THE COMPLEXITY OF MANIPULATING k-

APPROVAL

ELECTIONS

∗

Andrew Lin

lisano College of Computing and Information Sciences, Rochester Intitute of Technology, Rochester, NY 14623, U.S.A.

Keywords:

Elections manipulation approval complexity.

Abstract:

An important problem in computational social choice theory is the complexity of undesirable behavior among

agents, such as control, manipulation, and bribery in election systems, which are tempting at the individual

level but disastrous for the agents as a whole. Creating election systems where the determination of such

strategies is difﬁcult is thus an important goal.

An interesting set of elections is that of scoring protocols. Previous work in this area has demonstrated the

complexity of misuse in cases involving a ﬁxed number of candidates, and of speciﬁc election systems on

unbounded number of candidates such as Borda. In contrast, we take the ﬁrst step in generalizing the results

of computational complexity of election misuse to cases of inﬁnitely many scoring protocols on an unbounded

number of candidates.

We demonstrate the worst-case complexity of various problems in this area, by showing they are either

polynomial-time computable, NP-hard, or polynomial-time equivalent to another problem of interest. We

also demonstrate a surprising connection between manipulation in election systems and some graph theory

problems.

1 INTRODUCTION

Election systems are means for aggregating the pref-

erences of individuals to arrive at a decision that at-

tempts to maximize the collective welfare of the indi-

viduals. Situations needing such means can arise in

political science, such as choosing leaders or writing

laws, or also in computation, where a group of agents

must agree on an action, such as choose leaders in a

parallel algorithm.

An early unfortunate result in computational so-

cial choice theory is that all reasonable election sys-

tems are subject to misuses by the voters, termed ma-

nipulation. Manipulation occurs when one or more

agents report insincere preferences for their own ben-

eﬁt. A common manipulation involves burying ones

preference in the preference ordering.

Some election systems also encourage bribery, in

which an outside agent can convince, or bribe, some

of the agents to change their votes. This is a weakness

if the number of affected voters needed to alter the

outcome is small.

It is also possible for the administrators of an elec-

tion to control the outcome of an election by manipu-

∗

upported in part by NSF grant IIS-0713061

lating the set of voters or candidates (Bartholdi et al.,

1992). Ways to control the election include encour-

aging or discouraging potential voters or candidates

from participating, or partitioning the voter or candi-

date set, creating a multiple-round election.

Although it is desirable to design election systems

such that the potential for such exploitation is elim-

inated, several notable results (Gibbard, 1973; Sat-

terthwaite, 1975; Duggan and Schwartz, 2000) have

demonstrated that it is impossible to do so, and that

every interesting election is subject to manipulation

under some conditions.

It is challenged (Bartholdi et al., 1989) that the in-

evitable manipulation only constitutes a threat when

it is computationally easy, i.e. polynomial-time com-

putable, for one to determine for an election system.

Much work (Conitzer et al., 2002; Faliszewski et al.,

2006) has been done to characterize the worst-case

complexities of manipulating different election sys-

tems.

We examine an inﬁnite set of election systems,

approval-based families of scoring protocols, where

each candidate approvesof some function 0 ≤ f(m) ≤

m of the m candidates in the election, evaluating the

worst-case complexities of various forms of attack.

Essentially, we make the ﬁrst attempt at extending the

212

Lin A..

THE COMPLEXITY OF MANIPULATING Îž-APPROVAL ELECTIONS.

DOI: 10.5220/0003168802120218

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 212-218

ISBN: 978-989-8425-41-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

work of (Hemaspaandra and Hemaspaandra, 2007;

artholdi et al., 1992) to scoring protocols of an un-

bounded number of candidates, by looking at an in-

ﬁnite set of election systems. A characterization of

an inﬁnite set of election systems is also evaluated in

(Faliszewski et al., 2008). A more detailed descrip-

tion of some proofs in this paper can be found in the

archived version (Lin, 2010).

We show how some election systems lend them-

selves to manipulation by simple greedy algorithms,

while some are provably hard by reductions from Set

Cover or Hitting Set problems. We hope that from

our work one can gain a better understanding of what

properties of elections encourage or discourage ma-

nipulation.

1.1 Our Results

We summarize the most important complexity results

in this paper in the tables below, with new results in

bold.

All results are for constructive misuse.

