Article

Parameterized complexity of control problems in Maximin election

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

Elections are a central model in a variety of areas. This paper studies parameterized computational complexity of five control problems in the Maximin election. We obtain the following results: constructive control by adding candidates is W[2]-hard with respect to the parameter “number of added candidates”; both constructive and destructive control by adding/deleting voters are W[1]-hard with respect to the parameter “number of added/deleted voters”.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... There is a growing body of research regarding the parameterized complexity of voting problems (see, e.g., the survey of Betzler et al. (2012)), where typical parameters include the solution size (e.g., the number of candidates that can be added) and the election size (i.e., the number of candidates or the number of voters). For the solution size as the parameter, control problems usually turn out to be hard (Betzler and Uhlmann 2009;Liu et al. 2009;Liu and Zhu 2010). On the contrary, taking the number of candidates as the parameter almost always leads to FPT (fixed-parameter tractability) results (see, e.g., the papers of Faliszewski et al. (2011) and Hemaspaandra et al. (2013)). ...
Article
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.
... All FPT results are with respect to k. Results marked by ♣ are from [Faliszewski et al. 2011b], by ♦ from [Yang and Guo 2017], by ♥ from [Lin 2011], by ♠ from [Yang 2017b], by from [Russell 2007], by from [Brandt et al. 2015], by ¶ from [Faliszewski et al. 2009], by from [Bartholdi III et al. 1992], by § from [Faliszewski et al. 2011a], by £ from [Liu and Zhu 2013], and by from [Yang and Guo 2014a]. ...
Preprint
Full-text available
We investigate the complexity of {\sc{Constructive Control by Adding/Deleting Votes}} (CCAV/CCDV) for $r$-approval, Condorcet, Maximin and Copeland$^{\alpha}$ in $k$-axes and $k$-candidates partition single-peaked elections. In general, we prove that CCAV and CCDV for most of the voting correspondences mentioned above are NP-hard even when~$k$ is a very small constant. Exceptions are CCAV and CCDV for Condorcet and CCAV for $r$-approval in $k$-axes single-peaked elections, which we show to be fixed-parameter tractable with respect to~$k$. In addition, we give a polynomial-time algorithm for recognizing $2$-axes elections, resolving an open problem. Our work leads to a number of dichotomy results. To establish an NP-hardness result, we also study a property of $3$-regular bipartite graphs which may be of independent interest. In particular, we prove that for every $3$-regular bipartite graph, there are two linear orders of its vertices such that the two endpoints of every edge are consecutive in at least one of the two orders.
... Exploring parameterized complexity of various control problems has also gained a lot of interest. For example, [Betzler and Uhlmann, 2009] studied parameterized complexity of candidate control in elections and showed interesting connection with digraph problems, [Liu and Zhu, 2010;2013] studied parameterized complexity of control problem by deleting voters for many common voting rules, and so on [Dey et al., 2016;2019a]. Studying election control from a game theoretic approach using security games is also an active area of research. ...
Conference Paper
Full-text available
We study the parameterized complexity of the optimal defense and optimal attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a set of votes) and two integers k_a and k_d corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the optimal defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most k_d voter groups such that, no matter which k_a voter groups the attacker attacks, the out-come of the election does not change. In the optimal attack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking k_a voter groups such that, no matter which k_d voter groups the defender defends, the outcome of the election is always different from the original (without any attack) one. We show that both the optimal defense problem and the optimal attack problem are computationally intractable for every scoring rule and the Condorcet voting rule even when we have only3candidates. We also show that the optimal defense problem for every scoring rule and the Condorcet voting rule is W[2]-hard for both the parameters k_a and k_d, while it admits a fixed parameter tractable algorithm parameterized by the combined parameter (ka, kd). The optimal attack problem for every scoring rule and the Condorcet voting rule turns out to be much harder – it is W[1]-hard even for the combined parameter (ka, kd). We propose two greedy algorithms for the OPTIMAL DEFENSE problem and empirically show that they perform effectively on reasonable voting profiles.
... Exploring parameterized complexity of various control problems has also gained a lot of interest. For example, Betzler and Uhlmann [6] studied parameterized complexity of candidate control in elections and showed interesting connection with digraph problems, Liu and Zhu [41,42] studied parameterized complexity of control problem by deleting voters for many common voting rules, and so on [40,53,38,18,22]. Studying election control from a game theoretic approach using security games is also an active area of research. ...
