Weighted Electoral Control

Weighted Electoral Control

Abstract

Although manipulation and bribery have been extensively studied under weighted voting, there has been almost no work done on election control under weighted voting. This is unfortunate, since weighted voting appears in many important natural settings. In this paper, we study the complexity of controlling the outcome of weighted elections through adding and deleting voters. We obtain polynomial-time algorithms, NP-completeness results, and for many NP-complete cases, approximation algorithms. In particular, for scoring rules we completely characterize the complexity of weighted voter control. Our work shows that for quite a few important cases, either polynomial-time exact algorithms or polynomial-time approximation algorithms exist.

1 Introduction

In many real-world election systems the voters come with weights. Examples range from stockholder elections weighted by shares, to the US Electoral College, to the often-used example of the Nassau County Board of Supervisors, to (in effect) any parliamentary system in which the parties typically vote as blocks, to Sweden’s system of wealth-weighted voting instituted in 1866 (and no longer used) where “the wealthiest members of the rural communities received as many as 5,000 votes” and “in 10 percent of the districts the weighted votes of just three voters could be decisive” [Con11].

So it is not surprising that in the study of manipulative attacks on elections, weighted voting has been given great attention. For bribery and manipulation, two of the three most studied types of manipulative attacks on elections, study of the case of weighted voters has been extensively conducted. Yet for the remaining one of the three most studied types of attacks on elections, so-called control attacks, almost no attention has been given to the case of weighted voting; to the best of our knowledge, the only time this issue has been previously raised is in two M.S./Ph.D. theses [Rus07, Lin12]. This lack of attention is troubling, since the key types of control attacks, such as adding and deleting voters, certainly do occur in many weighted elections.

We study the complexity in weighted elections of arguably the most important types of control—adding and deleting voters—for various election systems. We focus on scoring rules, families of scoring rules, and (weak)Condorcet-consistent rules. Control by deleting (adding) voters asks whether in a given election a given candidate can be made to win by deleting (adding) at most a certain number of the voters (at most a certain number of the members of the pool of potential additional voters). These control types model issues that are found in many electoral settings, ranging from human to electronic. They are (abstractions of) issues often faced by people seeking to steer an election, such as experts doing campaign management, and deciding for example which people to offer rides to the polls.

Control was introduced (without weights) in 1992 in the seminal paper by Bartholdi, Tovey, and Trick [BTT92]. Control has been the subject of much attention since. That attention, and the present paper, are part of the line of work, started by Bartholdi, Orlin, Tovey, and Trick [BTT89, BO91, BTT92], that seeks to determine for which types of manipulative attacks on elections the attacker’s task requires just polynomial-time computation. For a more detailed discussion of this line of work, we point the reader to the related work section at the end of the paper and to the surveys [FHHR09, FHH10, BCE13].

Our main results are as follows (see Section 5 for tables summarizing our results). First, in Section 3.1 we provide a detailed study of the complexity of voter control under scoring protocols, for the case of fixed numbers of candidates. We show that both constructive control by adding voters and constructive control by deleting voters are in for -approval (and so this also covers plurality and -veto1) and are -complete otherwise. It is interesting to compare this result to an analogous theorem regarding weighted coalitional manipulation: There are cases where the complexities of voter control and manipulation are the same (e.g., for plurality or for Borda) but there are also cases where voter control is easier (-approval for , for elections with more than candidates). Is it ever possible that weighted voter control is harder than weighted voting manipulation? We show that weighted voter control is -hard for (weak)Condorcet-consistent rules with at least three candidates. Since weighted coalitional manipulation for the 3-candidate Llull system is in  [FHS08], together with the fact that Llull is weakCondorcet-consistent, this implies that there is a setting where weighted voter control is harder than weighted coalitional manipulation.

In Sections 3.2 and 3.3 we focus on the complexity of weighted voter control under -approval and -veto, for the case of unbounded numbers of candidates. At the start of Section 3.2, we will explain why these are the most interesting cases. In Section 3.2 we resolve six problems left open by Lin [Lin12]. We establish the complexity of weighted control by adding voters for -approval, -veto, and -approval, and of weighted control by deleting voters for -approval, -veto, and -veto. In Section 3.3, we give polynomial-time approximation algorithms for weighted voter control under -approval and -veto. Our algorithms seek to minimize the number of voters that are added or deleted.

