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# Parameterized complexity of control problems in Maximin election

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## 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”.

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... 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
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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. ...
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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]. ...
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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.
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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.
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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.
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Conference Paper
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Conference Paper
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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.
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