A Parameterized Perspective on Attacking and Defending Elections

We consider the problem of protecting and manipulating elections by recounting and changing ballots, respectively. Our setting involves a plurality-based election held across multiple districts, and the problem formulations are based on the model proposed recently by [Elkind et al, IJCAI 2019]. It turns out that both of the manipulation and protection problems are NP-complete even in fairly simple settings. We study these problems from a parameterized perspective with the goal of establishing a more detailed complexity landscape. The parameters we consider include the number of voters, and the budgets of the attacker and the defender. While we observe fixed-parameter tractability when parameterizing by number of voters, our main contribution is a demonstration of parameterized hardness when working with the budgets of the attacker and the defender.

Authors

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• A Parameterized Perspective on Protecting Elections

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• Protecting Elections by Recounting Ballots

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1 Introduction

Electoral fraud and errors in consolidation of large-scale voting data are fundamental issues in democratic societies. To counteract issues with malicious manipulations and accidental errors in the counting of votes, most electoral systems allow for strategic recounting of ballots to verify the election outcome. Recounting is generally an expensive and high-stakes process, and it would be desirable to formalize the problem so as to capture the relevant trade-offs and possibly pursue an algorithmic approach to finding an optimal recounting strategy. Such a framework was recently proposed by Elkind et al. [ElkindGORV19], where the authors considered the problems of protecting and manipulating elections by recounting and changing ballots, respectively. These problems are modeled as a Stackelberg game involving an attacker and a defender. Both players work with limited budgets (say and ), and the question is if the players can develop optimal strategies for their desired outcomes.

In this model, the election is spread out across multiple districts, with the voter preferences aggregated according to the plurality voting rule in one of two different ways, which we will explain explicitly in a moment. The manipulation problem is the following. The attacker has to optimize, typically with the goal of turning the election in favor of a particular candidate that he may have in mind, an attack strategy that involves manipulating the votes in at most

districts while ensuring that the impact of the attack persists even if the defender restores at most of these districts to their original state. In the recounting problem, the defender is given complete information about the original and manipulated voting profiles and she can restore the state of at most districts with the goal of making the “true winner” win the repaired election.

Known Results.

The results obtained in [ElkindGORV19] already demonstrate the hardness of the attacker’s and defender’s problems for two natural ways of aggregating the votes: (1) Plurality over Voters (PV), where districts are only used for the purpose of collecting the ballots and the winner is selected among the candidates that receive the largest number of votes in total, and (2) Plurality over Districts (PD), where each district selects a preferred candidate using the Plurality rule, and the overall winner is chosen among the candidates supported by the largest number of districts, or the set of districts with largest total weight if the districts have weights associated with them. We briefly recall the main highlights from [ElkindGORV19], since this provides the context for our contributions. It turns out that the recounting problem is NP-complete for both implementations of the plurality rule, even when there are only three candidates (this result assumes a succinct representation of the votes) or even when the votes are specified in unary. The problem is tractable when PD is employed over unweighted districts. On the other hand, the manipulation problem is NP-hard for PD even with unweighted districts, and in fact, -complete for PD with succinct input even when there are only three candidates. Further, it is NP-hard for PV, again even when there are only three candidates (in the setting of succinct input) or even when the votes are specified in unary.

Our Contributions.

Our main contribution is to establish the parameterized intractability of both the recounting and manipulation problems under both implementations of the plurality voting rule when parameterized by the budget of the players. Our contributions directly address a direction suggested by [ElkindGORV19]. In particular, we obtain the following results:

The PV-Rec and PD-Rec problems are W[2]-hard and W[1]-hard, respectively, when parameterized by the budget of the defender, and are FPT when parameterized by the number of districts.

The PV-Man and PD-Man problems are W[1]-hard when parameterized by the budget of the attacker (even when the defender budget is zero), and are FPT when parameterized by the number of voters.

