A Parameterized Perspective on Protecting Elections

05/28/2019 ∙ by Palash Dey, et al. ∙ IIT Kharagpur Indian Institute of Technology Kanpur IIT Gandhinagar 0

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.

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1 Introduction

The problem of election control asks if it is possible for an external agent, usually with a fixed set of resources, to influence the outcome of the election by altering its structure in some limited way. There are several specific manifestations of this problem: for instance, one may ask if it is possible to change the winner by deleting voter groups, presumably by destroying ballot boxes or rigging electronically submitted votes. Indeed, several cases of violence at the ballot boxes have been placed on record [7, 2], and in 2010, Halderman and his students exposed serious vulnerabilities in the electronic voting systems that are in widespread use in several states [1]. A substantial amount of the debates around the recently concluded presidential elections in the United States revolved around issues of potential fraud, with people voting multiple times, stuffing ballot boxes, etc. all of which are well recognized forms of election control. For example, Wolchok et al. [54] studied security aspects on Internet voting systems.

Parameters Optimal Defense Optimal Attack
Scoring rules Condorcet Scoring rules Condorcet

-hard [Observation 1] -hard [Observation 1] -hard [Observation 1] -hard [Observation 1]

-hard [Section 4] -hard [Section 4] -hard [Section 4] -hard [Section 4]

[Section 5]
No poly kernel [Section 4]

para--hard [Section 4] para--hard [Section 4]

Table 1: Summary of parameterized complexity results. the maximum number of voter groups that the defender can defend. the maximum number of voter groups that the attacker can attack. the number of candidates.

The study of controlling elections is fundamental to computational social choice: it is widely studied from a theoretical perspective, and has deep practical impact. Bartholdi et al [4] initiated the study of these problems from a computational perspective, hoping that computational hardness of these problems may suggest a substantial barrier to the phenomena of control: if it is, say -hard to control an election, then the manipulative agent may not be able to compute an optimal control strategy in a reasonable amount of time. This basic approach has been intensely studied in various other scenarios. For instance, Faliszewski et al. [27] studied the problem of control where different types of attacks are combined (multimode control), Mattei et al [44] showed hardness of a variant of control which just exercises different tie-breaking rules, Bulteau et al. [10] studied voter control in a combinatorial setting, etc [49, 52, 28, 11, 43, 31, 30, 29, 26, 45, 25, 24, 24, 34, 37, 33, 36, 32, 47, 48, 51, 14, 21, 20, 16, 17, 15].

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. See, for example, the works of An et al. and Letchford et al. [3, 39].

The broad theme of using computational hardness as a barrier to control has two distinct limitations: one is, of course, that some voting rules simply remain computationally vulnerable to many forms of control, in the sense that optimal strategies can be found in polynomial time. The other is that even

-hard control problems often admit reasonable heuristics, can be approximated well, or even admit efficient exact algorithms in realistic scenarios. Therefore, relying on

-hardness alone is arguably not a robust strategy against control. To address this issue, the work of Yin et al. [56] explicitly defined the problem of protecting an election from control, where in addition to the manipulative agent, we also have a “defender”, who can also deploy some resources to spoil a planned attack. In this setting, elections are defined with respect to voter groups rather than voters, which is a small difference from the traditional control setting. The voter groups model allows us to consider attacks on sets of voters, which is a more accurate model of realistic control scenarios.

In Yin et al. [56], the defense problem is modeled as a Stackelberg game in which limited protection resources (say ) are deployed to protect a collection of voter groups and the adversary responds by attempting to subvert the election (by attacking, say, at most groups). They consider the plurality voting rule, and show that the problem of choosing the minimal set of resources that guarantee that an election cannot be controlled is -hard. They further suggest a Mixed-Integer Program formulation that can usually be efficiently tackled by solvers. Our main contribution is to study this problem in a parameterized setting and provide a refined complexity landscape for it. We also introduce the complementary attack problem, and extend the study to voting rules beyond plurality. We now turn to a summary of our contributions.