Preprint
Full-text available
We study the parameterized complexity of the optimal defense and optimal attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a set of votes) and two integers $k_a$ and $k_d$ corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the optimal defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most $k_d$ voter groups such that, no matter which $k_a$ voter groups the attacker attacks, the outcome of the election does not change. In the optimal attack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking $k_a$ voter groups such that, no matter which $k_d$ voter groups the defender defends, the outcome of the election is always different from the original (without any attack) one.
... When election control is inevitable, how to prevent it from happening is largely ignored. In addition, the control is usually studied in situations where the voter preferences are fully known and the granularity is individual voters [2,5,12]. However, it is impossible for everyone to get precise voter preferences before the election and the manipulators can deploy a single attack such as a denial-of-service attack deleting some voter groups (e.g. ...
Article
Full-text available
Election control has always been an important issue of democratic institutions concerned. Considering that a voting rule is indeed susceptible to control by an external agent, it is natural to seek ways to protect elections. Much of prior work has focused on complete voter preferences and approached the problem from the perspective of the computational complexity of election control. However, it is impractical for everyone to have complete voter preferences in real-world scenarios. In addition, when given a voting rule, such as plurality which is widely used in our lives, is easy to control, how to design protection strategies to prevent the occurrence of election control is ignored. In this paper, we model the problem, where the attacker can deploy a single attack such as a denial-of-service attack to convert the voting result through deleting some voter groups, and the defender allocates the limited protection resources to prevent attacks on specific voter groups, as a Stackelberg game. Then we first use the minimax regret decision criterion for uncertainty about voter preferences in the game. We also propose heuristic algorithms to speed up computing minimax regret for the Stackelberg game. Finally, we conduct detailed experiments on both synthetic and real data, which show that our algorithms lead to much better solution quality than other algorithms in the literature.
... The one we mentioned here is the notion used by Brandt et al. [7] to establish their polynomial-time solvability results. [32]). The complexity of control problems in nearly single-peaked elections has been studied in the literature recently. ...
Article
Full-text available
We study the parameterized complexity of voter control problems in κ-peaked elections, where κ is a positive integer. In particular, we focus on the constructive/destructive control by adding/deleting votes for Condorcet, Maximin and Copelandα . It is known that in general elections all these problems are NP-hard, except for the destructive control by adding/deleting votes for Condorcet which is polynomial-time solvable. We strengthen these results by showing that, when restricted to κ-peaked elections where κ = 3,4, the above NP-hard problems not only remain NP-hard but also are W[1]-hard with respect to the number of added/deleted votes.
... Since the seminal paper by Bartholdi III et al. [3] on controlling an election by adding or deleting the fewest number of voters or candidates with the goal of making a specific candidate to win (constructive control ), a lot of research has been devoted to the study of control for different voting rules [16,14,24,23,4,21], on different control modes [17,18], or even on other controlling goals (e.g. aiming at several candidates' victory or a specific candidate's defeat) [20,26]. ...
Conference Paper
Voter control problems model situations in which an external agent tries toaffect the result of an election by adding or deleting the fewest number of voters. The goal of the agent is to make a specific candidate either win (\emph{constructive} control) or lose (\emph{destructive} control) the election. We study the constructive and destructive voter control problems whenadding and deleting voters have a \emph{combinatorial flavor}: If we add (resp.\ delete) a voter~$v$, we also add (resp.\ delete) a bundle~$\kappa(v) $ of voters that are associated with~$v$. While the bundle~$\kappa(v)$ may have more than one voter, a voter may also be associated with more than one voter. We analyze the computational complexity of the four voter control problems for the Plurality rule. We obtain that, in general, making a candidate lose is computationally easier than making her win. In particular, if the bundling relation is symmetric (i.e.\ $\forall w\colon w \in \kappa(v) \Leftrightarrow v \in \kappa(w) $), and if each voter has at most two voters associated with him, then destructive control is polynomial-time solvable while the constructive variant remains $\NP$-hard. Even if the bundles are disjoint (i.e.\ $\forall w\colon w \in \kappa(v) \Leftrightarrow \kappa(v) = \kappa(w) $), the constructive problem variants remain intractable. Finally, the minimization variant of constructive control by adding voters does not admit an efficient approximation algorithm, unless P=NP.