We believe that the complexity of weighted voter control, and more generally the complexity of attacks on weighted elections, is an important and interesting research direction that deserves much further study. In particular, our research suggests that it is worthwhile to seek -approximation results for weighted elections problems and that doing so can lead to interesting algorithms.

2 Preliminaries

We assume that the reader is familiar with the basic notions of computational complexity theory and the theory of algorithms. Below we provide relevant definitions and conventions regarding elections, election rules, and control in elections. We also review some -complete problems that we use in our reductions.

Elections

We take an election to be a pair , where is a set of candidates and is a collection of voters. Each voter has a preference order over the set . A preference order is a total, linear order that ranks the candidates from the most preferred one to the least preferred one. For example, if and some voter likes best, then , and then , then his or her preference order is . In weighted elections, each voter also has a positive integer weight . A voter of weight is treated by the election system as unweighted voters. Given two collections of voters, and , we write to denote their concatenation.

Election Rules

An election rule (or voting rule) is a function that given an election returns a subset , namely those candidates that are said to win the election.

An -candidate scoring rule is defined through a nonincreasing vector of nonnegative integers. For each voter , each candidate receives points, where is the position of in ’s preference order. The candidates with the maximum total score are the winners. Given an election and a voting rule that assigns scores to the candidates, we write to denote ’s total score in under . The voting rule used will always be clear from context. Many election rules are defined through families of scoring rules, with one scoring vector for each possible number of candidates. For example:

  1. Plurality rule uses vectors of the form .

  2. -approval uses vectors , where for each , and for . By -veto we mean the system that for candidates uses the -approval scoring vector. For -candidate -approval and -veto systems we will often treat each vote as a 0/1 -dimensional approval vector that indicates which candidates receive points from the vote. Naturally, such a vector contains exactly ones for -approval and exactly zeroes for -veto.2

  3. Borda’s rule uses vectors of the form , where is the number of candidates.

Given an election , a candidate is a (weak) Condorcet winner if for every other candidate it holds that more than half (at least half) of the voters prefer to . Note that it is possible that there is no (weak) Condorcet winner in a given election. We say that a rule is Condorcet-consistent if whenever there is a Condorcet winner he or she is the sole winner elected under . Analogously, a rule is weakCondorcet-consistent if it elects exactly the weak Condorcet winners whenever they exist. Every weakCondorcet-consistent system is Condorcet-consistent, but the converse does not always hold.

There are many Condorcet-consistent rules. We will briefly touch upon the Copeland family of rules and the Maximin rule. For a given election and two distinct candidates , we let be the number of voters that prefer to . Let be a rational number, . Under Copeland the score of candidate is defined as:

and under Maximin the score of candidate is defined as . The candidates with the highest score are winners. Llull is another name for Copeland. Clearly, Llull and Maximin are weakCondorcet-consistent.

Electoral Control

We focus on constructive control by adding/deleting voters in weighted elections. However, there are also other standard types of control studied in the literature (e.g., control by adding/deleting candidates and various forms of partitioning of candidates and voters; we point the reader to Section 4 for a discussion of related work).

Definition 2.1.

Let be a voting rule. In both weighted constructive control by adding voters under rule (-WCCAV) and weighted constructive control by deleting voters under rule (-WCCDV), our input contains a set of candidates , a collection of weighted voters (sometimes referred to as the registered voters) with preferences over , a preferred candidate , and a nonnegative integer . In -WCCAV we also have an additional collection of weighted voters (sometimes referred to as the unregistered voters) with preferences over . In these problems we ask the following questions:

  1. -WCCAV: Is there a subcollection of , of at most voters, such that ?

  2. -WCCDV: Is there a subcollection of , of at most voters, such that ?

Although in this paper we focus primarily on constructive control, Section 3.1 makes some comments about the so-called destructive variants of control problems. Given a voting rule , weighted destructive control by adding voters under rule (-WDCAV) and weighted destructive control by deleting voters under rule (-WDCDV) are defined analogously to their constructive variants, with the only difference being that the goal is to ensure that the distinguished candidate is not a winner.

Note that in the above definitions the parameter defines the number of voters that can be added/deleted, and not the total weight of the voters that can be added/deleted. This is a standard approach when modeling strategic behavior in weighted elections. For example, in the study of “-weighted-bribery” [FHH09], bribing each weighted voter has unit cost regardless of the voter’s weight.