Contribution:

We refer the reader to Section 2 for the relevant formal definitions, while focusing here on a high-level overview of our results. Recall that the Optimal Defense problem asks for a set of at most voter groups which, when protected, render any attack on at most voter groups unsuccessful. In this paper, we study the parameterized complexity of Optimal Defense for all scoring rules and the Condorcet voting rule (these are natural choices because they are computationally vulnerable to control - - the underlying “attack problem” can be resolved in polynomial time). We show that the problem of finding an optimal defense is tractable when both the attacker and the defender have limited resources. Specifically, we show that the problem is fixed-parameter tractable with the combined parameter by a natural bounded-depth search tree approach. We also show that the Optimal Defense problem is unlikely to admit a polynomial kernel under plausible complexity theoretic assumption. We observe that both these parameters are needed for fixed parameter tractability, as we show -hardness when Optimal Defense is parameterized by either or .

Another popular parameter considered for voting problems is , the number of candidates — as this is usually small compared to the size of the election in traditional application scenarios. Unfortunately, we show that Optimal Defense is -hard even when the election has only candidates, eliminating the possibility of fixed-parameter algorithms (and even XP algorithms). This strengthens a hardness result shown in Yin et al. [56]. Our hardness results on a constant number of candidates rely on a succinct encoding of the information about the scores of the candidates from each voter group. We also observe that the problem is polynomially solvable when only two candidates are involved.

We introduce the complementary problem of attacking an election: here the attacker plays her strategy first, and the defender is free to defend any of the attacked groups within the budget. The attacker wins if she is successful in subverting the election no matter which defense is played out. This problem turns out to be harder: it is already -hard when parameterized by both and , which is in sharp contrast to the Optimal Defense problem. This problem is also hard in the setting of a constant number of candidates — specifically, it is -hard for the plurality voting rule [Section 3] and the Condorcet voting rule [Section 3] even when we have only three candidates if every voter group is encoded as the number of plurality votes every candidate receives from that voter group. Our demonstration of the hardness of the attack problem is another step in the program of using computational intractability as a barrier to undesirable phenomenon, which, in this context, is the act of planning a systematic attack on voter groups with limited resources.

We finally propose two simple greedy algorithms for the Optimal Defense problem and empirically show that it may be able to solve many instances of practical interest.

2 Preliminaries

Let be a set of candidates and a set of voters. If not mentioned otherwise, we denote the set of candidates by , the set of voters by , the number of candidates by , and the number of voters by . Every voter has a preference or vote which is a complete order over . We denote the set of all complete orders over by . We call a tuple of preferences an -voter preference profile. Often it is convenient to view a preference profile as a multi-set consisting of its votes. The view we are taking will be clear from the context. A voting rule (often called voting correspondence) is a function which selects, from a preference profile, a nonempty set of candidates as the winners. We refer the reader to [9] for a comprehensive introduction to computational social choice. In this paper we will be focusing on two voting rules – the scoring rules and the Condorcet voting rule which are defined as follows.

Scoring Rule: A collection of

-dimensional vectors

with and for every naturally defines a voting rule — a candidate gets score from a vote if it is placed at the position, and the score of a candidate is the sum of the scores it receives from all the votes. The winners are the candidates with the highest score. Given a set of candidates , a score vector of length , a candidate , and a profile , we denote the score of in by . When the score vector is clear from the context, we omit from the superscript. A straight forward observation is that the scoring rules remain unchanged if we multiply every by any constant and/or add any constant . Hence, we assume without loss of generality that for any score vector , there exists a such that and for all . We call such a score vector a normalized score vector.

Weighted Majority Graph and Condorcet Voting Rule: Given an election and two candidates , let us define to be the number of votes where the candidate is preferred over . We say that a candidate defeats another candidate in pairwise election if . Using the election , we can construct a weighted directed graph as follows. The vertex set of the graph is the set of candidates . For any two candidates with , let us define the margin of from to be . We have an edge from to in if . Moreover, in that case, the weight of the edge from to is . A candidate is called the Condorcet winner of an election if there is an edge from to every other vertices in the weighted majority graph . The Condorcet voting rule outputs the Condorcet winner if it exists and outputs the set of all candidates otherwise.

Let be a voting rule. We study the -Optimal Defense problem which was defined by Yin et al. [56]. It is defined as follows. Intuitively, the -Optimal Defense problem asks if there is a way to defend voter groups such that, irrespective of which voter groups the attacker attacks, the output of the election (that is the winning set of candidates) is always same as the original one. A voter group gets deleted if only if it is attacked but not defended.