... Faliszewski, Hemaspaandra, and Hemaspaandra [23] were the first to study control in weighted elections (however the work of Baumeister et al. [5] is related to this topic). Other authors took a different perspective and, for example, studied parametrized complexity of control problems [6,36,37,50], or considered the complexity of control in elections where votes come from some restricted domains (e.g., the singlepeaked domain [8,25] or the single-crossing domain [38]). On the other hand, Wojtas and Faliszewski [48] studied counting variants of control problems, where instead of asking if someone can become a winner we ask for the probability that someone becomes a winner, given that a random control action is taken. ...
Article
Full-text available
We study the complexity of priced control in elections. Naturally, if a given control type is NP-hard for a given voting system ɛ then its priced variant is NP-hard for this rule as well. It is, however, interesting what effect introducing prices has on the complexity of those control problems that without prices are tractable. We show that for four prominent voting rules (plurality, approval, Condorcet, and Copeland) introducing prices does not increase the complexity of control by adding/deleting candidates/voters. However, we do show an example of a scoring rule for which such an effect takes place.
... There is a growing body of research regarding the parameterized complexity of voting problems (see, e.g., the survey by Betzler et al. [4]), where typical parameters include the solution size (e.g., the number of candidates that can be added) and the election size (i.e., the number of candidates or the number of voters). For the solution size as the parameter, control problems usually turn out to be hard Betzler and Uhlmann [2], Liu et al. [23], Liu and Zhu [22]. On the contrary, taking the number of candidates as the parameter almost always leads to FPT (fixed-parameter tractability) results (see, e.g., the papers by Faliszewski et al. [16] and by Hemaspaandra et al. [19]). ...
Article
Full-text available
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.
... Parameterized computation is a theory to cope with NP-hard problems [1] , which has been widely applied to solving problems in the fields of bioinformatics [2] , social science [3,4] , networks [5][6][7] , computation geometry [8] , etc. A parameterized problem Q is a language that is a subset of˙ N , where˙is a fixed alphabet and N is the set of all non-negative integers. ...
Article
Parameterized computation is a new method dealing with NP-hard problems, which has attracted a lot of attentions in theoretical computer science. As a practical preprocessing method for NP-hard problems, kernelizaiton in parameterized computation has recently become an active research area. In this paper, we discuss several kernelizaiton techniques, such as crown decomposition, planar graph vertex partition, randomized methods, and kernel lower bounds, which have been used widely in the kernelization of many hard problems.
... The kernelization algorithm of Wang et al. [17] achieves a polynomial problem kernel for CCDV-d-APPROVAL with d being a constant; note that, with unbounded d , CCDV-d-APPROVAL is WOE2-hard with respect to k [17] . It is conceivable that this reconstruction approach could lead to problem kernels for other fixedparameter tractable control problems [65] . However, compared to the diverse general tools for deriving polynomial (even linear) problem kernels for graphbased problems, the research on kernelization for such two-dimensional problems as voting problems seems little developed. ...
Article
Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context.
... These papers focused on the Plurality rule and the Condorcet rule (and the Approval rule, for the destructive case of Hemaspaandra et al. [24]). Since then, many other researchers extended this study to a number of other rules and models [3,16,17,19,28,27,31,33]. In all previous work on election control, the authors always assumed that one could affect each entity of the election at unit cost only. ...
Article
Voter control problems model situations such as an external agent trying to affect the result of an election by adding voters, for example by convincing some voters to vote who would otherwise not attend the election. Traditionally, voters are added one at a time, with the goal of making a distinguished alternative win by adding a minimum number of voters. In this paper, we initiate the study of combinatorial variants of control by adding voters: In our setting, when we choose to add a voter~$v$, we also have to add a whole bundle $\kappa(v)$ of voters associated with $v$. We study the computational complexity of this problem for two of the most basic voting rules, namely the Plurality rule and the Condorcet rule.
... From a technical standpoint, our research continues the line of work on the complexity of control. This line of work was initiated by Bartholdi, Tovey, and Trick [3], and then continued by Hemaspaandra, Hemaspaandra, and Rothe [28] (who introduced the destructive cases), by Meir et al. [37] (who considered multiwinner rules and who generalized the idea of the constructive and destructive cases), by Faliszewski, Hemaspaandra, and Hemaspaandra [19] (who introduced multimode model of control), by Faliszewski, Hemaspaandra, and Hemaspaandra [20] (who were first to consider control for weighted elections), by Rothe and Schend [46] (who initiated the empirical study of the complexity of control problems), and by many other researchers, who provided results for specific voting rules and who introduced various other novel means of studying control problems (see, e.g., the following papers [6,15,16,34,35,38,45]; we also point the readers to the survey [18]). ...
