condorcet winners for public goods

Annals of Operations Research 137, 229–242, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

Condorcet Winners for Public Goods

LIHUA CHENGuanghua School of Management, Peking University, Beijing 100080, P.R. China

XIAOTIE DENGDepartment of Computer Science, City University of Hong Kong, Hong Kong, P.R. China

QIZHI FANGDepartment of Mathematics, Ocean University of China, Qingdao 266071, P.R. China

FENG TIANInstitute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences,Beijing 100080, P.R. China

Abstract. In this work, we consider a public facility allocation problem decided through a voting processunder the majority rule. A location of the public facility is a majority rule winner if there is no other locationin the network where more than half of the voters would have been closer to than the majority rule winner. Wedevelop fast algorithms for interesting cases with nice combinatorial structures. We show that the computingproblem and the decision problem in the general case, where the number of public facilities is more than oneand is considered part of the input size, are all NP-hard. Finally, we discuss majority rule decision makingfor related models.

Keywords: public goods, Condorcet winner, majority equilibrium, complexity, algorithm

1. Introduction

Majority rule is arguably the best decision mechanism for public decision makings.Informally, a majority winner has the property that no other solutions would please morethan half of the voters in comparison to it. On the other hand, it is well known that amajority winner may not always exist as shown in the famous Condorcet paradox wherethree agents have three different orders of preferences, A > B > C , B > C > A,C > A > B among three alternatives A, B and C . In this work we are interested incomputational aspects for majority winners in public decision making.

We focus on a public facility location problem. Demange (1983) reviewed con-tinuous and discrete spatial models of collective choice, aiming at a characterization ofthe location problem of public services as a result of public voting process. To facilitatea rigorous study of the related problem, Demange proposed four types of Condorcetwinners and discussed corresponding results (Romero, 1978; Hansen and Thisse, 1981)concerning conditions for their existences. We consider a weighted version of the discrete

230 CHEN ET AL.

model of Demange, represented by a network G = ((V, w), (E, l)) linking communitiestogether. Each vertex i ∈ V represents a community, and w(i) represents the numberof voters that reside there. For each e ∈ E , l(e) > 0 represents the distance betweentwo ends of the road e = (i, j) connecting the two communities i and j . The loca-tion of a public facility such as library, community center, etc., is to be determined bythe public via a voting process under the majority rule. We consider a special type ofutility function: each member of the community is interested in minimizing the dis-tance of its location to that of the public facility. While each desires to have the publicfacility to be close to itself, the decision has to be agreed upon by a majority of thevotes.

Following Demange (1983), a location x ∈ V is a strong (resp. weak) Condorcetwinner if, for any other y ∈ V , the total weight of vertices that is closer to x than toy is more (resp. no less) than the total weight of vertices that is closer to y than to x .Similarly, it is a quasi-Condorcet winner if we change “closer to x than to y” to “closerto x than to y or of the same distance to x as to y”. Of the four types of majority winner,strong Condorcet winner is the most restrictive of all, and weak quasi-Condorcet winneris the least restrictive one and the other two are between them. For discrete modelsconsidered by Romero (1978), Hansen and Thisse (1981), it was known that, the orderinduced by strict majority relation (the weak Condorcet order) in a tree is transitive.Therefore, a weak Condorcet winner in any tree always exists. In addition, Demangeextended the existence condition of a weak Condorcet winner to all single peaked orderson trees (Demange, 1982). Note that an order on a tree is single peaked if, along eachsingle path on the tree, the preference strictly increases up to a peak and then strictlydecreases. Our study will focus on the algorithmic and complexity aspects of Condorcetwinners, and discuss trees and cycles as well as the cactus graph, that is, a connectedgraph in which each block is an edge or a cycle.

In general, a location choice rule is a systematic way to map preferences into lo-cations. Among many properties proposed for a choice rule, the incentives property ofstrategy-proofness (i.e., an agent should never be able to manipulate the choice rule bymisreporting his preferences to it) has been extensively studied (Moulin, 1980; Borderand Jordan, 1983; Barbera, Masso, and Serizawa, 1998). Schummer and Vohra has re-cently studied the public location choice rule on a continuous (non-discrete) networkmodel (Schummer and Vohra, 2002). They characterized a class of onto choice rules thatsatisfy the strategy-proofness condition when each agent’s preference over points on acontinuous network is determined by the distance between his location and the points. Inour model, the domain and the range of the majority rule are restricted to the set of verticesof a network. Therefore, our setting can be viewed as a discrete version of Schummer andVohra’s model. Our study distinguishes from previous work in our focus on algorithmicissues. Recently, there has been a growing research effort in re-examination of conceptsin human and social sciences via computational complexity approach, e.g., cooperativegames (Megiddo, 1978; Deng and Papadimitriou, 1994; Fang and Zhu, 2002), organiza-tional structure (Deng and Papadimitriou, 1999), arbitrage (Deng, Li, and Wang, 2000),as well as general equilibrium (Deng, Papadimitriou, and Safra, 2003).

