arXiv subjects

Joerg Rothe

Publications and source records attributed to Joerg Rothe.

At least 19 recordsLinked to original sources

arXiv2021

The Complexity of Online Bribery in Sequential Elections

Prior work on the complexity of bribery assumes that the bribery happens simultaneously, and that the briber has full knowledge of all votes. However, in many real-world settings votes come in sequentially, and the briber may have a use-it-or-lose-it moment to decide whether to alter a given vote, and when making that decision the briber may not know what votes remaining voters will cast. We introduce a model for, and initiate the study of, bribery in such an online, sequential setting. We show that even for election systems whose winner-determination problem is polynomial-time computable, an online, sequential setting may vastly increase the complexity of bribery, jumping the problem up to completeness for high levels of the polynomial hierarchy or even PSPACE. But we also show that for some natural, important election systems, such a dramatic complexity increase does not occur, and we pinpoint the complexity of their bribery problems.

cs.GT↗

arXiv2016

The Complexity of Controlling Candidate-Sequential Elections

Candidate control of elections is the study of how adding or removing candidates can affect the outcome. However, the traditional study of the complexity of candidate control is in the model in which all candidates and votes are known up front. This paper develops a model for studying online control for elections where the structure is sequential with respect to the candidates, and in which the decision regarding adding and deleting must be irrevocably made at the moment the candidate is presented. We show that great complexity---PSPACE-completeness---can occur in this setting, but we also provide within this setting polynomial-time algorithms for the most important of election systems, plurality.

cs.GT↗

arXiv2016

The Complexity of Online Voter Control in Sequential Elections

Previous work on voter control, which refers to situations where a chair seeks to change the outcome of an election by deleting, adding, or partitioning voters, takes for granted that the chair knows all the voters' preferences and that all votes are cast simultaneously. However, elections are often held sequentially and the chair thus knows only the previously cast votes and not the future ones, yet needs to decide instantaneously which control action to take. We introduce a framework that models online voter control in sequential elections. We show that the related problems can be much harder than in the standard (non-online) case: For certain election systems, even with efficient winner problems, online control by deleting, adding, or partitioning voters is PSPACE-complete, even if there are only two candidates. In addition, we obtain (by a new characterization of coNP in terms of weight-bounded alternating Turing machines) completeness for coNP in the deleting/adding cases with a bounded deletion/addition limit, and we obtain completeness for NP in the partition cases with an additional restriction. We also show that for plurality, online control by deleting or adding voters is in P, and for partitioning voters is coNP-hard.

cs.GT↗

arXiv2013

The Complexity of Online Manipulation of Sequential Elections

Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. We obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model. And we obtain what to the best of our knowledge are the first PNP[1]-completeness and PNP-completeness results in the field of computational social choice, in particular proving such completeness for, respectively, the complexity of 3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition manipulation of veto elections.

cs.GT↗

arXiv2012

The Complexity of Online Manipulation of Sequential Elections

Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. We obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model. And we obtain what to the best of our knowledge are the first P^NP[1]-completeness and P^NP-completeness results in the field of computational social choice, in particular proving such completeness for, respectively, the complexity of 3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition manipulation of veto elections.

cs.GT↗

arXiv2012

Control Complexity in Bucklin, Fallback, and Plurality Voting: An Experimental Approach

Walsh [Wal10, Wal09], Davies et al. [DKNW10, DKNW11], and Narodytska et al. [NWX11] studied various voting systems empirically and showed that they can often be manipulated effectively, despite their manipulation problems being NP-hard. Such an experimental approach is sorely missing for NP-hard control problems, where control refers to attempts to tamper with the outcome of elections by adding/deleting/partitioning either voters or candidates. We experimentally tackle NP-hard control problems for Bucklin and fallback voting. Among natural voting systems with efficient winner determination, fallback voting is currently known to display the broadest resistance to control in terms of NP-hardness, and Bucklin voting has been shown to behave almost as well in terms of control resistance [ER10, EPR11, EFPR11]. We also investigate control resistance experimentally for plurality voting, one of the first voting systems analyzed with respect to electoral control [BTT92, HHR07]. Our findings indicate that NP-hard control problems can often be solved effectively in practice. Moreover, our experiments allow a more fine-grained analysis and comparison-across various control scenarios, vote distribution models, and voting systems-than merely stating NP-hardness for all these control problems.