1.1.1 Unweighted Cases

1-app 2-app 3-app k-

app, k ≥ 4

Manip P P P P

Bribery P P NPC NPC

CCAV P P P NPC

CCDV P P NPC NPC

CCAC NPC NPC NPC NPC

CCDC NPC NPC NPC NPC

1-veto 2-veto 3-veto k-

veto, k ≥ 4

Manip P P P P

Bribery P P P NPC

CCAV P P NPC NPC

CCDV P P P NPC

CCAC NPC NPC NPC NPC

CCDC NPC NPC NPC NPC

1.1.2 Weighted Voter Cases

1-app 2-app 3-app k-

app, k ≥ 4

Manip P NPC NPC NPC

Bribery P NPC NPC NPC

CCAV P P sbw NPC

CCDV P sbw NPC NPC

CCAC NPC NPC NPC NPC

CCDC NPC NPC NPC NPC

n the table, swb is the complexity of Simple Weighted

b-Edge Cover of Multigraphs and sbw that of Simple b-Edge

Weighted Cover of Multigraphs. CCAV=constructive con-

trol by adding voters, CCDV=constructive control by delet-

ing voters, CCAC=constructive control by adding candidates,

CCDC=constructive control by deleting candidates

1-veto 2-veto 3-veto k-

veto, k ≥ 4

Manip NPC NPC NPC NPC

Bribery NPC NPC NPC NPC

CCAV P sbw NPC NPC

CCDV P P sbw NPC

CCAC NPC NPC NPC NPC

CCDC NPC NPC NPC NPC

1.1.3 Unweighted $Bribery Cases

1-app 2-app 3-app k-

app, k ≥ 4

$Bribery P swb NPC NPC

1-veto 2-veto 3-veto k-

veto, k ≥ 4

$Bribery P P swb NPC

2 PRELIMINARIES

.1 Deﬁnitions and Notations of

Election Systems

An election E = (C,V) is deﬁned as a pair, con-

taining m candidates C = {c

,...,c

} and n voters

V = {v

,...,v

}. Each voter presents a preference

proﬁle over the candidates given as a linear order.

In some cases, each voter v may have a pre-

determined integer weight, w(v). A vote of weight

w(v) is counted as w(v) votes in an unweighted elec-

tion. Additionally, in problems involving bribery, it is

also possible for voter v to also have a pre-determined

price, π(v), which the briber must pay to modify the

preferences of the voters, and the goal is to ﬁnd the

cheapest bribery.

An election system, E , speciﬁes how the prefer-

ences of the voters over the candidates are aggregated

to arrive at a winner or set of winners.

Scoring protocols are a class of election sys-

tems, and encompass a large number of common sys-

tems such as plurality, veto, Borda count, and some

approval-based systems for a ﬁxed number of candi-

dates. We formally deﬁne scoring protocols and some

examples here.

Deﬁnition 1. Let E = (C,V) be an election of m

candidates. A scoring protocol is an integer vec-

tor (α

,...,α

) with α

≥ ··· ≥ α

≥ 0 such that

each candidate c receives α

points for each voter that

ranks c as their i

favorite alternative.

A family of scoring protocols is an inﬁnite se-

ries (α

,...,α

,...) of scoring protocols where α

(α

,...,α

) is a scoring protocol of m candidates.

In plurality, each voter gives one point to his or

her favorite candidate, whereas in veto, each voter ve-

toes one candidate. In k-approval, each voter gives

THE COMPLEXITY OF MANIPULATING k-APPROVAL ELECTIONS

213

his or her k favorite candidates 1 point. Similarly, in

k-veto, each voter vetoes his or her least k favorite

candidates.

Although not a scoring protocol, another common

election system is approval. In approval voting, each

voter can approve as many or as few voters as he/she

chooses. In (Hemaspaandra et al., 2007), it is shown

that approval is resistant to many forms of misuse in-

volving the voter set, even with unweighted voters.

We will in fact examine the properties of this election

system that give rise to the resistance.

A generalization of k-approval and k-veto, f(m)-

approval, where f is a function of the number of

candidates m, is an election where each voter gives

1 point to each of his or her f(m) favorite candidates.

In this case, we will assume that f(m) is polynomial-

time computable with respect to the quantity m. Ab-

sent such assumption, such problems could be difﬁ-

cult simply because we cannot determine the outcome

of a manipulation itself.

2.2 Problems of Interest in

Computational Social Choice

Theory

In this paper, we study the complexity of manip-

ulation, bribery, and control of k-approval and k-

veto elections, which have been shown to have some

advantages over plurality (Brams and Herschbach,

2001). For each case, there are two problems: con-

structive, where the goal is to ensure victory of a spe-

ciﬁc candidate, and destructive, to ensure defeat of

such. In manipulation, we attempt to reach this goal

by giving preferences to a set of unestablished voters,

whereas in bribery we do so by changing the prefer-

ences of some established voters. There are two types

of control problems: One can alter either the voter set

(i.e., by adding or deleting) or the candidate set. In

some cases, voters may have weights and costs. We

make the formal deﬁnitions in this section.