[-Optimal Defense ] Given voter groups two integers and , does there exist an index set with such that, for every with , we have ? The integers and are called respectively attacker’s resource and defender’s resource. We denote an arbitrary instance of the -Optimal Defense problem by .

We also study the -Optimal Attack problem which is defined as follows. Intuitively, in the -Optimal Attack problem the attacker is interested to know if it is possible to attack voter groups such that, no matter which voter groups the defender defends, the outcome of the election is never same as the original (that is the attack is successful).

[-Optimal Attack ] Given voter groups two integers and , does there exist an index set with such that, for every with , we have ? We denote an arbitrary instance of the -Optimal Attack problem by .

Encoding of the Input Instance: In both the -Optimal Defense and -Optimal Attack problems, we assume that every input voter group is encoded as follows. The encoding lists all the different votes that appear in the voter group along with the number of times the vote appear in . Hence, if a voter group contains only different votes over candidates and consists of voters, then the encoding of takes bits of memory.

Parameterized complexity: In parameterized complexity, each problem instance comes with a parameter . Formally, a parameterized problem is a subset of , where is a finite alphabet. An instance of a parameterized problem is a tuple , where is the parameter. A central notion is fixed parameter tractability (FPT) which means, for a given instance , solvability in time , where is an arbitrary function of and is a polynomial in the input size

. Just as NP-hardness is used as evidence that a problem probably is not polynomial time solvable, there exists a hierarchy of complexity classes above FPT, and showing that a parameterized problem is hard for one of these classes is considered evidence that the problem is unlikely to be fixed-parameter tractable. The main classes in this hierarchy are:

We now define the notion of parameterized reduction [13]. Let be parameterized problems. We say that is fpt-reducible to if there exist functions , a constant and an algorithm which transforms an instance of into an instance of in time so that if and only if .

To show W-hardness in the parameterized setting, it is enough to give a parameterized reduction from a known hard problem. For a more detailed and formal introduction to parameterized complexity, we refer the reader to [13] for a detailed introduction to this paradigm.

[Kernelization] [50, 35] A kernelization algorithm for a parameterized problem is an algorithm that, given , outputs, in time polynomial in , a pair such that (a) if and only if and (b) , where is some computable function. The output instance is called the kernel, and the function is referred to as the size of the kernel. If , then we say that admits a polynomial kernel. For many parameterized problems, it is well established that the existence of a polynomial kernel would imply the collapse of the polynomial hierarchy to the third level (or more precisely, ). Therefore, it is considered unlikely that these problems would admit polynomial-sized kernels. For showing kernel lower bounds, we simply establish reductions from these problems.

[Polynomial Parameter Transformation] [8] Let and be parameterized problems. We say that is polynomial time and parameter reducible to , written , if there exists a polynomial time computable function , and a polynomial , and for all and , if , then if and only if , and . We call a polynomial parameter transformation (or a PPT) from to .

This notion of a reduction is useful in showing kernel lower bounds because of the following theorem.

[8, Theorem 3] Let and be parameterized problems whose derived classical problems are , respectively. Let be , and . Suppose there exists a PPT from to . Then, if has a polynomial kernel, then also has a polynomial kernel.

3 Classical Complexity Results

Yin et al. [56] showed that the Optimal Defense problem is polynomial time solvable for the plurality voting rule when we have only candidates. On the other hand, they also showed that the Optimal Defense problem is -complete when we have an unbounded number of candidates. We begin with improving their -completeness result by showing that the Optimal Defense problem becomes -complete even when we have only candidates and the attacker can attack any number of voter groups. Towards that, we reduce the -Sum problem to the Optimal Defense problem. The -Sum problem is defined as follows.

[-Sum] Given a set of positive integers and two positive integers and , does there exist an index set with such that ?

The -Sum problem can be easily proved to be -complete by modifying the -completeness proof of the Subset Sum problem in Cormen et al. [12]. We also need the following structural result for normalized scoring rules which has been used before [5, 19].

Let be a set of candidates and a normalized score vector of length . Let be any two arbitrary candidates. Then there exists a profile consisting of votes such that we have the following.