Our results rely on reductions from traditional problems such as Multi-Colored Clique and Dominating Set. Our hardness results work even when the input is specified in unary. It is reasonably natural to imagine that these parameters would be small in practice, since they correspond to real-world budget constraints. To that end, our results here bring mixed news: on the one hand, the hardness of mounting an attack may be viewed as a positive outcome, but on the other hand, it turns out that the problem of optimally reversing damages is hard as well. This triggers the natural question of whether the recounting problem admits good approximations when treated as an optimization problem, either on the criteria of the budget or on the criteria of the quality of the winning candidate that we are able to restore. The FPT algorithms that we present rely mostly on straightforward enumeration, and it would be interesting to improve the running times in question.

Related Work.

While we build most closely on the work of Elkind et al [ElkindGORV19], and much of the work on manipulation in the literature of social choice does not consider the possibility of a counter-attack, we note that some recent investigations have been carried out in a spirit that is similar to our present contribution. Dey et al. [ijcai2019-34] also consider a parameterized approach to protecting elections, where the voting rule in question is the Condorcet rule. They build on the work of Yin et al. [YinVAH16], who study a pre-emptive approach to protecting elections. In these models, the defender allocates resources to guard some of the electoral districts, so that the votes there cannot be influenced, and the attacker responds afterward. This is in contrast to our setting, where the defender makes the second move. The social choice literature is rich in studies of manipluation, control, and bribery. For a detailed overview we direct the reader to the surveys [ConitzerW16, FaliszewskiR16].

2 Preliminaries

We recall the setting from [ElkindGORV19]. We consider elections over a candidate set , where . There are voters who are partitioned into pairwise disjoint districts. The set of all districts is . For each , let . We note that in this context, denotes a subset of voters. For each , district has a weight , which is a positive integer. We say that an election is unweighted if for all . Each voter votes for a single candidate in . For each and each let denote the number of votes that candidate gets from voters in . We refer to the list as the vote profile.

Let be a linear order over ; indicates that is favored over . We consider the following two voting rules, which take the vote profile v as their input.

• Plurality over Voters (PV): We say that a candidate beats a candidate under PV if or and ; the winner is the candidate that beats all other candidates. Note that district weights are not relevant for this rule.

• Plurality over Districts (PD): For each the winner in is chosen from the set , with ties broken according to . Then, for each , , we set if , else . We say that a candidate beats a candidate under PD if or and . The winner is the candidate that beats all other candidates. Given the voting profile v, we take the winner in district to be . We shall omit the subscript if the voting profile to be used is clear from the context.

For PV and PD, we define the social welfare of a candidate as the total number of votes that gets and the total weight that gets, respectively:

 SWPV(c)=∑i∈Dvic,SWPD(c)=∑i∈Dwic

Hence, the winner under each voting rule is a candidate with the maximum social welfare. We now define some additional terminology that we use later in this paper:

• Score: Given a voting profile , under the PD rule, the score of a candidate is defined as the sum of weights of districts in which the candidate wins. Formally, (Here is the Kronecker delta function).

• Rivals: Given a voting profile , under the PD rule, we define the rivals of a candidate to be the set of candidates such that for all candidates , either or and .

We now consider the scenario where an election may be manipulated by an attacker, who wants to change the result of the election in favor of his preferred candidate . The attacker has a budget which enables him to change the voting profiles in at most districts. For each district , we define to be the number of votes that the attacker can manipulate in . After the manipulation we have a voting profile . We formalize the notion of a manipulation as a set and a voting profile such that where for all and for all it holds that and .

After the attacker, a socially minded defender with budget can demand a recount in at most districts. Formally, a recounting strategy is a set such that and . After the defender recounts, the vote counts of the districts in are restored to their original values. This results in a new voting profile where for all and for all . Then the voting rule is applied to the profile that is obtained to obtain a winner (let it be ) with ties broken according to . The defender’s objective is to maximize . It is a game of perfect information i.e. both entities know all information about the game.

We say that the attacker wins if he has a manipulation strategy such that after the defender moves optimally, the preferred candidate of attacker i.e. wins. We define the following two decision problems based on voting rule and the two entities:

• -Man: Given the voting rule , a voting profile v, the linear order , a preferred candidate , attacker budget , defender budget and weights and parameter for each district , does the attacker have a winning strategy?

• -Rec: Given the voting rule , a voting profile v, a manipulated voting profile , a preferred candidate , the linear order , defender budget and weights for each district , can the defender make win by recounting at most districts?