CONDORCET WINNER FOR PUBLIC GOODS 231

In Section 2, we introduce the formal formulation of the public facility locationproblem with a single facility in a network. Obviously, enumerating through all n loca-tions allows us to have a polynomial time algorithm to find a majority winner location.The issue here is how to improve the time complexity. In particular, we are interested inclassifying the types of networks for which a Condorcet winner can be found in lineartime. As a warm-up example, we present the solution for trees in Section 3. We present alinear algorithm for finding weak quasi-Condorcet winners of a tree with vertex-weightand edge-length functions; and prove that in this case, weak quasi-Condorcet winnersare the points which minimize the total weight-distance to individuals’ locations. InSection 4, we give a sufficient and necessary condition for a point to be a weak quasi-Condorcet winner for cycles in the case the edge-length function is a constant, andpresent a much more interesting linear time algorithm. This is further improved to ob-tain a linear time algorithm for cactus graphs (Section 5). We also discuss the propertyof (group) strategy-proofness for the majority rule of choosing weak (quasi) Condorcetwinners on a tree and a cycle respectively. In Section 6, we present NP-hard proofsfor the problem of finding a majority winner and the corresponding decision problemwhen the number of public facilities is taken as the input size, not a constant. Finally, inSection 7, our approach is extended to a more general public decision making process,the public road repair problem, pioneered by Tullock (1959) to study redistribution oftax revenue under a majority rule. We conclude with remarks and discussions on relatedissues.

2. Definition

Demange (1983) surveyed and discussed some spatial models of collective choice, andgave some results concerning the transitivity of the majority rule and the existence ofa majority winner. In this paper, we consider a public facility location problem with asingle facility in a graph. Let G = (V, E) be an undirected graph of order n with a weightfunction ω that assigns to each vertex v of G a non-negative weight ω(v), and a lengthfunction l that assigns to each edge e of G a positive length l(e). If P is a path of G,then we denote by l(P) the sum of lengths of all edges of P . We denote by dG(u, v) thelength of a shortest path joining two vertices u and v in G, and call it distance betweenu and v in G. For any R ⊆ V , we set ω(R) = ∑

v∈R ω(v). In particular, if R = V , wewrite ω(G) instead of ω(V ). A vertex v of G is said to be pendant if v has exactly oneneighbor in G.

Given a graph G = (V, E) with V = {v1, v2, . . . , vn}, each vi ∈ V has a preferenceorder ≥i on V induced by the distance on G. That is, we have x ≥i y if and only ifdG(vi , x) ≤ dG(vi , y) for two vertices x, y ∈ V . The following definition is an extensionof that given in Demange (1983).

Definition 1. Given a graph G = (V, E) and a profile (≥i )vi ∈V on V , a vertex v0 ∈ Vis called:

232 CHEN ET AL.

(1) Weak quasi-Condorcet winner, if for every u ∈ V distinct of v0,

ω({vi ∈ V : v0 ≥i u}) ≥ ω(G)

2, i.e. ω({vi ∈ V : u >i v0}) ≤ ω(G)

2.

(2) Strong quasi-Condorcet winner, if for every u ∈ V distinct of v0,

ω({vi ∈ V : v0 ≥i u}) >ω(G)

2, i.e. ω({vi ∈ V : u >i v0}) <

ω(G)

2.

(3) Weak Condorcet winner, if for every u ∈ V distinct of v0,

ω({vi ∈ V : v0 >i u}) ≥ ω({vi ∈ V : u >i v0}).(4) Strong Condorcet winner, if for every u ∈ V distinct of v0,

ω({vi ∈ V : v0 >i u}) > ω({vi ∈ V : u >i v0}).

Example. Denote by K2 and K3 the complete graphs of orders 2 and 3, respectively.Suppose that the length function on the edge set and the weight function on the vertexset in K2 and K3 are constant. Then K2 has weak Condorcet winners, and hence has alsoweak quasi-Condorcet winners, but has no strong Condorcet winners and strong quasi-Condorcet winners; K3 has strong quasi-Condorcet winners, weak Condorcet winnersand weak quasi-Condorcet winners, but has no strong Condorcet winners.

In this paper, we will mainly consider algorithms for finding weak quasi-Condorcetwinners of a tree, a cycle and, more generally, a cactus graph. The properties and al-gorithms for the other three types of Condorcet winners can be discussed in a similarway.