We will consider approximation algorithms for WCCAV and WCCDV under -approval and -veto. When doing so, we will assume that input instances do not contain the integer . Rather, the goal is simply to find (when success is possible at all) as small as possible a collection of voters to add/delete such that is a winner of the resulting election. For a positive integer , an -approximation algorithm for WCCAV/WCCDV is an algorithm that (when success is possible at all) always finds a solution that adds/deletes at most times as many voters as an optimal action does. The notion of an -approximation algorithm for WCCAV/WCCDV is defined analogously, where the argument to is some variable related to the problem or instance. And the meaning of -approximation algorithms will be similarly clear from context. It is natural to worry about how the above seemingly incomplete definitions interact with the possibility that success might be impossible regardless of how many votes one adds/deletes. However, for -approval WCCDV and -veto WCCDV (and indeed, for any scoring rule), it is always possible to ensure that is a winner, for example by deleting all the voters. For -approval WCCAV and -veto WCCAV, it is possible to ensure ’s victory through adding voters if and only if is a winner after we add all the unregistered voters that approve of . These observations make it particularly easy to discuss and study approximation algorithms for -approval and for -veto, because we can always easily check whether there is some solution. For voting rules that don’t have this easy-checking property, such an analysis might be much more complicated. (The reader may wish to compare our work with Brelsford et al.’s attempt at framing a general election-problem approximation framework [BFH08].)

In this paper we do not consider candidate-control cases (such as weighted constructive control by adding candidates and weighted constructive control by deleting candidates, WCCAC and WCCDC). The reason is that for a bounded number of candidates, when winner determination in the given weighted election system is in it holds that both WCCAC and WCCDC are in by brute-force search. On the other hand, if the number of candidates is not bounded then candidate control is already -hard for plurality (and -approval and -veto, in both the constructive setting and the destructive setting) even without weights [BTT92, HHR07, EFS11, Lin12]. Furthermore, many results for candidate control under Condorcet-consistent rules can be claimed in the weighted setting. For example, for the Maximin rule and for the Copeland family of rules, hardness results translate immediately, and it is straightforward to see that the existing polynomial-time algorithms for the unweighted cases also work for the weighted cases [FHH11b].

Weighted Coalitional Manipulation

One of our goals is to compare the complexity of weighted voter control with the complexity of weighted coalitional manipulation (WCM). WCM is similar to WCCAV in that we also add voters, but it differs in that (a) we have to add exactly a given number of voters, and (b) we can pick the preference orders of the added voters. It is quite interesting to see how the differences in these problems’ definitions affect their complexities.

Definition 2.2.

Let be a voting rule. In -WCM we are given a weighted election , a preferred candidate , and a sequence of positive integers. We ask whether it is possible to construct a collection of voters such that for each , , , and is a winner of the election . The voters in are called manipulators.

Computational Complexity

In our -hardness proofs we use reductions from the following -complete problems.

Definition 2.3.

An instance of Partition consists of a sequence of positive integers whose sum is even. We ask whether there is a set such that .

In the proof of Theorem 3.3 we will use the following restricted version of Partition, where we have greater control over the numbers involved in the problem.

Definition 2.4.

An instance of Partition consists of a sequence of positive integers, whose sum is even, such that (a)  is an even number, and (b) for each , , it holds that . We ask whether there is a set of cardinality such that .

Showing the -completeness of this problem is a standard exercise. (In particular, the -completeness of a variant of this problem is established as [FHH09, Lemma 2.3]; the same approach can be used to show the -completeness of Partition.) Our remaining hardness proofs are based on reductions from a restricted version of the well-known Exact-Cover-By-3-Sets problem. This restricted version is still NP-complete [GJ79].

Definition 2.5.

An instance of X3C consists of a set and a family of -element subsets of such that every element of occurs in at least one and in at most three sets in . We ask whether contains an exact cover for , i.e., whether there exist sets in whose union is .

3 Results

We now present our results. In Section 3.1 we focus on fixed numbers of candidates in scoring protocols and (weak)Condorcet-consistent rules. Then in Sections 3.2 and 3.3 we consider case of an unbounded number of candidates, for -approval and -veto.

3.1 Scoring Protocols and Manipulation Versus Control

It is well-known that weighted manipulation of scoring protocols is always hard, unless the scoring protocol is in effect plurality or triviality [HH07]. In contrast, weighted voter control is easy for -candidate -approval.

Theorem 3.1.

For all and , WCCAV and WCCDV for -candidate -approval are in .

Proof.