Name. Constructive E -Control by Adding Voters

Instance. A set C of candidates, a set V of estab-

lished voters, and V

′

of unestablished voters such

that V ∩ V

′

0, distinguished candidate p, and

non-negative integer quota q.

Question. Is there a subset V

′′

⊆ V

′

with ||V

′′

|| ≤ q

such that p is a winner of the E election with can-

didate set C and voter set V ∪V

′′

Another problem of interest is Constructive E -

Control by Deleting Voters, in which we ensure the

victory of p by deleting at most q voters.

Name. Constructive E -Control by Adding Candi-

dates

Instance. A set C of candidates, a set V of estab-

lished voters, and C

′

of unestablished candidates

such that C ∩ C

′

0, distinguished candidate p,

and non-negative integer quota q. In this case, V

provides a linear preference ordering overC ∪C

′

Question. Is there a subset C

′′

⊆ C

′

with ||C

′′

|| ≤ q

such that p is a winner of the E election with can-

didate set C ∪C

′′

and voter set V?

E -Control by Deleting Candidates is deﬁned sim-

ilarly.

In manipulation, a subset of voters working to-

gether seek to determine the outcome of an election.

Name. Constructive E -Manipulation

Instance. A set C of candidates, a set V of estab-

lished voters, and V

′

of unestablished voters such

that V ∩V

′

0, and distinguished candidate p.

Question. Is there an assignment of preference pro-

ﬁles for V

′

such that p is a winner of the E elec-

tion with candidate set C and voter set V ∪V

′

In all of these problems, another prob-

lem of interest is the cases where voters have

weights. In such case, we denote the problem

by E -weighted-constructive-manipulation and

E -weighted-destructive-manipulation.

Name. Constructive E -Bribery

Instance. A set C of candidates, a set V of voters,

distinguished candidate p, and non-negative inte-

ger quota q.

Question. Is it possible to make p a winner of the E

election by changing the preference proﬁles of at

most q voters in V.

As in the cases of control and manipulation,

bribery is also deﬁned for cases of weighted voters.

We denote this problem E -Weighted-Bribery. In ad-

dition, each voter can be assigned a price tag of a non-

negative integer. In this case, q is our budget, and

we want to achieve our goal (constructive or destruc-

tive bribery) by spending at most total cost q in our

bribery. We denote this problem by E -$Bribery and

E -Weighted-$Bribery if voters have both weights and

prices.

An election system E is resistant to misuse if de-

termining if such a misuse exists for a given elec-

tion is NP-hard. It is vulnerable if such a decision

is polynomial-time computable.

2.3 Some Important NP-Complete

Problems

Common NP-complete problems of choice for show-

ing NP-hardness of election systems include versions

of Set Cover, Knapsack, and Hitting Set.

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

214

In Exact 3-Set Cover (X3C) (Karp, 1972) (See

lso (Garey and Johnson, 1979, problem SP2)), a set

S = {s

,...,s

} and subsets T

,...,T

⊆ S such that

||T

|| = 3 are given. We wish to ﬁnd m such subsets

that exactly cover S. Reductions from similar prob-

lems were used in (Brelsford et al., 2008) to show the

resistance of the Borda count election to bribery.

In Hitting Set, a problem used to show hardness

of some manipulations in the candidate set, we are

given a set S = {s

,...,s

}, n subsets of S, T

,...,T

and positive integer 1 ≤ q ≤ m. We wish to deter-

mine if there is a subset of q elements of S that hit

each of the n given subsets. This problem is of in-

terest in the issue of control by adding and deleting

candidates (Bartholdi et al., 1989).

2.4 Edge Cover, b-Edge Cover and their

Relationship to Approval-based

Elections

Among other results, we demonstrate the relationship

between some approval-based election systems and

problems of edge coverings in graphs. In general,

this occurs in systems that distinguish two candidates

from the remaining as edges connect pairs of vertices.

The well known problem of Edge Cover, which is

polynomial-time computable, is deﬁned below.

Deﬁnition 2. An edge cover of a graph is a set of

edges such that every vertex of the graph is incident

to at least one edge of the set. In the decision problem

Edge Cover, we are given a graph G = (V,E) and

positive integer q, and wish to determine if there exists

an edge cover C ⊆ E for G of at most q edges.