Article
We consider the problem of predicting winners in elections, for the case where we are given complete knowledge about all possible candidates, all possible voters (together with their preferences), but where it is uncertain either which candidates exactly register for the election or which voters cast their votes. Under reasonable assumptions, our problems reduce to counting variants of election control problems. We either give polynomial-time algorithms or prove #P-completeness results for counting variants of control by adding/deleting candidates/voters for Plurality, k-Approval, Approval, Condorcet, and Maximin voting rules. We consider both the general case, where voters' preferences are unrestricted, and the case where voters' preferences are single-peaked.
... This parameterization has been studied in some voting systems and the relevant problems have often been found to be W[1]-hard or W[2]-hard, and thus very unlikely to be fixedparameter tractable. For example, under this parameterization, Betzler and Uhlmann [5] showed, for what are known as Copeland α elections, that constructive control by adding candidates and constructive control by deleting candidates are W[2]-complete, Liu and Zhu [25] proved, for maximin elections, that constructive control by adding candidates is W[2]-hard, and Liu and Zhu also achieved W[1]-hardness results for the relevant voter control problems. For additional control results parameterized on the problem's internal addition/deletion limit, see Table 8 of Betzler et al. [4]. ...
Article
Full-text available
Schulze and ranked-pairs elections have received much attention recently, and the former has quickly become a quite widely used election system. For many cases these systems have been proven resistant to bribery, control, or manipulation, with ranked pairs being particularly praised for being NP-hard for all three of those. Nonetheless, the present paper shows that with respect to the number of candidates, Schulze and ranked-pairs elections are fixed-parameter tractable to bribe, control, and manipulate: we obtain uniform, polynomial-time algorithms whose degree does not depend on the number of candidates.
Article
We study a set of voting problems where given an election E=(C,ΠV) (where C is the set of candidates and ΠV is a set of votes), and a non-empty subset of candidates J, the question under consideration is: Can we modify the election in a way so that none of the candidates in J wins the election? The modification operations allowed are that of either adding or deleting some candidates. Yang and Wang [AAMAS 2017] introduced these problems as the Resolute Control problem, a generalization of the destructive control problem where J is a singleton. They studied parameterized complexity of Resolute Control for voting rules Borda (both addition and deletion), Maximin (addition), and Copeland (both addition and deletion). They primarily consider |J| as parameter. In this paper we study Resolute Control parameterized by the other natural parameters viz., the number of candidates added or deleted. We show that the Resolute Control for Borda (both addition and deletion), Maximin (addition) and Copeland (deletion) are W[2]-hard. We complement this by showing that when the number of voters is odd, Copeland (deletion) is FPT parameterized by the sum of the number of deleted candidates and the size of the feedback arc set of the majority graph of the election.
Article
We study the parameterized complexity of the Optimal Defense and Optimal Attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a district consisting of a set of votes) and two integers ka and kd corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the Optimal Defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most kd voter groups such that, no matter which ka voter groups the attacker attacks, the outcome of the election does not change. In the Optimal Attack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking ka voter groups such that, no matter which kd voter groups the defender defends, the outcome of the election is always different from the original one (without any attack). We show that both the Optimal Defense problem and the Optimal Attack problem are computationally intractable for every scoring rule and the Condorcet voting rule even when we have only 3 candidates. We also show that the Optimal Defense problem for every scoring rule and the Condorcet voting rule is W[2]-hard for both the parameters ka and kd, while it admits a fixed parameter tractable algorithm parameterized by the combined parameter (ka,kd). The Optimal Attack problem for every scoring rule and the Condorcet voting rule turns out to be much harder – it is W[1]-hard even for the combined parameter (ka,kd). We propose two greedy algorithms for the Optimal Defense problem and empirically show that they perform effectively on many voting profiles.
Article
Full-text available
In order to achieve sustainability in agriculture, the need for modern water resources management in order to protect and optimize the use of this vital source can be an effective step in sustainability programs. Therefore, one of the ways to achieve agricultural water management, good governance Is a village In this research, we seek to investigate rural well-being in agricultural water management in the villages of Quchan city. The research method is analytical. For analyzing the data, SPSS software and Fuzzy Mamdani model have been used. The results of fuzzy model of Mamadani showed that in the result of good rural governance level in agricultural water management, the final naphasian output as the output of the fuzzy inference system in MATLAB software is equal to 0.39, which has a low utilization rate based on good rural governance indicators The study explains that among these indicators, after determining the membership function and normalizing the data, the range of significance is as follows: accountability index, accountability, collective agreement, equity and equality, legitimacy, transparency and openness and participation They have received it. In the following, using the path analysis test, direct and indirect effects of each of the good rural governance indicators on agricultural water management were studied. The results showed that the collective agreement index with the Beta coefficient was 0.67, the transparency index And openness with a Beta coefficient of 0.432 is the highest and lowest direct effect of good rural governance in managing agricultural water resources.