Next we state the definitions and parameterized hardness results for decision problems that are used throughout this paper, and refer the reader to [downey99parameterized, CyganEtAl] for a detailed introduction to parameterized complexity and the framework of parameterized reductions.

• Dominating Set: A set of vertices is a dominating set in graph if . Dominating Set asks that given a graph and a non-negative integer , does there exist a dominating set of size at most ? Dominating Set is known to be W[2]-hard parameterized by  [downey99parameterized].

• Multi-Colored Clique: Given a graph and a partition of the vertex set into color classes , Multi-Colored Clique asks whether there exists a clique of size with one vertex each from . Multi-Colored Clique is known to be W[1]-hard parameterized by  [FellowsHRV09, Pietrzak03].

3 Plurality over Voters (PV)

In this section, we analyze the parameterized complexity of PV-Rec and PV-Man with different parameters. It is easy to see that PV-Rec is FPT when parameterized by the number of districts (), since we may guess the districts to be recounted. Since , the problem is also FPT parameterized by the number of voters.

PV-Rec is FPT when parameterized by the no. of districts or the number of voters .

We now show that PV-Rec is W[2]-hard when parameterized by budget of the defender (). Before describing the construction formally, we briefly outline the main idea. We reduce from the Dominating Set, which is well-known to be W[2]-hard parameterized by the size of the solution, which we denote by . Let be an instance of Dominating Set. We create an instance of PV-Rec where we have candidates and districts corresponding to vertices of , along with a special candidate who is our desired winner. To begin with, we have an “immutable” district — one where the original and manipulated votes are identical — that sets the baseline score of the special candidate at . The number votes for any other candidate from this district is fixed to ensure that the total number of votes for is also . In a district corresponding to a vertex , every candidate corresponding to a vertex gets one vote. In the original scenario, all voters in these districts vote only for some dummy candidates. The key is that a “switch” in a district corresponding to a vertex reduces the vote count for all vertices in . Since receives no votes from any of the other districts, observe that the only way for to emerge as a unique winner is if all of the other candidates lose votes from the switches. It is not hard to infer from here that the defender has a valid switching strategy if and only if has a dominating set of size at most .

PV-Rec is W[2]-hard parameterized by , the defender’s budget.

Proof.

We present an FPT reduction from the Dominating Set problem. Let be an instance of Dominating Set. Let and . We begin by describing the construction of the reduced instance.

Districts: We introduce a baseline district . Further, for each vertex , we introduce a corresponding primary district .

Candidates: For each vertex we introduce a main candidate and a dummy candidate . We also have a special candidate .

Voting Outcomes: The voting outcomes are as follows. For ease of presentation, let be arbitrary but fixed.

1. The special candidate does not receive any votes from the primary districts in either the original or the manipulated settings. In particular, .

2. In the original election, the main candidates have no votes in the primary districts, that is, for all .

3. In the manipulated election, a main candidate has a single vote in its favor in the primary district if and only if . Formally,

 ¯¯¯vDv,cu={1if u∈NG[v]0otherwise
4. In the original election, a dummy candidate has a score of in the primary district corresponding to , and a score of zero everywhere else. Formally,

 vDv,du={d(v)+1if u=v0otherwise
5. The dummy candidates receive no votes in the primary districts in the manipulated elections.

6. In the baseline district, the score of the main candidates is defined to ensure that their total score in the manipulated election is . In particular, .

7. The dummy candidates receive no votes in the baseline district in both the original and manipulated elections.

8. The score of is in the baseline district in both the original and manipulated elections. In particular, .

To summarize, the primary districts corresponding to a vertex have voters, and all main candidates corresponding to vertices in get one vote each in the manipulated world; while the dummy candidate gets all the votes in the original world. Observe that in the manipulated election, all the candidates except the dummy candidates have a total score of , while the dummy candidates have a score of zero.

We set . The preferred candidate is . We also work with the following tie-breaking order: where the main candidates are preferred over the special candidate, but the special candidate dominates the dummy candidates. This completes the description of the constructed instance. We now turn to the proof of equivalence.