Let be an instance of WCCAV for -candidate -approval. We can assume that we add only voters who approve of . We can also assume that we add the heaviest voters with a particular set of approvals, i.e., if we add voters approving , we can assume that we added the heaviest voters approving . Since there are only —which is a constant—different sets of approvals to consider, it suffices to try all sequences of nonnegative integers whose sum is at most , and for each such sequence to check whether adding the heaviest voters of the th approval collection makes a winner.

The same argument works for WCCDV. Here, we delete only voters that do not approve of , and again we delete the heaviest voters for each approval collection. ∎

One might think that the argument above works for any scoring protocol, but this is not the case. For example, consider the 3-candidate Borda instance where consists of one weight-1 voter and consists of a weight-2 and a weight-1 voter with preference order . Then adding the weight-1 voter makes a winner, but adding the weight-2 voter does not. And, in fact, we have the following result.3

Theorem 3.2.

WCCAV and WCCDV for Borda are -complete. This result holds even when restricted to a fixed number of candidates.

Proof.

We reduce from Partition. Given a sequence of positive integers that sum to , construct an election with one registered voter of weight voting , and unregistered voters with weights voting . Set the addition limit to . It is easy to see that for to become a winner, ’s score (relative to ) needs to go down by at least , while ’s score (relative to ) should not go up by more than . It follows that has a partition if and only if can be made a winner.

We use the same construction for the deleting voters case. Now, all voters are registered and the deletion limit is . Since we can’t delete all voters, and since our goal is to make a winner, we can’t delete the one voter voting . The rest of the argument is identical to the adding voters case. ∎

Interestingly, it is possible to extend the above proof to work for all scoring protocols other than -approval (the main idea stays the same, but the technical details are more involved). And so, regarding the complexity of WCCAV and WCCDV for scoring protocols with a fixed number of candidates, the cases of Theorem 3.1 are the only P cases (assuming P NP).

Theorem 3.3.

For each scoring protocol , if there exists an , , such that , then WCCAV and WCCDV for are -complete.

Proof.

Let be a scoring protocol such that there is an such that . Let be the third largest value in the set . We will show that WCCAV and WCCDV are -complete for scoring protocol . While formally we have defined scoring protocols to contain only nonnegative values, using simplifies our construction and does not affect the correctness of the proof. To further simplify notation, given some candidates , by we mean a fixed preference order that ensures, under , that each , , is ranked at a position that gives points. (The candidates not mentioned in the notation are ranked arbitrarily.) We let , , and be the three highest values in the set . Clearly, .

We give a reduction from Partition to -WCCAV (the membership of -WCCAV in is clear); let be an instance of Partition, i.e., a sequence of positive integers that sum to . We form an election where and where the collection contains the following three groups of voters (for the WCCAV part of the proof below, we set ; for the WCCDV part of the proof we will use the same construction but with a larger value of ):

  1. A group of voters, each with weight and preference order .

  2. A group of voters, each with weight and preference order .

  3. For each , there are collections of voters, one collection for each permutation of ; the voters in each collection have weight and preference order .

Let be the number of points that each of , , and receive from the third group of voters (each of these candidates receives the same number of points from these voters). For each and each , receives at least points more than from the voters in the third group (in each vote in the third group, receives at least as many points as , and there are two collections of voters where receives points and receives points). Thus it holds that our candidates have the following scores:

  1. has points,

  2. has points,

  3. has points, and

  4. each candidate has at most points.

As a result, is the unique winner. There are unregistered voters with weights , each with preference order . We set the addition limit to be . It is clear that irrespective of which voters are added, none of the candidates in becomes a winner.

If there is a subcollection of that sums to , then adding corresponding unregistered voters to the election ensures that all three of , , and are winners. On the other hand, assume that there are unregistered voters of total weight , whose addition to the election ensures that is among the winners. For to have score at least as high as , we must have that . However, for not to have score higher than , it must be that . This means that . Thus it is possible to ensure that is a winner of the election by adding at most unregistered voters if and only if there is a subcollection of that sums to . And, completing the proof, we note that the reduction can be carried out in polynomial time.

Let us now move on to the case of WCCDV. We will use the same construction, but with the following modifications:

  1. Our reduction is now from Partition. Thus without loss of generality we can assume that is an even number and that for each , , it holds that .

  2. We set (the reasons for this choice of will become apparent in the course of the proof; intuitively it is convenient to think of as of a large value that, nonetheless, is polynomially bounded with respect to ).