For any two candidates , we use to denote the profile as defined in Section 3. We are now ready to present our -completeness result for the Optimal Defense problem for the scoring rules even in the presence of candidates only. In the interest of space, we will provide only a sketch of a proof for a several results.

The Optimal Defense problem is -complete for every scoring rule even if the number of candidates is and the attacker can attack any number of the voter groups.

Proof.

The Optimal Defense problem for every scoring rule can be shown to belong to by using a defense strategy  (a subset of at most voter groups) as a certificate. The fact that the certificate can be validated in polynomial time involves checking if there exists a successful attack despite protecting all groups in . This can be done in polynomial time, but due to space constraints, we defer a detailed argument to a full version of this manuscript. We now turn to the reduction from -Sum.

Let be any normalized score vector of length . The Optimal Defense problem for the scoring rule based on belongs to . Let be an arbitrary instance of the -Sum problem. We can assume, without loss of generality, that divides and for every ; if this is not the case, we replace and by respectively and for every which clearly is an equivalent instance of the original instance. Let us also assume, without loss of generality, that (if not then add enough copies of to ) and (since otherwise, it is a trivial No instance). We construct the following instance of the Optimal Defense problem for the scoring rule based on . Let be an integer such that and divides . We have candidates, namely , , and . We have the following voter groups.

  • [topsep=0pt,itemsep=0pt,leftmargin=*]

  • For every , we have a voter group consisting of copies of (as defined in Section 3) and copies of . Hence, we have the following.

  • We have one voter group consisting of copies of , copies of , and copies of . We have the following.

Let be the resulting profile; that is . We have . Since and , we have and . Thus the candidate wins the election uniquely. We define , the maximum number of voter groups that the defender can defend, to be . We define , the maximum number of voter groups that the attacker can attack, to be . This finishes the description of the Optimal Defense instance. We claim that the two instances are equivalent.

In the forward direction, let the -Sum instance be a Yes instance and with be an index set such that . Let us consider the defense strategy where the defender protects the voter groups for every . Since , we have . Let be the profile of voter groups corresponding to the index set ; that is, . Let be the profile remaining after the attacker attacks some voter groups. Without loss of generality, we can assume that the attacker does not attack the voter group since otherwise the candidate continues to win uniquely. We thus obviously have . We have and . Since the candidate receives as much score as any other candidate in the voter group for every , we have and . Hence, the candidate wins uniquely in the resulting profile after the attack and thus the defense is successful.

In the other direction, let the Optimal Defense instance be a Yes instance. Without loss of generality, we can assume that the attacker does not attack the voter group and thus the defender does not defend the voter group . We can also assume, without loss of generality, that the defender defends exactly voter groups since the candidate receives as much score as any other candidate in the voter group for every . Let with such that defending all the voter groups is a successful defense strategy. We claim that . Suppose not, then let us assume that . Since, is divisible by and positive for every and is divisible by , we have . Let be the profile of voter groups corresponding to the index set ; that is, . We have . Hence attacking the voter groups makes the score of strictly less than the score of . This contradicts our assumption that defending all the voter groups is a successful defense strategy. Hence we have . We now claim that . Suppose not, then let us assume that . Since, is divisible by and positive for every and is divisible by , we have . Let be the profile of voter groups corresponding to the index set ; that is, . We have . Hence attacking the voter groups makes the score of strictly less than the score of . This contradicts our assumption that defending all the voter groups is a successful defense strategy. Hence we have . Therefore we have and thus the -Sum instance is a Yes instance. ∎

In the proof of Section 3, we observe that the reduced instance of the Optimal Defense problem viewed as an instance of the Optimal Attack problem is a No instance if and only if the -Sum instance is a Yes instance. Hence, the same reduction as in the proof of Section 3 gives us the following result for the Optimal Attack problem.

The Optimal Attack problem is -hard for every scoring rule even if the number of candidates is and the attacker can attack any number of voter groups.

We now prove a similar hardness result as of Section 3 for the Condorcet voting rule.

The Optimal Defense problem is -complete for the Condorcet voting rule even if the number of candidates is and the attacker can attack any number of voter groups.

Proof.