Forward Direction. Let a dominating set of size at most be given. Now, select the districts for all to recount. After recounting, for every vertex in the dominating set, the votes of for all will decrease by one. Since , all candidates corresponding to vertices will lose at least one vote each. Also, no dummy candidate can gain more than votes after recounting. So, has more votes than any main candidate and at least as many votes as any of the dummy candidates. Therefore, will win after recounting.

Reverse Direction. Conversely, suppose that the defender has a valid recounting strategy that results in making the winner. Since , so at most districts can be recounted. Observe that any optimal solution will not recount the baseline district, since it does not affect the outcome, so without loss of generality, every recounted district corresponds to a vertex . We claim that the vertices corresponding to the recounted districts, which we denote by , constitute a dominating set in .

We argue this by contradiction — indeed, assume that is not a dominating set for . Then there exists at least one vertex such that . Then, observe that the score of the candidate remains after the recounting. In particular, has the same score as the special candidate after recounting, since the construction ensures that it is not possible to change the score of by recounting. Since the main candidates dominate the special candidate in the tie-breaking order, we have that wins the election, which contradicts our assumption that was a valid recounting strategy. ∎

We now turn our attention to PV-Man. First, we prove that PV-Man is FPT parameterized by the number of voters. This follows by first observing that without loss of generality, since at most candidates can have a non-trivial score across the original and manipulated instances combined, and the remaining candidates are irrelevant to the instance. The algorithm can then proceed by guessing a manipulation strategy — note that the space of all possible strategies is bounded once the candidates are bounded — and then invoking the PD-Rec algorithm from the previous section as a subroutine to verify the validity of the guessed strategy. Thus, we have the following.

PV-Man is FPT parameterized by , the number of voters.

We now show that PV-Man is W[1]-hard parameterized by budget of the attacker (). Before describing the construction formally, we briefly outline the main idea. We reduce from the Multi-Colored Clique problem, which is well-known to be W[1]-hard parameterized by the number of color classes, which we denote by . Let be an instance of Multicolored-Clique. In the reduced instance, we introduce a special candidate who is the preferred candidate of the defender. The rival candidates are candidates corresponding to color classes

, ordered pairs of color classes

and vertices . We also introduce some dummy candidates. We introduce districts corresponding to each and each . Also, there exists a special district which is “immutable”, which sets up the initial scores of all candidates such that they are equal to a large number (say ). Initially, has votes. The scores are set up in such a way that the attacker has to transfer votes of districts to to make her win. The scores are engineered to ensure that the attacker has a successful manipulation strategy if and only if these districts correspond to vertices and edges that form a multicolored clique in .

PV-Man is W[1]-hard parameterized by , the attacker’s budget.

Proof.

We demonstrate a parameterized reduction from the Multi-Colored Clique problem. Let be an instance of Multi-Colored Clique. We begin by describing the construction.

Districts: There are two types of districts. We have a primary district for each vertex and two secondary districts and for each edge . Apart from these, there is a baseline district .

Candidates: For each vertex we will have a main candidate . Also, we have challenger candidates corresponding to color classes ’s and ordered pairs of color classes ’s. We introduce some dummy candidates of an unspecified number, whose role is equalize the number of votes across primary and secondary districts. Finally, we have a special candidate .

Voting profiles: We introduce the following voting outcomes for the candidates.

1. The score of the special candidate is zero in all districts.

2. A main candidate for has a score of in the primary district corresponding to , and a score of zero in all other primary districts.

3. A main candidate for has a score of one in every secondary district corresponding to an edge that it is not incident to, provided and share the same color class. In particular, if and there are edges incident on , then has a score of one in secondary districts. Formally, assuming , we have:

 vDuw,cv={1if u≠v and% u∈Vi0otherwise
4. A challenger candidate corresponding to color class has one vote from any primary district corresponding to a vertex and a score of zero from all other primary districts. In other words:

 vDv,Ri={1if v∈Vi0otherwise
5. A challenger candidate corresponding to an ordered pair of color classes has one vote from any secondary district corresponding to an edge whose endpoints and are in color classes and respectively, and a score of zero from all other secondary districts. Note that the candidates and receive scores of one from distinct secondary districts. Specifically, we have: if and .

Now, let be the size of the largest — in terms of the number of voters — among the primary and secondary districts constructed so far. For every primary or secondary district with voters, we add dummy voters and dummy candidates, and we let each dummy voter vote for a distinct dummy candidate. We let .

