We focus on the concept of \emph{competitiveness}, recently introduced by Bogomolnaia and Moulin [6], and contribute a polynomial-time algorithm that determines competitiveness, resolving their open question. Additionally, we explore two optimization variants of congested assignments by examining the problem of finding envy-free or maximally competitive assignments that guarantee a certain amount of social welfare for every agent, termed \emph{top-guarantees} [6]. While we prove that both problems are NP-hard, we develop parameterized algorithms with respect to the number of agents or resources.

We study variants of the Optimal Refugee Resettlement problem where a set F of refugee families need to be allocated to a set P of possible places of resettlement in a feasible and optimal way. Feasibility issues emerge from the assumption that each family requires certain services (such as accommodation, school seats, or medical assistance), while there is an upper and, possibly, a lower quota on the number of service units provided at a given place. Besides studying the problem of finding a feasible assignment, we also investigate two natural optimization variants. In the first one, we allow families to express preferences over P, and we aim for a Pareto-optimal assignment. In a more general setting, families can attribute utilities to each place in P, and the task is to find a feasible assignment with maximum total utilities. We study the computational complexity of all three variants in a multivariate fashion using the framework of parameterized complexity. We provide fixed-parameter algorithms for a handful of natural parameterizations, and complement these tractable cases with tight intractability results.

We introduce a multi-winner reconfiguration model to examine how to transition between subsets of alternatives (aka. committees) through a sequence of minor yet impactful modifications, called reconfiguration path. We analyze this model under four approval-based voting rules: Chamberlin-Courant (CC), Proportional Approval Voting (PAV), Approval Voting (AV), and Satisfaction Approval Voting (SAV). The problem exhibits computational intractability for CC and PAV, and polynomial solvability for AV and SAV. We provide a detailed multivariate complexity analysis for CC and PAV, demonstrating that although the problem remains challenging in many scenarios, there are specific cases that allow for efficient parameterized algorithms.

In many applications, we want to influence the decisions of independent agents by designing incentives for their actions. We revisit a fundamental problem in this area, called GAME IMPLEMENTATION: Given a game in standard form and a set of desired strategies, can we design a set of payment promises such that if the players take the payment promises into account, then all undominated strategies are desired? Furthermore, we aim to minimize the cost, that is, the worst-case amount of payments. We study the tractability of computing such payment promises and determine more closely what obstructions we may have to overcome in doing so. We show that GAME IMPLEMENTATION is NP-hard even for two players, solving in particular a long-standing open question and suggesting more restrictions are necessary to obtain tractability results. We thus study the regime in which players have only a small constant number of strategies and obtain the following. First, this case remains NP-hard even if each player’s utility depends only on three others. Second, we repair a flawed efficient algorithm for the case of both small number of strategies and small number of players. Among further results, we characterize sets of desired strategies that can be implemented at zero cost as a generalization of Nash equilibria.

We investigate winner determination for two popular proportional representation systems: the Monroe and Chamberlin-Courant (abbrv. CC) systems. Our study focuses on (nearly) single-peaked resp. single-crossing preferences. We show that for single-crossing approval preferences, winner determination of the Monroe rule is polynomial, and for both rules, winner determination mostly admits FPT algorithms with respect to the number of voters to delete to obtain single-peaked or single-crossing preferences. Our results answer some complexity questions from the literature [19, 29, 22].

We study four NP-hard optimal seat arrangement problems which each have as input a set of n agents, where each agent has cardinal preferences over other agents, and an n-vertex undirected graph (called the seat graph). The task is to assign each agent to a distinct vertex in the seat graph such that either the sum of utilities or the minimum utility is maximized, or it is envy-free or exchange-stable. Aiming at identifying hard and easy cases, we extensively study the algorithmic complexity of the four problems by looking into natural graph classes for the seat graph (e.g., paths, cycles, stars, or matchings), problem-specific parameters (e.g., the number of non-isolated vertices in the seat graph or the maximum number of agents towards whom an agent has non-zero preferences), and preference structures (e.g., non-negative or symmetric preferences). For strict preferences and seat graphs with disjoint edges and isolated vertices, we correct an error in the literature and show that finding an envy-free arrangement remains NP-hard in this case.

Participatory budgeting (PB) is a democratic process where citizens jointly decide on how to allocate public funds to indivisible projects. In this work, we focus on PB processes where citizens may provide additional money to projects they want to see funded. We introduce a formal framework for this kind of PB with donations. Our framework also allows for diversity constraints, meaning that each project belongs to one or more types, and there are lower and upper bounds on the number of projects of the same type that can be funded. We propose three general classes of methods for aggregating the citizens’ preferences in the presence of donations and analyze their axiomatic properties. Furthermore, we investigate the computational complexity of determining the outcome of a PB process with donations and of finding a citizen’s optimal donation strategy.