Remark 1. Barycenter of a graph.

For each vertex u of a graph G = (V, E), denote

sG(u) =∑v∈V

ω(v)dG(u, v).

Intuitively, sG(u) is the weighted total distance from members of other communities to u.A vertex u0 is called a barycenter if sG(u0) = minv∈V sG(v). Though “weak Condorcetwinner” and “barycenter” are two different concepts for general graphs, we will showthat they are the same for trees.

Remark 2. Strategy-proofness and group-strategy-proofness.

Let G = (V, E) be an undirected graph and N = {1, 2, . . . , n} be a set of n voters. Ingeneral, when the preference on V of each voter is single peaked and uniquely determinedby its peak on G, a public facility choice rule is a function π : V n → V mapping voters’


peaks into vertices on the graph. In our model, there is a weight function ω that assigns toeach voter i ∈ N a positive weight ω(i) representing the voter i’s decision power. Givena profile of the peaks (residencies) of n voters 〈p〉 = (p1, p2, . . . , pn) ∈ V n , π (〈p〉) isdefined to be the corresponding majority rule winners. In the rest of this paper, we denoteby 〈p; p′

k, k〉 the profile obtained by substituting voter k’s peak pk with p′k in 〈p〉, and

denote by 〈p; p′S, S〉 the profile obtained by substituting the peaks pS = {pi : i ∈ S ⊆ N }

with peaks p′S = {p′

i : i ∈ S ⊆ N } in 〈p〉.Among many properties of choice rules, we are interested in the well known

incentives property of strategy-proofness: any agent should never be able to manip-ulate the choice rule by misreporting his preferences to it. Formally, a choice ruleπ : V n → V is strategy-proof if for every voter i ∈ N , and for every profile ofpeaks 〈p〉 = (p1, p2, . . . , pn) ∈ V n ,

∀p′i ∈ V, dG(pi , π (〈p〉)) ≤ dG(pi , π (〈p; p′

i , i〉)).

A choice rule π is group-strategy-proof if for every coalition S ⊆ N , and for everyprofile of peaks 〈p〉 = (p1, . . . , pn) ∈ V n ,

∃ p′S ∈ V |S|, such that for any i ∈ S, dG(pi , π (〈p; p′

S, S〉)) ≥ dG(pi , π (〈p〉)),

and at least one of the above inequalities is strict.The concepts of strategy-proofness and group-strategy-proofness can be viewed

as corresponding to the stability of Nash equilibrium and strong equilibrium. We willdiscuss these properties for majority rules in our models.

3. Weak quasi-Condorcet winner of a tree

Romero (1978) and Hansen and Thisse (1981) pointed out that the family of ordersinduced by a distance on a tree guarantees the existence of weak Condorcet winners.Furthermore, weak Condorcet winners are the points that minimize the total distanceto the individuals’ locations (Demange, 1983). In this section we propose a linear al-gorithm for finding weak quasi-Condorcet winners on a tree with vertex-weight andedge-length functions; and prove that in this model, weak quasi-Condorcet winnersare the same as points which minimize the total weight-distance to the individuals’locations.

Given two vertices v, x ∈ V , the set of quasi-friend vertices of v relative to x isdefined as

FG(v, x) = {u : dG(u, v) ≤ dG(u, x)};

and the set of hostile vertices of v relative to x is defined as

HG(v, x) = {u : dG(u, v) > dG(u, x)}.

234 CHEN ET AL.

By the definition of weak quasi-Condorcet winner, a vertex v0 ∈ V is a weak quasi-Condorcet winner of G, if for any vertex x = v0,

ω(FG(v0, x)) ≥ 1

2ω(G), or equivalently, ω(FG(v0, x)) ≥ ω(HG(v0, x)).

Theorem 3.1. Every tree has one weak quasi-Condorcet winner, or two adjacent weakquasi-Condorcet winners. We can find it or them in linear time.

Proof. Let T = (V, E) be a tree of order n, and ω(v) and l(e) be the weight functionand length function on V and E , respectively. Without loss of generality, we assume thatω(v) > 0 for each vertex v ∈ V . We prove the theorem by induction on n.

When n = 1, the conclusion is trivial. When n = 2, we set T = ({u, v}, uv). Ifω(u) = ω(v), assuming without loss of generality, that ω(u) < ω(v), then it is easy tosee that v is the unique weak quasi-Condorcet winner of T . If ω(u) = ω(v), then both ofu and v are weak quasi-Condorcet winners of T .