Article
Election control encompasses attempts from an external agent to alter the structure of an election in order to change its outcome. This problem is both a fundamental theoretical problem in social choice, and a major practical concern for democratic institutions. Consequently, this issue has received considerable attention, particularly as it pertains to different voting rules. In contrast, the problem of how election control can be prevented or deterred has been largely ignored. We introduce the problem of optimal defense against election control, including destructive and constructive control, where manipulation is allowed at the granularity of groups of voters (e.g., voting locations) through a denial-of-service attack, and the defender allocates limited protection resources to prevent control. We consider plurality voting, and show that it is computationally hard to prevent both types of control, though destructive control itself can be performed in polynomial time. For defense against destructive control, we present a double-oracle framework for computing an optimal prevention strategy. We show that both defender and attacker best response subproblems are NP-complete, and develop exact mixed-integer linear programming approaches for solving these, as well as fast heuristic methods. We then extend this general approach to develop effective algorithmic solutions for defense against constructive control. Finally, we generalize the model and algorithmic approaches to consider uncertainty about voter preferences. Experiments conducted on both synthetic and real data demonstrate that the proposed computational framework can scale to realistic problem instances.
Article
A natural generalization of the single-peaked elections is the k-peaked elections, where at most k peaks are allowed in each vote. Motivated by NP-hardness in general and polynomial-time solvability in single-peaked elections, we aim at establishing a complexity dichotomy of several control problems for r-approval voting in fc-peaked elections with respect to fc. It turns out that most NP-completeness results in general also hold in k-peaked elections, even for k = 2, 3. On the other hand, we derive polynomial-time algorithms for certain control problems for k = 2. In addition, we also study the problems from the viewpoint of parameterized complexity and achieve both FPT and W-hardness results. Several of our results apply to approval voting and sincere-strategy preference-based approval voting as well. Copyright © 2014, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved.
Article
Electoral control models ways of changing the outcome of an election via such actions as adding, deleting, or partitioning either candidates or voters. To protect elections from such control attempts, computational complexity has been used to establish so-called resistance results. We show that fallback voting, an election system proposed by Brams and Sanver [12] to combine Bucklin with approval voting, displays the broadest control resistance currently known to hold among natural election systems with a polynomial-time winner problem. We also study the control complexity of Bucklin voting and show that it performs almost as well as fallback voting in terms of control resistance. Furthermore, we investigate the parameterized control complexity of Bucklin and fallback voting, according to several parameters that are often likely to be small for typical instances. In a companion paper [28], we challenge our worst-case complexity results from an experimental point of view.
Conference Paper
We study computational aspects of various forms of manipulation and control in judgment aggregation, with a focus on the premise-based procedure. For manipulation, we in particular consider incomplete judgment sets and the notions of top-respecting and closeness-respecting preferences introduced by Dietrich and List [13]. This complements previous work on the complexity of manipulation in judgment aggregation that focused on Hamming-distance-induced preferences [14,6], which we also study here. Regarding control, we introduce the notion of control by bundling judges and show that the premise-based procedure is resistant to it in terms of NP-hardness.
Article
Full-text available
We review NP-hard voting problems together with their status in terms of parameterized complexity results. In addition, we survey standard techniques for achieving fixed-parameter (in)tractability results in voting.
Article
Elections are an important preference aggregation model in a variety of areas. Given a pool of n potential voters, the chair may strategically selecting k voters from the pool to feed to an election system, in order to control the final outcome of the election system. This type of control, called control by voter selection, is closely related to two already well-studied types of control, i.e., control by voter addition and control by voter deletion. This paper studies parameterized complexity of control by voter selection for five election systems, i.e., Maximin, Copeland α (0≤α≤1), Borda, Bucklin, and Approval. We prove that constructive/destructive control of Maximin, constructive/destructive control of Copeland α , constructive control of Borda, constructive control of Bucklin, and constructive control of Approval are all W[2]-hard, with respect to the parameter “number of selected voters”.