We investigate the Euclidean $d$-Dimensional Stable Roommates problem, which asks whether a given set~$V$ of $d \cdot n$ points from the 2-dimensional Euclidean space can be partitioned into $n$ disjoint (unordered) subsets $Π=\{V_1,\ldots,V_{n}\}$ with $|V_i|=d$ for each $V_i\in Π$ such that $Π$ is stable. Here, stability means that no point subset $W\subseteq V$ is blocking $Π$ and $W$ is said to be blocking $Π$ if $|W|= d$ such that $\sum_{w'\in W}δ(w,w') < \sum_{v\in Π(w)}δ(w,v)$ holds for each point $w\in W$, where $Π(w)$ denotes the subset $V_i\in Π$ which contains $w$ and $δ(a,b)$ denotes the Euclidean distance between points $a$ and $b$. Complementing the existing known polynomial-time result for $d=2$, we show that such polynomial-time algorithms cannot exist for any fixed number $d \ge 3$ unless P=NP. Our result for $d=3$ answers a decade-long open question in the theory of Stable Matching and Hedonic Games [17, 1, 9, 25, 20].

We study generalizations of stable matching in which agents may be matched fractionally; this models time-sharing assignments. We focus on the so-called ordinal stability and cardinal stability, and investigate the computational complexity of finding an ordinally stable or cardinally stable fractional matching which either maximizes the social welfare (i.e., the overall utilities of the agents) or the number of fully matched agents (i.e., agents whose matching values sum up to one). We complete the complexity classification of both optimization problems for both ordinal stability and cardinal stability, distinguishing between the marriage (bipartite) and roommates (non-bipartite) cases and the presence or absence of ties in the preferences. In particular, we prove a surprising result that finding a cardinally stable fractional matching with maximum social welfare is NP-hard even for the marriage case without ties. This answers an open question and exemplifies a rare variant of stable marriage that remains hard for preferences without ties. We also complete the picture of the relations of the stability notions and derive structural properties.

We present a data structure that in a dynamic graph of treedepth at most d, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time , which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time f(d) for some non-elementary (i.e. tower-exponential) function f. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth d from doubly-exponential in d to dO(d). As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter k and a dynamic graph G, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: Does G contain a simple path on k vertices? Does G contain a simple cycle on at least k vertices? In the first case, the data structure achieves amortized update time . In the second case, the amortized update time is . In both cases we assume access to a dictionary on the edges of G.

Adaptivity to changing environments and constraints is key to success in modern society. We address this by proposing “incrementalized versions” of Stable Marriage and Stable Roommates. That is, we try to answer the following question: for both problems, what is the computational cost of adapting an existing stable matching after some of the preferences of the agents have changed. While doing so, we also model the constraint that the new stable matching shall be not too different from the old one. After formalizing these incremental versions, we provide a fairly comprehensive picture of the computational complexity landscape of Incremental Stable Marriage and Incremental Stable Roommates. To this end, we exploit the parameters “degree of change” both in the input (difference between old and new preference profile) and in the output (difference between old and new stable matching). We obtain both hardness and tractability results, in particular showing a fixed-parameter tractability result with respect to the parameter “distance between old and new stable matching”.

We investigate the following many-to-one stable matching problem with diversity constraints (SMTI-DIVERSE): Given a set of students and a set of colleges which have preferences over each other, where the students have overlapping types, and the colleges each have a total capacity as well as quotas for individual types (the diversity constraints), is there a matching satisfying all diversity constraints such that no unmatched student-college pair has an incentive to deviate? SMTI-DIVERSE is known to be NP-hard. However, as opposed to the NP-membership claims in the literature [Aziz et al., AAMAS 2019; Huang,SODA 2010], we prove that it is beyond NP: it is complete for the complexity class Σ^P_2. In addition, we provide a comprehensive analysis of the problem’s complexity from the viewpoint of natural restrictions to inputs and obtain new algorithms for the problem.

The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity-Stable Roommates remains NP-complete.

In this paper we consider the $p$-Norm Hamming Centroid problem which asks to determine whether some given binary strings have a centroid with a bound on the $p$-norm of its Hamming distances to the strings. Specifically, given a set of strings $S$ and a real $k$, we consider the problem of determining whether there exists a string $s^*$ with $\big(\sum_{s \in S}d^p(s^*,s)\big)^{1/p} \leq k$, where $d(,)$ denotes the Hamming distance metric. This problem has important applications in data clustering, and is a generalization of the well-known polynomial-time solvable \textsc{Consensus String} $(p=1)$ problem, as well as the NP-hard \textsc{Closest String} $(p=\infty)$ problem. Our main result shows that the problem is NP-hard for all fixed rational $p > 1$, closing the gap for all rational values of $p$ between $1$ and $\infty$. Under standard complexity assumptions the reduction also implies that the problem has no $2^{o(n+m)}$-time or $2^{o(k^{\frac{p}{(p+1)}})}$-time algorithm, where $m$ denotes the number of input strings and $n$ denotes the length of each string, for any fixed $p > 1$. Both running time lower bounds are tight. In particular, we provide a $2^{k^{\frac{p}{(p+1)}+\varepsilon}}$-time algorithm for each fixed $\varepsilon > 0$. In the last part of the paper, we complement our hardness result by presenting a fixed-parameter algorithm and a factor-$2$ approximation algorithm for the problem.