cs.GT↗

arXiv2011

Taking the Final Step to a Full Dichotomy of the Possible Winner Problem in Pure Scoring Rules

The Possible Winner problem asks, given an election where the voters' preferences over the candidates are specified only partially, whether a designated candidate can become a winner by suitably extending all the votes. 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).

cs.CC↗

arXiv2009

Degrees of Guaranteed Envy-Freeness in Finite Bounded Cake-Cutting Protocols

Cake-cutting protocols aim at dividing a ``cake'' (i.e., a divisible resource) and assigning the resulting portions to several players in a way that each of the players feels to have received a ``fair'' amount of the cake. An important notion of fairness is envy-freeness: No player wishes to switch the portion of the cake received with another player's portion. Despite intense efforts in the past, it is still an open question whether there is a \emph{finite bounded} envy-free cake-cutting protocol for an arbitrary number of players, and even for four players. We introduce the notion of degree of guaranteed envy-freeness (DGEF) as a measure of how good a cake-cutting protocol can approximate the ideal of envy-freeness while keeping the protocol finite bounded (trading being disregarded). We propose a new finite bounded proportional protocol for any number n \geq 3 of players, and show that this protocol has a DGEF of 1 + \lceil (n^2)/2 \rceil. This is the currently best DGEF among known finite bounded cake-cutting protocols for an arbitrary number of players. We will make the case that improving the DGEF even further is a tough challenge, and determine, for comparison, the DGEF of selected known finite bounded cake-cutting protocols.

cs.GT↗

arXiv2009

The Cost of Stability in Coalitional Games

A key question in cooperative game theory is that of coalitional stability, usually captured by the notion of the \emph{core}--the set of outcomes such that no subgroup of players has an incentive to deviate. However, some coalitional games have empty cores, and any outcome in such a game is unstable. In this paper, we investigate the possibility of stabilizing a coalitional game by using external payments. We consider a scenario where an external party, which 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 their coalition(s). The sum of this payment plus the actual gains of the coalition(s) may then be divided among the agents so as to promote stability. We define the \emph{cost of stability (CoS)} as the minimal external payment that stabilizes the game. We provide general bounds on the cost of stability in several classes of games, and explore its algorithmic properties. To develop a better intuition for the concepts we introduce, we provide a detailed algorithmic study of the cost of stability in weighted voting games, a simple but expressive class of games which can model decision-making in political bodies, and cooperation in multiagent settings. Finally, we extend our model and results to games with coalition structures.

cs.GT↗

arXiv2009

The Complexity of Computing Minimal Unidirectional Covering Sets

Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal upward or downward covering set. For both problems, we raise this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size covering sets are hard or complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer's result that minimal bidirectional covering sets (i.e., sets that are both minimal upward and minimal downward covering sets) are polynomial-time computable.

cs.CC↗

arXiv2009

Sincere-Strategy Preference-Based Approval Voting Fully Resists Constructive Control and Broadly Resists Destructive Control

We study sincere-strategy preference-based approval voting (SP-AV), a system proposed by Brams and Sanver [Electoral Studies, 25(2):287-305, 2006], and here adjusted so as to coerce admissibility of the votes (rather than excluding inadmissible votes a priori), with respect to procedural control. In such control scenarios, an external agent seeks to change the outcome of an election via actions such as adding/deleting/partitioning either candidates or voters. SP-AV combines the voters' preference rankings with their approvals of candidates, where in elections with at least two candidates the voters' approval strategies are adjusted--if needed--to approve of their most-preferred candidate and to disapprove of their least-preferred candidate. This rule coerces admissibility of the votes even in the presence of control actions, and hybridizes, in effect, approval with pluralitiy voting. We prove that this system is computationally resistant (i.e., the corresponding control problems are NP-hard) to 19 out of 22 types of constructive and destructive control. Thus, SP-AV has more resistances to control than is currently known for any other natural voting system with a polynomial-time winner problem. In particular, SP-AV is (after Copeland voting, see Faliszewski et al. [AAIM-2008, Springer LNCS 5034, pp. 165-176, 2008]) the second natural voting system with an easy winner-determination procedure that is known to have full resistance to constructive control, and unlike Copeland voting it in addition displays broad resistance to destructive control.