We are now ready to specify the voting outcomes from the baseline district. these are simply designed to ensure that all primary candidates get votes and the challenger candidates get votes, which is easy to verify from the proposed outcomes below:

1. )

Note that apart from the above, the dummy candidates get vote each, and the special candidate has votes. Also, the attacker has no room to manipulate in the baseline district, that is, . On the other hand, the attacker can modify up to votes in the primary and secondary districts. Further, we set and . The preferred candidate is . Finally, we impose the following tie-breaking order:

 …Ri…≻…Rij…≻…cv…≻w≻… dummies….

This completes the description of the constructed instance. We now turn to the proof of equivalence.

Forward Direction. Let a multi-colored clique of size be given. The attacker chooses the primary and secondary districts corresponding to the vertices and edges of , and transfers all the votes in these districts to the desired candidate . The score of is now . Further, note that that the scores of all challenger candidates has decreased by one to , and the scores of all main candidates have decreased by as well, but for different reasons: for main candidates corresponding to vertices of the clique, the drop is directly from the recounting in the primary districts, while for any other main candidate, the drop is cumulative across relevant secondary districts. In particular, suppose . Consider any such that , and observe that had a score of one in the following districts:

 Dviv1,…,Dvivi−1,Dvivi+1,…,Dvivk,

which have indeed been attacked, and therefore the score of reduces by . This leaves all candidates ranked ahead of the special candidate in the tie-breaking order with a score less than the final score of , and the scores of the dummy candidates is either zero or one, thus they pose no threat to . Therefore, wins the election under this attack, concluding the argument in the forward direction.

Reverse Direction. Suppose that the attacker has a valid manipulation strategy that results in making the winner. With a budget of and manipulable districts with voters, recalling that , we observe that the maximum score that an attacker an achieve for the special candidate is . Since challenger candidates have a score of and are ahead of in the tie-breaking order, we conclude that for each , must contain at least one primary district corresponding to a vertex from — indeed, if not, the attack does not influence the score of challenger candidates corresponding to some color class, and attack will not result in a win for . We claim that these vertices correspond to a clique in . We argue this claim by contradiction. To begin with, we note that analogous to the vertex-based challenger candidates, the attack is forced to have a certain structure to account for the edge-based challeger candidates. In particular, for each ordered pair of color classes , the attack must recount in a secondary district such that and . Note that the existence of such a district implies that . Also, recalling the argument made earlier for the existence of primary districts corresponding to each color class in the attack, and combining this with the available budget of , it is easy to see that any successful attack has the following specific form: it involves exactly primary districts corresponding to vertices from different color classes, and secondary districts corresponding to each ordered pair of color classes.

Now, let the vertices derived from the attack be where , and suppose, for the sake of contradiction, that . The challenger candidates and force the attacker to recount in two secondary districts corresponding to edges that have endpoints in and . Suppose the secondary district recounted to account for was , with and . Suppose, without loss of generality, that . We know from the original score of and the maximum possible final score of that a successful attack needs to ensure that the score of reduces by at least . However, note that this decrease cannot be addressed by a primary district in the attack , since this is already “used up” for . Also, among the remaining secondary districts chosen, it is easy to verify that there are at most that give a score of one to the candidate , and therefore, we can only hope to reduce the score of by at most overall. Note that we are in this situation because has a score of zero in the district , which might be intuitively viewed as a “lost opportunity” for reducing the score of . This establishes that the attack in question is futile in it’s effectiveness towards making win, clearly contrary to our assumption. Therefore, must choose primary districts which correspond to a multi-colored clique in , and this completes the argument in the reverse direction. ∎

4 Plurality over Districts (PD)

In this section, we analyze the parameterized complexity of PD-Rec and PD-Man. As with PV-Rec, it is easy to see that PD-Rec is also FPT when parameterized by the number of districts (), since we may guess the districts to be recounted. Since , it is also FPT parameterized by the number of voters.

PD-Rec is FPT when parameterized by the number of districts or the number of voters .