The classic Stable Roommates problem (the non-bipartite generalization of the well-known Stable Marriage problem) asks whether there is a stable matching for a given set of agents, i.e. a partitioning of the agents into disjoint pairs such that no two agents induce a blocking pair. Herein, each agent has a preference list denoting who it prefers to have as a partner, and two agents are blocking if they prefer to be with each other rather than with their assigned partners. Since stable matchings may not be unique, we study an NP-hard optimization variant of Stable Roommates, called Egal Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost gamma, i.e. the sum of the dissatisfaction of the agents is minimum. The dissatisfaction of an agent is the number of agents that this agent prefers over its partner if it is matched; otherwise it is the length of its preference list. We also study almost stable matchings, called Min-Block-Pair Stable Roommates, which seeks to find a matching with a minimum number beta of blocking pairs. Our main result is that Egal Stable Roommates parameterized by gamma is fixed-parameter tractable, while Min-Block-Pair Stable Roommates parameterized by beta is W[1]-hard, even if the length of each preference list is at most five.

Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time k^{O(k + d)} s^{O(1)} for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d = 3.

We thoroughly study a generalized version of the famous Stable Marriage problem, now based on multi-modal preference lists. The central twist herein is to allow each agent to rank its potentially matching counterparts based on more than one "evaluation mode" (e.g., more than one criterion); thus, each agent is equipped with multiple preference lists, each ranking the counterparts in a possibly different way. We introduce and study three natural concepts of stability, investigate their mutual relations and focus on computational complexity aspects with respect to computing stable matchings in these new scenarios. Mostly encountering computational hardness (NP-hardness), we can also spot few islands of tractability and make a surprising connection to the Graph Isomorphism problem.

Consider an important meeting to be held in a team-based organization. Taking availability constraints into account, an online scheduling poll is being used in order to decide upon the exact time of the meeting. Decisions are to be taken during the meeting, therefore each team would like to maximize its relative attendance (i.e. the proportional number of its team members attending the meeting). We introduce a corresponding game, where each team can declare a lower total availability in the scheduling poll in order to improve its relative attendance—the pay-off. We are especially interested in situations where teams can form coalitions. We provide an efficient algorithm that, given a coalition, finds an optimal way for each team in a coalition to improve its pay-off. In contrast, we show that deciding whether such a coalition exists is NP-hard. We also study the existence of Nash equilibria: Finding Nash equilibria for various small sizes of teams and coalitions can be done in polynomial time while it is coNP-hard if the coalition size is unbounded.

We study computational problems for two popular parliamentary voting procedures: the amendment procedure and the successive procedure. They work in multiple stages where the result of each stage may influence the result of the next stage. Both procedures proceed according to a given linear order of the alternatives, an agenda. We obtain the following results for both voting procedures: On the one hand, deciding whether one can make a specific alternative win by reporting insincere preferences by the fewest number of voters, the Manipulation problem, or whether there is a suitable ordering of the agenda, the Agenda Control problem, takes polynomial time. On the other hand, our experimental studies with real-world data indicate that most preference profiles cannot be manipulated by only few voters and a successful agenda control is typically impossible. If the voters' preferences are incomplete, then deciding whether an alternative can possibly win is NP-hard for both procedures. Whilst deciding whether an alternative necessarily wins is coNP-hard for the amendment procedure, it is polynomial-time solvable for the successive procedure.

We study the computational complexity of candidate control in elections with few voters, that is, we consider the parameterized complexity of candidate control in elections with respect to the number of voters as a parameter. We consider both the standard scenario of adding and deleting candidates, where one asks whether a given candidate can become a winner (or, in the destructive case, can be precluded from winning) by adding or deleting few candidates, as well as a combinatorial scenario where adding/deleting a candidate automatically means adding or deleting a whole group of candidates. Considering several fundamental voting rules, our results show that the parameterized complexity of candidate control, with the number of voters as the parameter, is much more varied than in the setting with many voters.

We study the computational complexity of candidate control in elections with few voters (that is, we take the number of voters as a parameter). We consider both the standard scenario of adding and deleting candidates, where one asks if a given candidate can become a winner (or, in the destructive case, can be precluded from winning) by adding/deleting some candidates, and a combinatorial scenario where adding/deleting a candidate automatically means adding/deleting a whole group of candidates. Our results show that the parameterized complexity of candidate control (with the number of voters as the parameter) is much more varied than in the setting with many voters.