In the variation b-Edge Cover, each vertex v is to

be covered by a minimum of some number, b(v), of

edges. There are several interesting versions: Each

edge can be chosen only once (Simple b-Edge Cover),

an arbitrary number of times (b-Edge Cover), or have

a capacity and be chosen up to that many times (Ca-

pacitated b-Edge Cover) (see (Schrijver, 2003) 34.1,

34.7, 34.8), all of which are polynomial-time com-

putable (Pulleyblank, 1973; Cunningham and III,

1978; Gabow, 1983; Anstee, 1987).

Theorem1. Simple b-Edge Cover for Multigraphs, in

which we are given a multigraph G with b-values b(v)

deﬁned for each vertex v ∈ V(G) and integer q, and

wish to determine if G can be simple b-edge covered

by at most q edges, is polynomial-time computable.

This follows from Capacitated b-Edge

Cover (Schrijver, 2003), with edge capacity cor-

responding to the number of edges between each two

vertices.

We introduce two additional versions of b-Edge

Coverof Multigraphs, involvingweights. These prob-

lems are of interest in some weighted election manip-

ulations.

In Simple Weighted b-Edge Cover of Multi-

graphs, a weight is assigned to each edge in the multi-

graph. We are interested in whether a cover exists of

at most weight q.

In Simple b-Edge Weighted Cover of Multi-

graphs, a weight is assigned to each edge. We are

interested in ﬁnding a cover C ⊆ E of at most q ver-

tices such that each vertex v ∈ V is incident to edges

in C of total weight at least b(v).

We introduce these two problems to convert the

otherwise obscure problems of election manipulation

to that of more natural graph theory-related problems.

It is hoped that algorithms and heuristics available for

similar problems can be applied toward these two ver-

sions, and thus to some problems in manipulation.

The complexity of either of these variations is left as

an open problem.

3 MISUSES OF APPROVAL

BASED ELECTIONS IN

UNWEIGHTED CASES

We examine the problems of manipulation,

bribery, and control, for f(m)-approval elections, and

also for the special cases of k-approval and k-veto, in

unweighted elections. This extends the known results

of 1-approval and 1-veto (Bartholdi et al., 1989; Fal-

iszewski et al., 2006). We wish to examine how the

difﬁculty of these problems increase as we generalize

the scoring protocol.

3.1 Destructive Misuses

In several previous results (Hemaspaandra and

Hemaspaandra, 2007; Russell, 2007), it has been

shown that destructive misuse is vastly easier than

constructive misuse for many election systems. The

principle is as follows: To keep p from winning the

election, it sufﬁces to ensure some candidate p

′

6= p

beats p. However, this can occur iff bribing voters

giving p the greatest lead against p

′

sufﬁces.

We conclude that all unweighted f(m)-approval

elections are vulnerable to destructive bribery, manip-

ulation, and control by adding or deleting voters.

THE COMPLEXITY OF MANIPULATING k-APPROVAL ELECTIONS

215

3.2 Manipulation of Approval-Based

Scoring Protocols

We show that manipulation of all families of scoring

protocols of the form f(m)-approval is easy. This ob-

servation was made independently by Procaccia (Pro-

caccia, 2009).

Theorem 2. Unweighted f(m)-approval elections

are vulnerable to constructive manipulation.

Proof. Consider an f(m)-approval election E =

(V,C), distinguished candidate p ∈ C, and set of non-

established voters V

′

. A simple greedy algorithm that

ﬁnds a constructive manipulation is to iteratively ap-

prove p and the f(m) − 1 non-distinguished candi-

dates currently with the lowest scores for each voter

in V

′

Correctness of this algorithm can be shown by in-

duction on ||V

′

||, by proving that a manipulation ex-

ists iff there exists one such that at least one voter ap-

proves of p and the f(m) − 1 non-distinguished can-

didates currently with the lowest scores.

3.3 Bribery in Approval-based Scoring

Protocols

The goal of a briber is to determine whether it is possi-

ble to ensure victory of a desired candidate by altering

voter preferences. Unweighted 1-approval and 1 veto

elections are known to be vulnerable to unpriced con-

structive bribery, by simple greedy algorithms (Fal-

iszewski et al., 2006).

Theorem 3. Unweighted 2-veto elections are vulner-

able to unpriced constructive bribery.