  3. We include the unregistered voters as “the fourth group of voters.”

  4. We set the deletion limit to .

By the same reasoning as in the WCCAV case, it is easy to see that if there is a size- subcollection of that sums to , then deleting the corresponding voters ensures that is among the winners (together with and ). We now show that if there is a way to delete up to voters to ensure that is among the winners, then the deleted voters must come from the fourth group, must have total weight , and there must be exactly of them. For the sake of contradiction, let us assume that it is possible to ensure ’s victory by deleting up to voters, of whom fewer than come from the fourth group. Let be the number of deleted voters from the fourth group () and let be a real number such that is their total weight. We have that is at most:

That is, we have . Prior to deleting any voters, has points more than . After deleting the voters from the fourth group, this difference decreases to . If we additionally delete up to voters from the first three groups of voters, each with weight , then the difference between the scores of and decreases, at most, to the following value (note that in each deleted vote both and are ranked at positions where they receive , or points):

The final inequality follows by our choice of . The above calculation shows that if there is a way to ensure ’s victory by deleting up to voters then it requires deleting exactly voters from the fourth group. The same reasoning as in the case of WCCAV shows that these deleted voters must correspond to a size- subcollection of that sums to . ∎

As a side comment, we mention that WDCAV and WDCDV for scoring protocols (that is, the destructive variants of WCCAV and WCCDV) have simple polynomial-time algorithms: It suffices to loop through all candidates , , and greedily add/delete voters to boost the score of relative to as much as possible.

Combining Theorems 3.1 and 3.3, we obtain the following corollary, which we contrast with an analogous result for WCM [HH07].

Corollary 3.4.

For each scoring protocol the problems WCCAV and WCCDV are -complete if and are in otherwise.

Theorem 3.5 (Hemaspaandra and Hemaspaandra [Hh07]).

For each scoring protocols , , WCM is -complete if and is in otherwise.

We see that for scoring protocols with a fixed number of candidates, either WCM is harder than WCCAV and WCCDV (for the case of -approval with ), or the complexity of WCM, WCCAV, and WCCDV is the same (-membership for plurality and triviality, and -completeness for the remaining cases). For other voting rules, it is also possible that WCM is easier than WCCAV and WCCDV.

Theorem 3.6.

For every weakCondorcet-consistent election system and for every Condorcet-consistent election system, WCCAV and WCCDV are -hard. This result holds even when restricted to a fixed number of candidates.

Proof.

To show that WCCAV is -hard, we reduce from Partition. Given a sequence of positive integers that sum to , construct an election with two registered voters, one voter with weight 1 voting and one voter with weight voting , and unregistered voters with weights voting . Set the addition limit to . Suppose we add unregistered voters to the election with a total vote weight equal to .

  • If , then is the Condorcet winner, and thus the unique winner of the election.

  • If , then is the Condorcet winner, and thus the unique winner of the election.

  • If , then is the Condorcet winner, and thus the unique winner of the election.

The WCCDV case uses the same construction. Now, all voters are registered and the deletion limit is . Since we can delete at most of our voters, and since our goal is to make a winner, we can’t delete the sole voter voting , since then would be the Condorcet winner. The rest of the argument is similar to the adding voters case. ∎

Let Condorcet be the election system whose winner set is exactly the set of Condorcet winners. Let weakCondorcet be the election system whose winner set is exactly the set of weak Condorcet winners.

Corollary 3.7.

For Condorcet and weakCondorcet, WCM is in and WCCAV and WCCDV are -complete. This result holds even when restricted to a fixed number of candidates.

Proof.

It is immediate that WCM for Condorcet and weakCondorcet are in P. To see if we have a “yes”-instance of WCM, it suffices to check whether letting all the manipulators rank (the preferred candidate) first and ranking all the remaining candidates in some arbitrary order ensures ’s victory. -completeness of WCCAV and WCCDV follows directly from Theorem 3.6. ∎

Condorcet and weakCondorcet do not always have winners. For those who prefer their voting systems to always have at least one winner, we note that WCM for -candidate Llull is in P [FHS08].

Corollary 3.8.

For 3-candidate Llull, WCM is in and WCCAV and WCCDV are -complete.

The main results of this section are also presented in Table 1 of Section 5.