We introduce a graph-theoretic vertex dissolution model that applies to a number of redistribution scenarios, such as gerrymandering in political districting or work balancing in an online situation. The central aspect of our model is the deletion of certain vertices and the redistribution of their load to neighboring vertices in a completely balanced way. We investigate how the underlying graph structure, the knowledge of which vertices should be deleted, and the relation between old and new vertex loads influence the computational complexity of the underlying graph problems. Our results establish a clear borderline between tractable and intractable cases.

The NP-hard Subset Interconnection Design problem, also known as Minimum Topic-Connected Overlay, is motivated by numerous applications including the design of scalable overlay networks and vacuum systems. It has as input a finite set $V$ and a collection of subsets $V_1, V_2, \ldots, V_m \subseteq V$, and asks for a minimum-cardinality edge set $E$ such that for the graph $G=(V,E)$ all induced subgraphs $G[V_1], G[V_2], \ldots, G[V_m]$ are connected. We study Subset Interconnection Design in the context of polynomial-time data reduction rules that preserve the possibility of constructing optimal solutions. Our contribution is threefold: First, we show the incorrectness of earlier polynomial-time data reduction rules. Second, we show linear-time solvability in case of a constant number $m$ of subsets, implying fixed-parameter tractability for the parameter $m$. Third, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. To achieve our results, we elaborate on polynomial-time data reduction rules which also may be of practical use in solving Subset Interconnection Design.

In the Shift Bribery problem, we are given an election (based on preference orders), a preferred candidate p, and a budget. The goal is to ensure that p wins by shifting p higher in some voters' preference orders. However, each such shift request comes at a price (depending on the voter and on the extent of the shift) and we must not exceed the given budget. We study the parameterized computational complexity of Shift Bribery with respect to a number of parameters (pertaining to the nature of the solution sought and the size of the election) and several classes of price functions. When we parameterize Shift Bribery by the number of affected voters, then for each of our voting rules (Borda, Maximin, Copeland) the problem is W[2]-hard. If, instead, we parameterize by the number of positions by which p is shifted in total, then the problem is fixed-parameter tractable for Borda and Maximin, and is W[1]-hard for Copeland. If we parameterize by the budget for the cost of shifting, then the results depend on the price function class. We also show that Shift Bribery tends to be tractable when parameterized by the number of voters, but that the results for the number of candidates are more enigmatic.

Assume that each of n voters may or may not approve each of m issues. If an agent (the lobby) may influence up to k voters, then the central question of the NP-hard Lobbying problem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals. This problem can be modeled as a simple matrix modification problem: Can one replace k rows of a binary n x m-matrix by k all-1 rows such that each column in the resulting matrix has a majority of 1s? Significantly extending on previous work that showed parameterized intractability (W[2]-completeness) with respect to the number k of modified rows, we study how natural parameters such as n, m, k, or the "maximum number of 1s missing for any column to have a majority of 1s" (referred to as "gap value g") govern the computational complexity of Lobbying. Among other results, we prove that Lobbying is fixed-parameter tractable for parameter m and provide a greedy logarithmic-factor approximation algorithm which solves Lobbying even optimally if m < 5. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that Lobbying is LOGSNP-complete for constant values g>0, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.

We extend work by Christian et al. [Review of Economic Design 2007] on lobbying in multiple referenda by first providing a more fine-grained analysis of the computational complexity of the NP-complete Lobbying problem. Herein, given a binary matrix, the columns represent issues to vote on and the rows correspond to voters making a binary vote on each issue. An issue is approved if a majority of votes has a 1 in the corresponding column. The goal is to get all issues approved by modifying a minimum number of rows to all-1-rows. In our multivariate complexity analysis, we present a more holistic view on the nature of the computational complexity of Lobbying, providing both (parameterized) tractability and intractability results, depending on various problem parameterizations to be adopted. Moreover, we show non-existence results concerning efficient and effective preprocessing for Lobbying and introduce natural variants such as Restricted Lobbying and Partial Lobbying.

In distributed virtual environments, hosts typically have to react to events within a time span which is less than the network latency. As a consequence, hosts do routinely take actions although the system is in an inconsistent state. This has a noticeable influence on the perceived quality of these actions and their effect on the application. We argue that the level of this influence depends on the degree of inconsistency. In this paper, we tackle two fundamental questions: How does the degree of inconsistency influence the perceived quality of the users' actions? How can the degree of inconsistency be quantified? We propose a benchmark test for comparing different consistency algorithms with each other which consists of two measures of inconsistency and a sample scenario. For two different consistency algorithms, we compare the results of our benchmark test with the results of a user evaluation test and a simple yield measure.