We now show that PD-Rec is W[1]-hard parameterized by budget of the defender . Before describing the construction formally, we briefly outline the main idea. We reduce from the Multicolored-Clique Problem, which is well-known to be W[1]-hard parameterized by the no. of color classes, which we denote by . Let be an instance of Multicolored-Clique. In the reduced instance, we introduce a special candidate who is the preferred candidate of the defender. The rival candidates are candidates corresponding to color classes and ordered pairs of color classes . Further, we introduce candidates encoding the vertices and edges . We also introduce some dummy candidates. We introduce two districts corresponding to each and five districts corresponding to each . Also, there exists a baseline district for each candidate which is “immutable”, which sets up the initial scores of all candidates such that they are equal to a large number (say ). The scores are set up in such a way that there is no way to increase the score of , thus we require to reduce the score of all the rivals by at least one while not increasing the scores of other candidates. But the districts and voting profiles are engineered so as to enforce that any recounting solution must have a certain structure, from which we can draw a correspondence to a subset of vertices which must in fact form a multi-colored clique of size in .

PD-Rec is W[1]-hard parameterized by , the defender’s budget.

Proof.

This hardness result follows from the following reduction from the Multi-Colored Clique problem. The given instance is the graph and the number of unique color classes, , where denotes the color class. We begin by describing the construction of the reduced instance.

Candidates: For every color class there is a candidate corresponding to . Further, for every pair of color classes such that , we introduce two candidates and . These will be the rival candidates of the reduced instance. Now we introduce candidates that encode the vertices and edges of the graph . To begin with, for each vertex we introduce two candidates and , which we will refer to as the main and dummy candidates, respectively. Also, for every edge , we introduce two candidates and , which we refer to as the helper and auxiliary candidates, respectively. Finally, we have a special candidate . To summarize, the overall set of candidates is:

Districts: We introduce the following districts.

1. For each we introduce a primary district with weight one and a critical district with weight .

2. For each , we introduce two edge districts and , one support district , and two transfer districts and . The support districts have weight two, while the remaining districts have weight one.

3. For each candidate , we introduce a baseline district with a weight of , which will be specified in due course.

Voting outcomes: The voting outcomes in the original and manipulated districts are depicted in the table below.

Note that the voting outcome in the baseline districts is the same in the original and manipulated settings. It only remains to specify explicitly the weights of the baseline districts. We let . Recall that this is the weight of the baseline district corresponding to the special candidate. For any non-dummy candidate , let be its score from the manipulated districts. We then set . With these weights, we ensure that all the non-dummy candidates tie for the same score (i.e, ) in the manipulated election, and all dummy candidates have a score of zero. We note that all the weights introduced here are polynomially bounded. We set . The preferred candidate is .

We enforce the following tie-breaking order:

This completes the description of the constructed instance. We now turn to the proof of equivalence.

Forward Direction. Let a multi-colored clique of size be given, and without loss of generality, we assume that . Now, recount in districts and for all . Also, for every edge in , we recount in the districts , , , and .

We claim that this recounting strategy leads to a win for . To begin with, observe that the score of every rival candidate drops by one. Indeed, for all , the score of reduces by one having lost the primary district . Further, for all , the scores of and reduce by one each, having lost in the edge districts and , respectively. It is easy to see that the score of and any candidate corresponding to vertices and edges not involved in the clique remain unchanged.

Now, consider the auxiliary and helper candidates corresponding to an edge in . The score of the helper candidate increases by two in the edge districts used to beat the rival candidates, but also decreases by two because of the recounting in the support district corresponding to . This causes the score of the auxiliary candidate to increase by two, but the recounting the two transfer districts again decreases its score by two. Therefore, in terms of total score, the auxiliary and helper candidates are back to where they started.

Meanwhile, the recounting across all transfer districts causes the scores of all the main candidates corresponding to vertices of the clique to increase by , and their score also increases by one in the recounted primary districts used to decrease the score of the rival candidates corresponding to vertices. However, recounting in the critical districts reduces their score by , and we conclude that the net change of score is zero for the main candidates as well. The recounting across the critical districts causes the scores of some dummy candidates to increase by . The scores of all other dummy candidates remains unchanged at zero. Based on the tie-breaking order, it is easy to verify that wins under this score profile (all rival candidates ranked ahead score less than and all other candidates that are tied with are ranked below in the tie-breaking order). This concludes the argument in the forward direction.