Proof. Consider a 2-veto elections. We iteratively

bribe a voter v vetoing {p,c} such that no other voter

′

vetoes {p, c

′

} such that c currently has fewer ve-

toes than c

′

, and give the vetoes to the candidates with

currently the fewest vetoes. Correctness is shown by

induction on the bribery size.

Our next result shows the connection between Set

Cover and approval-based elections.

Theorem 4. Unweighted k-approval elections for k ≥

3 and k-veto for k ≥ 4 are resistant to unpriced con-

structive bribery.

Proof. Consider the following reduction from X3C to

bribery in 3-approval elections.

Let S = {s

,...,s

} and T

,...,T

be 3-subsets

of S. Consider the following 3-approval election of

C = {p, p

′

, p

′′

}∪{s

,...,s

}∪{b

,...,b

6(nm+m−n)

For each set T

= {t

i,1

i,2

i,3

}, voter T

approves

i,1

i,2

i,3

, and for 1 ≤ i ≤ 3(nm+ m− n) voter S

ap-

provesb

2i−1

, b

, and one of s

,...,s

, such that each

receives exactly n+1 approvals. Finally, n−m vot-

ers approve p, p

′

, p

′′

. We set our bribery quota q to m.

A successful bribery must remove one point from

each candidate of S, corresponding to a solution to

X3C. Buffer candidates can be added for cases of k-

approval for k ≥ 4. For 4-veto elections we veto p

and the three candidates correspondingto each subset.

Theorem 5. Unweighted 2-approval and 3-veto elec-

tions are vulnerable to unpriced constructive bribery.

Proof. Consider bribery in 2-approval elections.

Clearly, we only bribe voters not approving p to ap-

prove p.

For c ∈ C, let Approvals(c) be the number of

voters currently approving c. Following bribery, p

will receive Approvals(p) + q approvals. Also deﬁne

Deﬁcit(c) = Approvals(p)+ q− Approvals(c), as the

number of excess approvals p ends with relative to c.

Consider the following instance of Sim-

ple b-Edge Cover of Multigraphs. Let

X =

∑

c6=p

max(0,−Deﬁcit(c)) be the number of

”excess” approvals that must be removed. If

X > 2q, then bribery is not possible. Similarly,

let D =

∑

c6=p

max(0,Deﬁcit(c)) be the number of

approvals that we can give to the non-distinguished

candidates. We can bribe at most min(D,q) voters.

Construct G as follows. Let V(G) = C\{p}. For

every voter approving u and v such that p /∈ {u, v} ,

we add edge (u,v). We set the b-values to b(v) =

max(0,−Deﬁcit(c)), and our covering quota to q

′

min(D,q). We can show that G has a simple b-edge

covering of q

′

= min(D,q) edges iff there exists a

bribery of q voters making p a winner.

In a similar construction for 3-veto elections,

votes vetoing candidates p,c

, and c

correspond to

an edge between c

and c

in G.

3.4 Controlling an Election via Voters

Algorithms and reductions similar to cases of bribery

can be applied to control by adding or deleting voters.

Theorem 6. Unweighted 1 and 2-approval and 1 and

2-veto elections are vulnerable to control by adding

or deleting voters. 3-approval is also vulnerable to

control by adding voters, and 3-veto to deleting vot-

ers.

Proof. Unweighted 3-approval and 2-veto elections

are vulnerable to constructive control by adding vot-

ers, and 2-approval and 3-veto elections to deleting

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

216

voters using Simple b-Edge Cover of Multigraphs.

he others are due to greedy algorithms.

Theorem 7. U

nweighted k-approval elections for k ≥

4 and k-veto for k ≥ 3 are resistant to constructive

control by adding voters, and k-approval for k ≥ 3

and k-veto for k ≥ 4 by deleting voters, due to reduc-

tion from X3C.

3.5 Controlling an Election via

Candidates

Reductions from Hitting Set (Bartholdi et al., 1992;

Hemaspaandra et al., 2007) have shown that plurality

and veto are resistant to both constructiveand destruc-

tive control by adding and deleting candidates. The

reductions are from Hitting Set, and buffer candidates

can be added for the cases of k-approval and k-veto.

These systems are thus resistant to control by adding

or deleting candidates.

4 ON WEIGHTED AND PRICED

CASES OF ELECTION MISUSE

From the results in (Hemaspaandra and Hemaspaan-

dra, 2007), we conclude that 1-approval is the only

approval-based family of scoring protocol that is

vulnerable to weighted manipulation and weighted

bribery.

Theorem 8. Unweighted 1-approval, 1-veto, and 2-

veto elections are vulnerable to constructive

Bribery

(i.e., bribery with priced voters).