Conference Paper
Full-text available
We review NP-hard voting problems together with their status in terms of parameterized complexity results. In addition, we survey standard techniques for achieving fixed-parameter (in)tractability results in voting.
Conference Paper
Full-text available
In the context of voting, manipulation and control refer to attempts to influence the outcome of elections by either setting some of the votes strategically (i.e., by reporting untruthful preferences) or by altering the structure of elections via adding, deleting, or partitioning either candidates or voters. Since by the celebrated Gibbard–Satterthwaite theorem (and other results expanding its scope) all reasonable voting systems are manipulable in principle and since many voting systems are in principle susceptible to many control types modeling natural control scenarios, much work has been done to use computational complexity as a shield to protect elections against manipulation and control. However, most of this work has merely yielded NP-hardness results, showing that certain voting systems resist certain types of manipulation or control only in the worst case. Various approaches, including studies of the typical case (where votes are given according to some natural distribution), pose serious challenges to such worst-case complexity results and might allow successful manipulation or control attempts, despite the NP-hardness of the corresponding problems. We survey and discuss some recent results on these challenges to complexity results for manipulation and control, including typical-case analyses and experiments, fixed-parameter tractability, domain restrictions (single-peakedness), and approximability.
Article
In 1992, Bartholdi, Tovey, and Trick opened the study of control attacks on elections---attempts to improve the election outcome by such actions as adding/deleting candidates or voters. That work has led to many results on how algorithms can be used to find attacks on elections and how complexity-theoretic hardness results can be used as shields against attacks. However, all the work in this line has assumed that the attacker employs just a single type of attack. In this paper, we model and study the case in which the attacker launches a multipronged (i.e., multimode) attack. We do so to more realistically capture the richness of real-life settings. For example, an attacker might simultaneously try to suppress some voters, attract new voters into the election, and introduce a spoiler candidate. Our model provides a unified framework for such varied attacks, and by constructing polynomial-time multiprong attack algorithms we prove that for various election systems even such concerted, flexible attacks can be perfectly planned in deterministic polynomial time. Comment: 41 pages, 2 tables
Article
We study the behavior of Range Voting and Normalized Range Voting with respect to electoral control. Electoral control encompasses attempts from an election chair to alter the structure of an election in order to change the outcome. We show that a voting system resists a case of control by proving that performing that case of control is computationally infeasible. Range Voting is a natural extension of approval voting, and Normalized Range Voting is a simple variant which alters each vote to maximize the potential impact of each voter. We show that Normalized Range Voting has among the largest number of control resistances among natural voting systems.
Article
Full-text available
Abstract This supplement,provides a brief introduction to the field of fixed-parameter tractability and parameterized complexity. Some basic notions are explained and some related results are presented, with a focus on problems arising in the field of com putational social choice. 1 Fixed-Parameter Tractability and Parameterized Complexity The study of fixed-parameter tractability and parameterize d complexity has emerged,as a new field
Conference Paper
Full-text available
Recent work by Procaccia, Rosenschein and Zohar [14] established some results regarding the complexity of manipulation and control in elections with multiple winners, such as elections of an assembly or committee; that work provided an initial understanding of the topic. In this paper, we paint a more complete picture of the topic, investigating four prominent multi-winner voting rules. First, we characterize the complexity of manipulation and control in these voting rules under various kinds of formalizations of the manipulator's goal. Second, we extend the results about complexity of control to various well-known types of control. This work enhances our comprehension of which multi-winner voting rules should be employed in various settings.
Conference Paper
Full-text available
We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lobby") seeks to influence voters' preferences of voting for or against multiple issues when the voters' preferences are represented in terms of probabilities. In particular, we provide two evaluation criteria and two bribery methods to formally describe these models, and we consider the resulting forms of lobbying with and without issue weighting. We provide a formal analysis for these problems of lobbying in a stochastic environment, and determine their classical and parameterized complexity depending on the given bribery/evaluation criteria and on various natural parameterizations. Specifically, we show that some of these problems can be solved in polynomial time, some are NP-complete but fixed-parameter tractable, and some are W[2]-complete. Finally, we provide approximability and inapproximability results for these problems and several variants.
Article
Full-text available
In this paper we show that lobbying in conditions of “direct democracy” is virtually impossible, even in conditions of complete information about voters preferences, since it would require solving a very computationally hard problem. We use the apparatus of parametrized complexity for this purpose.