Reverse Direction. Conversely, suppose that the defender has a recounting strategy that results in making the winner. Since the rival candidates dominate in the tie-breaking order and have the same score as among the manipulated districts, it is imperative for the defender to force every rival candidate to lose at least one of the districts that it wins in the manipulated setting. In particular, must include at least primary districts and edge districts. Let us say that a solution is well-formed if it consists of primary districts, critical districts, and edge districts, support districts and transfer districts, and further, that for any :

 Duv∈Z⟺Dvu∈Z⟺Se∈Z⟺Te,u∈Z and Te,v∈Z.

A well-formed solution naturally corresponds to a subset of edges, which we will now refer to as the affected edges. Note that any vertex incident to an affected edge emerges as a threat to because of the recounting in the transfer districts and , which are won by and respectively in the original election. Now, observe that the only way to “fix” this is to recount in a critical district corresponding to the vertex in question. Therefore, if the affected edges span vertices, we are forced to recount at least critical districts. Recall that the rival candidates corresponding to color classes impose a requirement of recounting primary districts. Thus, given our overall budget of , we conclude that the affected edges derived from any well-formed solution must in fact span exactly vertices, and it is straightforward to verify that these vertices will correspond to a multi-colored clique in .

From the discussion above, it suffices to show that any valid solution must, in fact, be a well-formed solution. To begin with, note that the rival candidates corresponding to pairs of color classes force a recount in of the edge districts. We first argue that these must correspond to distinct edges, in other words, . Indeed, suppose not. Consider the set of all edges corresponding to the recounted edge districts in , and note that our assumption implies that .

Now observe that the recounting in the edge districts turns the corresponding helper edge candidates into rivals. The only way to reverse this damage is to recount in the support districts corresponding to these edges, which are the only districts that are won by the helper candidates in the manipulated election. However, these recounts in turn cause the “partner” auxiliary candidates to emerge as new rivals whose scores are two more than the score of — recall that the support districts had a weight of two. Again, based on the voting outcomes, we see that we are now forced to recount in both the transfer districts corresponding to the auxiliary candidates, since these are again the only districts that are won by the auxiliary candidates in question. As a result of the recounting in the transfer districts, the score of every vertex incident to any edge in has strictly increased. Note, however, that since , necessarily spans more than vertices in , each of which are now a threat to . This forces a recount of as many critical districts for to win, but this contradicts the available budget. Therefore, we conclude that . It is easy to see that the recounting of the corresponding support and transfer districts are forced based on the voting outcomes and the arguments made earlier in this discussion. Thus we conclude that any valid solution is well-formed, which in turn leads to a natural selection of vertices corresponding to a clique in , completing the argument in the reverse direction. ∎

We now turn our attention to PD-Man. We observe that PD-Man can also be shown to be W[1]-hard by a reduction from PD-Rec. We briefly sketch the main idea: to begin with, we switch the roles of the manipulated profiles and the original ones, set the defender budget to zero and the attacker budget to the defender budget. We would also set up the votes in the districts to be such that the only meaningful manipulation by the attacker is to move the manipulated profile to the original one. The equivalence is based on repurposing a recounting strategy to an attacking one and vice-versa.

5 Concluding Remarks

Our main contribution was to settle the parameterized complexity for the problems of recounting and manipulation when parameterized by the player budgets, for both the PD and PV implementations of the plurality voting rules. We also observed that these problems are FPT when parameterized by the number of voters, and that the recounting problem is FPT when parameterized by the number of districts as well.

We make some remarks about directions for future work. In the setting of succinct input, the problems of recounting and manipulation are already para-NP-hard because of the NP-completeness for three candidates. When the votes are specified in unary is an interesting direction for future work. The dynamic programming algorithm proposed by [ElkindGORV19]

already shows that the problem is in XP, parameterized by the number of candidates, and we leave open the issue of whether the problem is FPT. The problem of manipulation parameterized by the number of districts is another unresolved case. More broadly, it would be interesting to challenge the theoretical hardness results obtained here against heuristics employed on real world data sets. The issue of identifying and working with structural parameters is also an interesting direction for further thought.