cs.GT↗

arXiv2008

Llull and Copeland Voting Computationally Resist Bribery and Control

The only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. We study a parameterized version of Copeland voting, denoted by Copeland^α, where the parameter αis a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. 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^α, for all rational α, 0 < α< 1. Copeland voting is the first natural election system proven to have full resistance to constructive control. We also prove that both Copeland^1 (Llull elections) and Copeland^0 are resistant to all standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational α, 0 \leq α\leq 1, Copeland^αvoting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^α. We also study Copeland^αelections under more flexible models such as microbribery and extended control and we integrate the potential irrationality of voter preferences into many of our results.

cs.GT↗

arXiv2008

Hybrid Elections Broaden Complexity-Theoretic Resistance to Control

Electoral control refers to attempts by an election's organizer ("the chair") to influence the outcome by adding/deleting/partitioning voters or candidates. The groundbreaking work of Bartholdi, Tovey, and Trick [BTT92] on (constructive) control proposes computational complexity as a means of resisting control attempts: Look for election systems where the chair's task in seeking control is itself computationally infeasible. We introduce and study a method of combining two or more candidate-anonymous election schemes in such a way that the combined scheme possesses all the resistances to control (i.e., all the NP-hardnesses of control) possessed by any of its constituents: It combines their strengths. From this and new resistance constructions, we prove for the first time that there exists an election scheme that is resistant to all twenty standard types of electoral control.

cs.GT↗

arXiv2008

Frequency of Correctness versus Average-Case Polynomial Time and Generalized Juntas

We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. We also study some features of probability weight of correctness with respect to generalizations of Procaccia and Rosenschein's junta distributions [PR07b].

cs.CC↗

arXiv2008

Satisfiability Parsimoniously Reduces to the Tantrix(TM) Rotation Puzzle Problem

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) 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(TM) rotation puzzle problem. In particular, this reduction preserves the uniqueness of the solution, which implies that the unique Tantrix(TM) 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.

cs.CC↗

arXiv2008

The Three-Color and Two-Color Tantrix(TM) Rotation Puzzle Problems are NP-Complete via Parsimonious Reductions

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem with four colors is NP-complete, and they showed that the infinite variant of this problem is undecidable. In this paper, we study the three-color and two-color Tantrix(TM) rotation puzzle problems (3-TRP and 2-TRP) and their variants. Restricting the number of allowed colors to three (respectively, to two) reduces the set of available Tantrix(TM) tiles from 56 to 14 (respectively, to 8). We prove that 3-TRP and 2-TRP are NP-complete, which answers a question raised by Holzer and Holzer in the affirmative. Since our reductions are parsimonious, it follows that the problems Unique-3-TRP and Unique-2-TRP are DP-complete under randomized reductions. We also show that the another-solution problems associated with 4-TRP, 3-TRP, and 2-TRP are NP-complete. Finally, we prove that the infinite variants of 3-TRP and 2-TRP are undecidable.

cs.CC↗

arXiv2007

Enforcing and Defying Associativity, Commutativity, Totality, and Strong Noninvertibility for One-Way Functions in Complexity Theory

Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist one-way functions (i.e., p-time computable, honest, p-time noninvertible functions; this paper is in the worst-case model, not the average-case model) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve the sufficient condition to ``P does not equal NP.'' More generally, in this paper we completely characterize which types of one-way functions stand or fall together with (plain) one-way functions--equivalently, stand or fall together with P not equaling NP. We look at the four attributes used in Rabi and Sherman's seminal work on algebraic properties of one-way functions (see [RS97,RS93]) and subsequent papers--strongness (of noninvertibility), totality, commutativity, and associativity--and for each attribute, we allow it to be required to hold, required to fail, or ``don't care.'' In this categorization there are 3^4 = 81 potential types of one-way functions. We prove that each of these 81 feature-laden types stand or fall together with the existence of (plain) one-way functions.

cs.CC↗

arXiv2007

On Approximating Optimal Weighted Lobbying, and Frequency of Correctness versus Average-Case Polynomial Time

We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant--optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a ``frequently self-knowingly correct algorithm.'' We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein's junta distributions [PR07].

cs.GT↗