Conference Paper
Full-text available
In pattern recognition, there is a growing use of multiple classifier combinations with the goal to increase recognition performance. In many cases, plurality voting is a part of the combination process. In this article, we discuss and test several well known voting methods from politics and economics on classifier combination in order to see if an alternative to the simple plurality vote exists. We found that, assuming a number of prerequisites, better methods are available, that are comparatively simple and fast.
Article
Full-text available
When autonomous agents attempt to coordinate action, it is often necessary that they reach some kind of consensus. Reaching consensus has traditionally been dealt with in the Distributed Artificial Intelligence literature via negotiation.
Book
Contents 1. Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Keep the Parameter Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Preliminaries and Agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Parameterized Complexity---a Brief Overview . . . . . . . . . . . . . . 6 1.3.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Interpreting Fixed-Parameter Tractability . . . . . . . . . . . 9 1.4 Vertex Cover -- an Illustrative Example . . . . . . . . . . . . . . . . . 11 1.4.1 Parameterize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Specialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 Generalize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.4 Count or Enumerate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Article
We show how computational complexity might protect the integrity of social choice. We exhibit a voting rule that efficiently computes winners but is computationally resistant to strategic manipulation. It is NP-complete for a manipulative voter to determine how to exploit knowledge of the preferences of others. In contrast, many standard voting schemes can be manipulated with only polynomial computational effort.
Article
Voting systems are common tools in a variety of areas. This paper studies parameterized computational complexity of control of Plurality, Condorcet and Approval voting systems, respectively. The types of controls considered include adding or deleting candidates or voters, under constructive or destructive setting. We obtain the following results: (1) constructive control by adding candidates in Plurality voting is W[2]-hard with respect to the parameter “number of added candidates”, (2) destructive control by adding candidates in Plurality voting is W[2]-hard with respect to the parameter “number of added candidates”, (3) constructive control by adding voters in Condorcet voting is W[1]-hard with respect to the parameter “number of added voters”, (4) constructive control by deleting voters in Condorcet voting is W[1]-hard with respect to the parameter “number of deleted voters”, (5) constructive control by adding voters in Approval voting is W[1]-hard with respect to the parameter “number of added voters”, and (6) constructive control by deleting voters in Approval voting is W[2]-hard with respect to the parameter “number of deleted voters”.
Article
Preference aggregation in a multiagent setting is a central issue in both human and computer contexts. In this paper, we study in terms of complexity the vulnerability of preference aggregation to destructive control. In particular, we study the ability of an election's chair to, through such mechanisms as voter/candidate addition/suppression/partition, ensure that a particular candidate (equivalently, alternative) does not win. And we study the extent to which election systems can make it impossible, or computationally costly (NP-complete), for the chair to execute such control. Among the systems we study—plurality, Condorcet, and approval voting—we find cases where systems immune or computationally resistant to a chair choosing the winner nonetheless are vulnerable to the chair blocking a victory. Beyond that, we see that among our studied systems no one system offers the best protection against destructive control. Rather, the choice of a preference aggregation system will depend closely on which types of control one wishes to be protected against. We also find concrete cases where the complexity of or susceptibility to control varies dramatically based on the choice among natural tie-handling rules.
Article
Some voting schemes that are in principle susceptible to control are nevertheless resistant in practice due to excessive computational costs; others are vulnerable. We illustrate this in detail for plurality voting and for Condorcet voting.
Conference Paper
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 [BTT92]. An election system in which such an agent's computational task is NP-hard is said to be resistant to the given type of control. The only election systems known to be resistant to all the standard control types are highly artificial election systems created by hybridization [HHR07]. In this paper, we prove that an election system developed by the 13th century mystic Ramon Llull and the well-studied Copeland election system are both resistant to all the standard types of (constructive) electoral control other than one variant of addition of candidates. This is the most comprehensive resistance to control yet achieved by any natural election system. In addition, we show that Llull and Copeland voting are very broadly resistant to bribery attacks, and we integrate the potential irrationality of voter preferences into many of our results.