The Optimal Defense problem for the Condorcet voting rule clearly belongs to . To show -hardness, we reduce an arbitrary instance of the -Sum problem to the Optimal Defense problem for the Condorcet voting rule. Let be an arbitrary instance of the -Sum problem. We construct the following instance of the Optimal Defense problem for the Condorcet voting rule. Let . We have candidates, namely , , and . We have the following voter groups.

  • [topsep=0pt,itemsep=0pt]

  • For every , we have a voter group where , and .

  • We have one voter group where the candidates and receive respectively , and .

We define , the maximum number of voter groups that the defender can defend, to be . We define , the maximum number of voter groups that the attacker can attack, to be . We observe that the candidate is the Condorcet winner of the election. This finishes the description of the Optimal Defense instance. We claim that the two instances are equivalent.

In the forward direction, let the -Sum instance be a Yes instance and with be an index set such that . Let us consider the defense strategy where the defender protects the voter groups for every . Since , we have . Without loss of generality, we can assume that the attacker does not attack the voter group . We observe that the candidate is the Condorcet winner of the election even when the attacker attacks all the voter groups . Hence the Optimal Defense instance is a Yes instance.

In the other direction, let the Optimal Defense instance be a Yes instance. Without loss of generality, we can assume that the attacker does not attack the voter group and thus the defender does not defend the voter group . We can also assume, without loss of generality, that the defender defends exactly voter groups since the candidate continues to be the Condorcet winner if the attacker attacks at most voter groups. Let with such that defending all the voter groups is a successful defense strategy. We claim that . Suppose not, then let us assume that . Then attacking the voter groups makes the candidate defeat the candidate in pairwise election. This contradicts or assumption that defending all the voter groups is a successful defense strategy. Hence we have . We now claim that . Suppose not, then let us assume that . Then attacking the voter groups makes the candidate defeat the candidate in pairwise election. This contradicts or assumption that defending all the voter groups is a successful defense strategy. Hence we have . Therefore we have and thus the -Sum instance is a Yes instance. ∎

In the proof of Section 3, we observe that the reduced instance of Optimal Defense viewed as an instance of the Optimal Attack problem is a No instance if and only if the -Sum instance is a Yes instance. Hence, the same reduction as in the proof of Section 3 gives us the following result for the Optimal Attack problem.

The Optimal Attack problem is -hard for the Condorcet voting rule even if the number of candidates is and the attacker can attack any number of voter groups.

4 W-Hardness Results

In this section, we present our hardness results for the Optimal Defense and the Optimal Attack problems in the parameterized complexity framework. We consider the following parameters for both the problems – number of candidate (), defender’s resource (), and attacker’s resource (). From Sections 3, 3, 3 and 3 we immediately have the following result for the Optimal Defense and Optimal Attack problems parameterized by the number of candidates for both the scoring rules and the Condorcet voting rule.

The Optimal Defense problem is para--hard parameterized by the number of candidates for both the scoring rules and the Condorcet voting rule. The Optimal Attack problem is para--hard parameterized by the number of candidates for both the scoring rules and the Condorcet voting rule.

The -completeness proof for the Optimal Defense problem for the plurality voting rule by Yin et al. [56] is actually a parameter preserving reduction from the Hitting Set problem parameterized by the solution size. The Hitting Set problem is defined as follows.

[Hitting Set ] Given a universe , a set of subsets of , and a positive integer which is at most , does there exist a subset with such that for every . We denote an arbitrary instance of Hitting Set by . Since the Hitting Set problem parameterized by the solution size is known to be -complete [23], the following result immediately follows from Theorem 2 of Yin et al. [56].

Observation 1 ([56]).

The Optimal Defense problem for the plurality voting rule is -hard parameterized by .

We now generalize Observation 1 to any scoring rule by exhibiting a polynomial parameter transform from the Hitting Set problem parameterized by the solution size.

The Optimal Defense problem for every scoring rule is -hard parameterized by .

Proof.

Let be an arbitrary instance of Hitting Set. Let . Without loss of generality, we assume that for every since otherwise the instance is a No instance. Let be a normalized score vector of length . We construct the following instance of the Optimal Defense problem for the scoring rule based on . The set of candidates . We have the following voter groups.

  • [topsep=0pt,itemsep=0pt]

  • For every , we have a voter group . For every with we have copies of in .

  • We have one group where we have copies of for every and copies of .