3.2 -Approval and -Veto with an Unbounded Number of Candidates

Let us now look at the cases of -approval and -veto rules, for an unbounded number of candidates. The reason we focus on these is that these are the most interesting families of scoring protocols whose complexity has not already been resolved in the previous section. The reason we say that is that Theorem 3.3 shows that whenever we have at least three distinct values in a scoring vector, we have NP-completeness. And so any family that at even one number of candidates has three distinct values in its scoring vector is NP-hard for WCCAV and WCCDV. Thus the really interesting cases are indeed -approval and -veto.

Our starting point here is the work of Lin [Lin12], which showed that for , WCCAV for -approval and WCCDV for -veto are -complete, and that for , WCCDV for -approval and WCCAV for -veto are -complete. These results hold even for the unweighted case. It is also known that the remaining unweighted cases are in P [BTT92, Lin12] and that WCCAV and WCCDV for plurality and veto are in P [Lin12]. In this section, we look at and solve the remaining open cases, WCCAV for -approval, -approval, and -veto, and WCCDV for -approval, -veto, and -veto. We start by showing that -approval-WCCAV is in .

Theorem 3.9.

WCCAV for 2-approval is in .

Proof.

We claim that Algorithm 1 solves -approval-WCCAV in polynomial time. (In this algorithm and the proof of correctness, whenever we speak of the heaviest voters in voter set , we mean the heaviest voters in .)

Input:
forall do  let .   Delete from all voters that do not approve of .
repeat  forall do  if the sum of the weights of the heaviest voters in that do not approve of is less than then reject
// It is impossible to get by adding less than or equal to voters from .
  forall and do  if the sum of the weights of the heaviest voters in that do not approve of is less than then  delete from all voters approving except for the heaviest such voters.
// We need to add at least voters that do not approve of , and so we can add at most voters approving .
  until no more changes.
if then accept // We can make a winner by adding the heaviest voters from .
if then  if adding all of will make a winner then accept else reject  
Algorithm 1 -approval-WCCAV

It is easy to see that we never reject incorrectly in the repeat-until, assuming that we don’t incorrectly delete voters from . It is also easy to see that if we add voters approving , we may assume that we add the heaviest voters approving (this is also crucial in the proof of Theorem 3.1), and so we never delete voters incorrectly in the second for loop in the repeat-until.

If we get through the repeat-until without rejecting, and we have fewer than voters left in , then adding all of is the best we can do (since all voters in approve ).

Finally, if we get through the repeat-until, and we have at least voters left in , then adding the heaviest voters from will make a winner. Why? Let be a candidate in . Let be the number of voters from that are added and that approve of . Since we made it through the repeat-until, we know that [the sum of the weights of the heaviest voters in that do not approve of ] is at least . We will show that after adding the voters, , which implies that is a winner. If , = - [the sum of the weights of the heaviest voters in ] . If , then [the sum of the weights of the heaviest voters in that do not approve of ] is at least (for otherwise we would have at most voters approving left in ). And so = - [the sum of the weights of the heaviest voters in that do not approve of ] . ∎

Theorem 3.10.

WCCDV for 2-veto is in .

Instead of proving this theorem directly, we show a more general relation between the complexity of -approval/-veto WCCAV and WCCDV.

Theorem 3.11.

For each fixed , it holds that -veto-WCCDV (-approval-WCCDV) polynomial-time many-one reduces to -approval-WCCAV (-veto-WCCAV).

Proof.

We first give a reduction from -veto-WCCDV to -approval-WCCAV. The idea is that deleting a -veto vote from -veto election is equivalent, in terms of net effect on the scores, to adding a -approval vote to this election, where approves exactly of the candidates that disapproves of. The problem with this approach is that we are to reduce -veto-WCCDV to -approval-WCCAV and thus we have to show how to implement -veto scores with -approval votes.

Let be an instance of -veto-WCCDV, where . Let . Let be the highest weight of a vote in . We set to be a set of up to new candidates, such that is a multiple of . We set to be a collection of -approval votes, where each vote has weight and each candidate in is approved in exactly one of the votes. For each vote in we create a set of candidates and we create a collection of voters . Each voter , , has weight and approves of the th candidate approved by and of the candidates .

We form an election , where and . For each candidate , let be ’s -veto score in ; it is easy to see that ’s -approval score in is . Furthermore, each candidate has -approval score at most in .

We form an instance of -approval-WCCAV, where , and for each , , , and approves exactly of those candidates that disapproves of. It is easy to see that adding voter to -approval election has the same net effect on the scores of the candidates in as does deleting from -veto election .