Proof. The case of $Bribery for 1-approval is shown

in (Faliszewski et al., 2006). In 1-veto, we iteratively

bribe the cheapest voters vetoing p, giving the veto to

the candidate currently having the fewest voters.

In the case of 2-veto, we will bribe only voters

vetoing p, and for each candidate c 6= p, we will bribe

the cheapest voters vetoing {p,c}. For each c ∈ C,

let Vetoes(c) be the total number of vetoes given to c

initially.

Suppose that a total of s voters are bribed, so that

p ends with Vetoes(p) − s vetoes. Let C

= {c ∈

C\ {p}|Vetoes(c) < Vetoes(p) − s} and C

= {c ∈

C\ {p}|Vetoes(c) ≥ Vetoes(p)− s}. We need to make

sure that each candidate c 6= p ends up with at least

Vetoes(p) − s vetoes.

Let s = s

such that s

of the bribed voters

veto p and a candidate in C

and s

+ s

voters veto p

and a candidate in C

, such that at least s

voters veto

{p, c} where no more than Vetoes(c) − (Vetoes(p) −

s) voters veto {p,c}.

For each case, we choose the cheapest s

voters

vetoing p and a candidate in C

. We then iteratively

choose the cheapest s

voters vetoing p and a candi-

date in C

, such that for each c ∈ C

, no more than

Vetoes(c) − (Vetoes(p) − s) voters vetoing {p,c} are

chosen. We ﬁnally choose the cheapest s

remaining

voters vetoing p and a candidate in C

to bribe. We

try this for each s

+ s

≤ ||V||.

Theorem 9. W

eighted 1-approval and 1-veto elec-

tions are vulnerable to constructive control by adding

and deleting voters. 2-approval is vulnerable to

adding voters, and 2-veto to deleting voters.

Proof. The cases of 1-approval and 1-veto follow

from greedy algorithms, for example, iteratively

adding the heaviest voters approving p.

Consider an addition of voters in a weighted 2-

approval election that makes p a winner. Clearly we

only add voters approving p. Let v be an unestab-

lished voter that approves of {p,c} such that no other

unestablished voter v

′

approves {p,c

′

} with c

′

having

more approvals initially than c. We may assume that

at least one voter approving of {p,c} is added. It also

sufﬁces to add the heaviest voter approving {p,c}. A

correct algorithm is to add such voters iteratively.

The cases of constructivecontrol by adding (delet-

ng) voters in 3-approval and 2-veto (2-approval

and 3-veto), as well as priced bribery in 2-approval

and 3-veto, are polynomial-time equivalent to some

weighted variations of b-Edge Cover as follows: The

weights of voters correspond to the weighted cover-

ings, while the prices in bribery problems correspond

to weighted edges. The complexity of these two prob-

lems is left as an open problem.

5 RESULTS AND DISCUSSION

These results show the variance of complexity in mis-

use among different problems in election systems in

elections of the form k-approval, k-veto, and f(m)-

approval, and give the ﬁrst results of complexity for

inﬁnitely many scoring protocols of an unbounded

number of candidates. There are a few interesting

cases: These manipulations can either be easy by a

simple greedy algorithm, equivalent to a correspond-

ing variation of b-Edge Cover, which is easy for the

unweighted and unpriced cases but unknown for the

weighted or priced variations, or hard by reduction

from Set Cover. Manipulations involving the candi-

date sets are always difﬁcult as a result of Hitting Set.

These results demonstrate the strengths and weak-

nesses of approval-based election systems, and we

THE COMPLEXITY OF MANIPULATING k-APPROVAL ELECTIONS

217

hope these results can lead to further generalizations

and possible developments of systems that better re-

sist such attacks.

It is important to realize that NP-completeness

only addresses the worst-case complexity of a given

problem, and does not take into consideration the dis-

tribution of problems that might be given. Some sim-

ple distributions were considered in (Walsh, 2009;

Friedgut et al., 2008), and it may be of interest to char-

acterize the complexity of more interesting and real-

istic distributions, depending upon the application..

This model also makes the assumption that in a k-

approval election, each voter may vote for any com-

bination of the k candidates independently. We know

that in practice, most elections do not follow this

principle. It may thus be of interest to character-

ize these properties in a more realistic distribution of

voter preferences.

ACKNOWLEDGEMENTS

We wish to offer our special thanks to Dr. E. Hemas-

paandra for pointing out the connection between b-

Edge Cover and elections as well as proofreading.

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