Let be the resulting profile; that is . We define the defender’s resource to be and attacker’s resource to be . This finishes the description of the Optimal Defense instance. Since for every , we have for every . We also have . Hence the candidate is the unique winner of the profile . We now prove that the Optimal Defense instance is equivalent to the Hitting Set instance .

In the forward direction, let us suppose that the Hitting Set instance is a Yes instance. Let be such that and . We claim that the defender’s strategy of defending the voter groups for every and results in a successful defense. Let be the profile of voter groups corresponding to the index set ; that is, . Let be the profile remaining after the attacker attacks some voter groups. We thus obviously have . Since forms a hitting set, we have for every . Also since the voter group is defended, we have . Hence the candidate continues to win uniquely even after the attack and hence the Optimal Defense instance is a Yes instance.

In the other direction, let the Optimal Defense instance be a Yes instance. Without loss of generality, we can assume that the defender defends the voter group since otherwise the attacker can attack the voter group which makes the score of the candidate more than the score of the candidate and thus defense would fail. We can also assume, without loss of generality, that the defender defends exactly voter groups. Let with such that defending all the voter groups and is a successful defense strategy. Let us consider . We claim that must form a hitting set. Indeed, otherwise let us assume that there exists a such that . Consider the situation where the attacker attacks voter groups for every . We observe that . This contradicts our assumption that defending all the voter groups and is a successful defense strategy. Hence forms a hitting set and thus the Hitting Set instance is a Yes instance. ∎

In the proof of Observation 1, we observe that the reduced instance of Optimal Defense viewed as an instance of the Optimal Attack problem is a No instance if and only if the -Sum instance is a Yes instance. Hence, the same reduction as in the proof of Observation 1 gives us the following result for the Optimal Attack problem.

The Optimal Attack problem for every scoring rule is -hard parameterized by .

We now show -hardness of the Optimal Defense problem for the Condorcet voting rule parameterized by . Towards that, we need the following lemma which has been used before [46, 55]. For any function , such that

  1. .

  2. is even,

there exists a voters’ profile such that for all , defeats with a margin of . Moreover,

Next, we show the -hardness of the Optimal Defense problem for the Condorcet voting rule parameterized by . This is also a parameter-preserving reduction from the Hitting Set problem.

The Optimal Defense problem for the Condorcet voting rule is -hard parameterized by .

Proof.

Let be an arbitrary instance of Hitting Set. Let . Without loss of generality, we assume that for every since otherwise the instance is a No instance. We construct the following instance of the Optimal Defense problem for the Condorcet voting rule. The set of candidates . For every , we have a voter group . For every with we have . Let be the resulting profile; that is . We define the defender’s resource to be and attacker’s resource to be . This finishes the description of the Optimal Defense instance. Since for every , we have for every . Hence the candidate is the Condorcet winner of the profile . We now prove that the Optimal Defense instance is equivalent to the Hitting Set instance .

In the forward direction, let us suppose that the Hitting Set instance is a Yes instance. Let be such that and . We claim that the defender’s strategy of defending the voter groups for every results in a successful defense. Let be the profile of voter groups corresponding to the index set ; that is, . Let be the profile remaining after the attacker attacks some voter groups. We thus obviously have . Since forms a hitting set, we have for every . Hence the candidate continues to win uniquely even after the attack and hence the Optimal Defense instance is a Yes instance.

In the other direction, let the Optimal Defense instance be a Yes instance. We can also assume, without loss of generality, that the defender defends exactly voter groups. Let with such that defending all the voter groups is a successful defense strategy. Let us consider . We claim that must form a hitting set. Indeed, otherwise let us assume that there exists a such that . Consider the situation where the attacker attacks voter groups for every . We observe that and hence the candidate is not the Condorcet winner. This contradicts our assumption that defending all the voter groups is a successful defense strategy. Hence forms a hitting set and thus the Hitting Set instance is a Yes instance. ∎

In the proof of Observation 1, we observe that the reduced instance of Optimal Defense viewed as an instance of the Optimal Attack problem is a No instance if and only if the -Sum instance is a Yes instance. Hence, the same reduction as in the proof of Observation 1 gives us the following result for the Optimal Attack problem.