Let us now give a reduction from -approval-WCCDV to -veto-WCCAV. The idea is the same as in the previous reduction; the main part of the proof is to show how to implement -approval scores with -veto votes. Let be an instance of -approval-WCCDV, where . Let and let be the highest weight of a vote in . We set to be a set of candidates such that and for some integer , (note that for our setting to not be trivial it must be that ). We set to be a collection of votes, each with weight ; each candidate from is approved in all these votes whereas each candidate from is disapproved in at least half of them (since , it is easy to construct such votes). For each vote in , we create a collection of votes satisfying the following requirements: (a) each candidate approved in is also approved in each of the votes in , and (b) each candidate not approved in , is approved in exactly votes in . (Such votes are easy to construct: We always place the top candidates from in the top positions of the vote; for the remaining positions, in the first vote we place the candidates in some arbitrary, easily computable order, and in each following vote we shift these candidates cyclically by positions with respect to the previous vote.) Each vote in has weight .

We form an election , where and . For each candidate , let be ’s -approval score in ; it is easy to see that ’s -veto score in is . Furthermore, each candidate from has -veto score at most in .

We form an instance of -veto-WCCAV, where , and for each , , , and disapproves of exactly those candidates that approves of. It is easy to see that adding voter to -veto election has the same net effect on the scores of candidates in as deleting voter from -approval election has. Furthermore, since each candidate in has at least fewer points than each candidate in , the fact that adding increases scores of candidates in does not affect the correctness of our reduction. ∎

All other remaining cases (WCCDV for 2-approval, WCCAV for 3-approval, WCCAV for 2-veto, and WCCDV for 3-veto) are -complete. Interestingly, in contrast to many other -complete weighted election problems, we need only a very limited set of weights to make the reductions work.

Theorem 3.12.

WCCAV for 2-veto and 3-approval and WCCDV for 2-approval and 3-veto are -complete.

Proof.

Membership in NP is immediate, so it suffices to prove NP-hardness. We will first give the proof for WCCDV for 2-approval. By Theorem 3.11 this also immediately gives the result for WCCAV for 2-veto. We will reduce from X3C from Definition 2.5. Let and let be a family of 3-element subsets of such that every element of occurs in at least one and in at most three sets in . We construct the following instance of WCCDV for 2-approval. We set ( are dummy candidates that are used for padding). For , let be the number of sets in that contain . Note that . consists of the following voters:

weight preference order
2 for all and
1
1
1
2
for all such that .

Note that , , , , and . We set and we claim that contains an exact cover if and only if can become a winner after deleting at most voters.

: Delete the weight-2 voters corresponding to the sets not in the cover and delete the weight-1 voters corresponding to the sets in the cover. Then the score of does not change, the score of each decreases by 2, the score of each decreases by at least 1, and the score of each decreases by 1. So, is a winner.

: We need to delete voters to decrease the score of every voter by 1. After deleting these voters, there are at most values of , , such that the score of and the score of are at most 2.

If there are exactly values of , , such that the score of and the score of are at most 2, then these values of correspond to a cover. If there are less than values of , , such that the score of and the score of are at most 2, then the remaining voters that are deleted, and there are at most of them, need to decrease the score of and/or for more than values of , . But that is not possible, since there is no voter that approves of both or and or for .

Note that this construction uses only weights 1 and 2. In fact, we can establish NP-completeness for WCCDV for 2-approval for every set of allowed weights of size at least two (note that if the set of weights has size one, the problem is in P, since this is in essence the unweighted case [Lin12]). Since the reductions of Theorem 3.11 do not change the set of voter weights, we have the same result for WCCAV for 2-veto.

So, suppose our weight set contains and , . We modify the construction above as follows. We keep the same set of candidates and we change the voters as follows.

# weight preference order
1 for all and
1
1
1
2 if
1 if
for all .

Here, is the smallest integer such that . Note that and so is never negative. Note that , , , , and . The same argument as above shows that contains an exact cover if and only if can become a winner after deleting at most voters.

We now turn to the proof for WCCDV for 3-veto. Our construction will use only weights 1 and 3. Since the reductions of Theorem 3.11 do not change the set of voter weights, weights 1 and 3 also suffice to get NP-completeness for WCCAV for 3-approval. Given the instance of X3C described above, we construct the following instance of WCCDV for 3-veto. We set ( and are dummy candidates that are used for padding) and consists of the following voters:

# weight preference order
1 3 for all and
1 1
1 1
1 1
1
1 for all
1 for all .

It is more convenient to count the number of vetoes for each candidate than to count the number of approvals. Note that , , , , and . We claim that contains an exact cover if and only if can become a winner (i.e., have a lowest number of vetoes) after deleting at most voters.