Article
In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown,that all general voting protocols are manipulable. One could try to avoid manipulation by using protocols where determining a beneficial manipulation is hard. Especially among,computational agents, it is reasonable to measure this hardness by computational complexity. Some earlier work has been done in this area, but it was assumed that the number of voters and candidates is unbounded. Such hardness results lose relevance when the number of candidates is small, because manipulation algorithms that are exponential only in the number,of candidates (and only slightly so) might be available. We give such an algorithm for an individual agent to manipulate the Single Transferable V. Conitzer and T. Sandholm were funded by the National Science Foundation under CAREER Award
Article
There are different ways for an external agent to influence the outcome of an election. We concentrate on “control ” by adding or deleting candidates. Our main focus is to investigate the parameterized complexity of various control problems for different voting systems. To this end, we introduce natural digraph problems that may be of independent interest. They help in determining the parameterized complexity of control for different voting systems including Llull, Copeland, and plurality voting. Devising several parameterized reductions, we provide an overview of the parameterized complexity of the digraph and control problems with respect to natural parameters such as adding/deleting only a bounded number of candidates or having only few voters.
Conference Paper
We consider the problem of combining ranking results from various sources. In the context of the Web, the main applications include building meta-search engines, combining ranking functions, selecting documents based on multiple criteria, and improving search precision through word associations. We develop a set of techniques for the rank aggregation problem and compare their performance to that of well-known methods. A primary goal of our work is to design rank aggregation techniques that can effectively combat "spam," a serious problem in Web searches. Experiments show that our methods are simple, efficient, and effective.
Article
In 1992, Bartholdi, Tovey, and Trick opened the study of control attacks on elections---attempts to improve the election outcome by such actions as adding/deleting candidates or voters. That work has led to many results on how algorithms can be used to find attacks on elections and how complexity-theoretic hardness results can be used as shields against attacks. However, all the work in this line has assumed that the attacker employs just a single type of attack. In this paper, we model and study the case in which the attacker launches a multipronged (i.e., multimode) attack. We do so to more realistically capture the richness of real-life settings. For example, an attacker might simultaneously try to suppress some voters, attract new voters into the election, and introduce a spoiler candidate. Our model provides a unified framework for such varied attacks, and by constructing polynomial-time multiprong attack algorithms we prove that for various election systems even such concerted, flexible attacks can be perfectly planned in deterministic polynomial time. Comment: 41 pages, 2 tables
Article
We show that the parameterized problem Perfect Code belongs to W[1]. This result closes an old open question, because it was often conjectured that Perfect Code could be a natural problem having complexity degree intermediate between W[1] and W[2]. This result also shows W[1]-membership of the parameterized problem Weighted Exact CNF Satisfiability.
Article
The growth of Internet commerce has stimulated the use of collaborative filtering (CF) algorithms as recommender systems. Such systems leverage knowledge about the behavior of multiple users to recommend items of interest to individual users. CF methods have been harnessed to make recommendations about such items as web pages, movies, books, and toys. Researchers have proposed several variations of the technology. We take the perspective of CF as a methodology for combining preferences. The preferences predicted for the end user is some function of all of the known preferences for everyone in a database. Social Choice theorists, concerned with the properties of voting methods, have been investigating preference aggregation for decades. At the heart of this body of work is Arrow's result demonstrating the impossibility of combining preferences in a way that satisfies several desirable and innocuous-looking properties. We show that researchers working on CF algorithms of...
Article
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. Comment: 26 pages
Article
We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by a certain amount of bribing voters a specified candidate can be made the election's winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k-approval, and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P.
Conference Paper
Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Faliszewski et al. [9] proved that Llull voting (which is here denoted by Copeland1) and a variant (here denoted by Copeland0) of Copeland voting are computationally resistant to many, yet not all, types of constructive control and that they also provide broad resistance to bribery. We study a parameterized version of Copeland voting, denoted by Copelandα , where the parameter α is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates in Copeland elections. For each rational α, 0 < α< 1, and each previously studied control scenario, we either prove that Copelandα is computationally vulnerable to control in that scenario (i.e., we give a P-time algorithm that determines whether control is possible, and if so, determines exactly how to exert the control) or we prove that Copelandα is computationally resistant to control in that scenario (i.e., we prove that control problem to be NP-hard). In particular, we prove that Copeland0.5, the system commonly referred to as “Copeland voting,” provides full resistance to constructive control. Among systems with a polynomial-time winner problem, this is the first natural election system proven to have full resistance to constructive control. Looking at rational α, 0 < α< 1, we give a broad set of results on bribery and on the fixed-parameter tractability of bounded-case control for Copelandα (previously only Copeland0 and Copeland1 had been studied), and we introduce and obtain fixed-parameter tractability results even in a new, more flexible model of control (that we dub “extended control”).
Perfect code is W [1]-complete, Information Processing Let-ters
  • M Cesati
M. Cesati, Perfect code is W [1]-complete, Information Processing Let-ters 81 (2002) 163–168.