The Optimal Attack problem for the Condorcet voting rule is -hard parameterized by .

We now show that the Optimal Defense problem for scoring rules is -hard parameterized by also by exhibiting a parameter preserving reduction from a problem closely related to Hitting Set, which is Set Cover problem parameterized by the solution size. The Set Cover problem is defined as follows. This is a -complete problem [23]. We now present our -hardness proof for the Optimal Defense problem for scoring rules parameterized by , by a reduction from Set Cover.

[Set Cover ] Given an universe , a set of subsets of , and a non-negative integer which is at most , does there exists an index set with such that . We denote an arbitrary instance of Set Cover by .

The Optimal Defense problem for every scoring rule and Condorcet rule is -hard parameterized by .

The Optimal Defense problem for every scoring rule is -hard parameterized by .

Proof.

Let be an arbitrary instance of Set Cover. Let . We assume that since otherwise the Set Cover instance is polynomial time solvable. For , let be the number of such that ; that is, . We assume, without loss of generality, that for every , by adding at most empty sets in SS. We construct the following instance of the Optimal Defense problem for the scoring rule induced by the score vector rule. The set of candidates . Let be any normalized score vector of length . We have the following voter groups.

  • [topsep=0pt,itemsep=0pt]

  • For every , we have a voter group . For every and with , we have copies of .

  • We have another voter group where, for every , we have copies of and copies of .

We define attacker resource to be and the defender’s resource to be . This finishes the description of the Optimal Defense instance. We first observe that the score of the candidate is strictly less than the score of every other candidate. We now observe that the candidate is the unique winner of the election since the score of the candidate is more than the score of the candidate for every . We now prove that the Optimal Defense instance is equivalent to the Set Cover instance .

In the forward direction, let us suppose that the Set Cover instance is a Yes instance. Let be such that and . We claim that the defender’s strategy of defending the voter groups for every results in a successful defense. To see this, we first observe that, if the attacker attacks the voter group , then the candidate continues to uniquely win the election irrespective of what other voter groups the attacker attacks. Indeed, since for every , the score of the candidate is strictly less than the score of the candidate irrespective of what other voter groups the attacker attacks. Since, for every and , the score of the candidate is not more than the score of the candidate in the voter group , we may assume that the attacker attacks the voter group for every (since they are the only voter groups unprotected except ). Now, since forms a set cover of , after deleting the voter groups , the score of the candidate increases by at most from the original election for every . Hence, after deleting the voter groups , the score of the candidate is still strictly less than the score of the candidate . Hence the candidate continues to win and thus the defense is successful. Hence the Optimal Defense instance is a Yes instance.

In the other direction, let us suppose that the Optimal Defense instance is a Yes instance. We assume, without loss of generality, that the defender protects exactly voter groups. We argued in the forward direction that we can assume, without loss of generality, that the attacker never attacks the voter group . Hence, we can also assume, without loss of generality, that the defender also does not defend the voter group . Let be such that and the defender defends the voter group for every . We claim that the sets forms a set cover of . Suppose not, then let be an element in which is not covered by . We observe that attacking the voter groups for every increases the score of the candidate by which makes the candidate lose in the resulting election (after deleting the voter groups for every ) since the score of is strictly more than the score of . This contradicts our assumption that defending the voter group for every is a successful defense strategy. Hence forms a set cover of and thus the Set Cover instance is a Yes instance. ∎

We now present our -hardness proof for the Optimal Defense problem for the Condorcet voting rule parameterized by . The Optimal Defense problem for the Condorcet voting rule is -hard parameterized by .

Proof.

Let be an arbitrary instance of Set Cover. Let . We assume that since otherwise the Set Cover instance is polynomial time solvable. For , let be the number of such that ; that is, . We assume, without loss of generality, that for every , by adding at most empty sets in SS. We construct the following instance of the Optimal Defense problem for the Condorcet voting rule. The set of candidates . We have the following voter groups.

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  • For every , we have a voter group . For every and , we have if and otherwise. We also have for every with .

  • We have another voter group where, for every , we have . We also have for every with .

We define attacker resource to be and the defender’s resource to be . This finishes the description of the Optimal Defense instance. We first observe that the candidate is a Condorcet winner of the resulting election. We now prove that the Optimal Defense instance