: Delete the weight-3 voters corresponding to the sets not in the cover and delete the weight-1 voters that veto and that correspond to the sets in the cover. Then and . So, is a winner.

: We can assume that we delete only voters that veto . Suppose we delete weight-1 voters and weight-3 voters, . After this deletion, , , and . In order for to be a winner, we need . This implies that . We also need . Since , it follows that . So we delete weight-1 votes and weight-3 votes, and after deleting these voters . In order for to be a winner, we can delete at most one veto for each and at most three vetoes for each . This implies that the set of deleted weight-1 voters corresponds to a cover. ∎

3.3 Approximation and Greedy Algorithms

When problems are computationally difficult, such as being NP-complete, it is natural to wonder whether good polynomial-time approximation algorithms exist. So, motivated by the NP-completeness results discussed earlier in this paper for most cases of WCCAV/WCCDV for -approval and -veto, this section studies greedy and other approximation algorithms for those problems. (Recall that WCCAV is NP-complete for -approval, , and for -veto, , and WCCDV is NP-complete for -approval, , and for -veto, .) Although we are primarily interested in constructing good approximation algorithms, we are also interested in cases where particular greedy strategies can be shown to fail to provide good approximation algorithms, as doing so helps one eliminate such approaches from consideration and sheds light on the approach’s limits of applicability. First, we will establish a connection to the weighted multicover problem, and we will use it to obtain approximation results. Then we will obtain an approximation algorithm that will work by direct action on our problem. Table 3 in Section 5 summarizes our results on approximation algorithms for -approval/-veto WCCAV/WCCDV.

A Weighted Multicover Approach

Let us first consider the extent to which known algorithms for the Set-Cover family of problems apply to our setting. Specifically, we will use the following multicover problem.

Definition 3.13.

An instance of Weighted Multicover (WMC) consists of a set , a sequence of nonnegative integers (covering requirements), a collection of subsets of , and a sequence of positive integers (weights of the sets in ). The goal is to find a minimum-cardinality set such that for each it holds that , or to declare that no such set exists.

That is, given a WMC instance we seek a smallest collection of subsets from that satisfies the covering requirements of the elements of (keeping in mind that a set of weight covers each of its elements times). WMC is an extension of Set-Cover with unit costs. We will not define here the problem known as covering integer programming (CIP) (see [KY05]). However, that problem will be quite important to us here. The reason is that we observe that WMC is a special case of CIP (with multiplicity constraints but) without packing constraints; footnote 4 below is in effect describing how to embed our problem in that problem. An approximation algorithm of Kolliopoulos and Young [KY05] for CIP (with multiplicity constraints but) without packing constraints, applied to the special case of WMC, gives the following result.4

Theorem 3.14 (Kolliopoulos and Young [Ky05]).

There is a polynomial-time algorithm that when given an instance of WMC in which each set contains at most elements gives an -approximation.

For -approval both WCCAV and WCCDV naturally translate to equivalent WMC instances. We consider WCCAV first. Let be an instance of -approval-WCCAV, where is the collection of voters that we may add. We assume without loss of generality that each voter in ranks among its top candidates (i.e., approves of ).

We form an instance of WMC as follows. We set . For each , we set its covering requirement to be , where . For each vote , let be the set of candidates that does not approve of. By our assumption regarding each voter ranking among its top candidates, no contains . We set and we set . It is easy to see that a set is a solution to this instance of WMC (that is, satisfies all covering requirements) if and only if adding the voters to the election ensures that is a winner. The reason for this is the following: If we add voter to the election then for each candidate , the difference between the score of and the score of decreases by , and for each candidate this difference does not change. The covering requirements are set to guarantee that ’s score will match or exceed the scores of all candidates in the election.

We stress that in the above construction we did not assume to be a constant. Indeed, the construction applies to -veto just as well as to -approval. So using Theorem 3.14 we obtain the following result.

Theorem 3.15.

There is a polynomial-time -approximation algorithm for -approval-WCCAV. There is a polynomial-time algorithm that when given an instance of -veto-WCCAV () gives an -approximation.

Proof.

It suffices to use the reduction of -approval/-veto t