Now suppose that n ≥ 3. Let u1, u2, . . . , uk (k ≥ 2) be all the pendant vertices ofT . Assume, without loss of generality, that u1 is one vertex with minimum weight amongthese pendant vertices. Hence w(u1) < 1

2ω(T ). Let u1v1 be the corresponding pendantedge. Denote T ∗ = T − u1, and define the weight function ω∗ on T ∗ as follows.

ω∗(v) ={ω(u1) + ω(v1), if v = v1;

ω(v), if v = v1.

Note that FT (u1, v1) = {u1} and HT (u1, v1) = V (G) − {u1}. Because ω(u1) < 12ω(T ),

we have ω(FT (u1, v1)) < ω(HT (u1, v1)), and hence u1 is not a weak quasi-Condorcetwinner of T . We show the following

Claim. If v0 is a weak quasi-Condorcet winner of T ∗, then v0 is also a weak quasi-Condorcet winner of T .

Proof of Claim. Notice that for any vertex u = u1, dT (u, v1) = dT ∗(u, v1) anddT (u, u1) = dT (u, v1) + l(u1v1). Thus,

(i) v1 ∈ FT (v0, u) if and only if v1 ∈ FT ∗(v0, u); Similarly, v1 ∈ HT (v0, u) if and onlyif v1 ∈ HT ∗(v0, u).

(ii) If v1 ∈ FT ∗(v0, u), then u1 ∈ FT (v0, u); Similarly, if v1 ∈ HT ∗(v0, u), then u1 ∈HT (v0, u).

(iii) ω∗(FT ∗(v0, u)) = ω(FT (v0, u)); ω∗(HT ∗(v0, u)) = ω(HT (v0, u)).

By the definition of weak quasi-Condorcet winner, if v0 is a weak quasi-Condorcet winnerof T ∗, then ω∗(FT ∗(v0, u)) ≥ ω∗(HT ∗(v0, u)). Thus ω(FT (v0, u)) ≥ ω(HT (v0, u)). Thatis, v0 is also a weak quasi-Condorcet winner of T . Theorem 3 now follows from the Claim.


Thus we can propose the following linear algorithm for finding weak quasi-Condorcetwinners of a tree T of order n:

Step 1. Take the pendant vertex v of T such that w(v) < 12w(T );

Step 2. T − v ⇒ T , w(v) + w(u) ⇒ w(u), where u is the (unique) neighbor of v;Step 3. n − 1 ⇒ n; if n = 1, or, n = 2 and the two vertices have the same weights, then

stop; otherwise go to Step 1. �

Theorem 3.2. Let T be a tree. Then v0 is a weak quasi-Condorcet winner of T if andonly if v0 is a barycenter of T .

Proof. Let T1, T2, . . . , Tk be the subtrees of T −v0 and let ui be the (unique) vertex of Ti

adjacent to v0 (i = 1, 2, . . . , k). Obviously, FT (v0, ui ) = V (T ) − V (Ti ), HT (v0, ui ) =V (Ti ) for all i = 1, 2, . . . , k.

(i) By the definition of weak quasi-Condorcet winner, we get that v0 is a weak quasi-Condorcet winner of T if and only if for each i = 1, 2, . . . , k, ω(V (T ) − V (Ti )) ≥ω(Ti ), i.e., ω(Ti ) ≤ 1

2ω(T ). Denote Ti = T − V (Ti ), i = 1, 2, . . . , k. It is easy tosee that

sT (v0) = sTi(v0) + sTi (ui ) + ω(Ti )dT (v0, ui ) and

sT (ui ) = sTi(v0) + sTi (ui ) + ω(Ti )dT (v0, ui ).

(ii) By the definition of barycenter, we get that v0 is a barycenter of T if and only ifω(Ti ) ≤ ω(Ti ), i.e., ω(Ti ) ≤ 1

2ω(T ) for each i = 1, 2, . . . , k.

From (i) and (ii), we get the conclusion of Theorem 3.2.

Given a set of voters N and a profile of peaks 〈p〉 of N on a tree T , denote the setof weak quasi-Condorcet winners by C〈p〉, which contains one vertex or two adjacentvertices according to Theorem 1. Let π : V n → V denote the majority rule of choosingweak quasi-Condorcet winners. Then for any profile of peaks 〈p〉 of N , π (〈p〉) may beany member of C〈p〉. In the following theorem we define dT (v, π (〈p〉)) as the distancebetween vertex v and the set C〈p〉, that is, dT (v, π (〈p〉)) = mint∈C〈p〉 dT (v, t) for anyvertex v of T .

Theorem 3.3. Let T = (V, E) be a tree, N = {1, 2, . . . , n} be a set of voters withpositive weight ω : N → R+. The majority rule π : V n → V of choosing weakquasi-Condorcet winners of T satisfies the property of (group) strategy-proofness.

236 CHEN ET AL.

Proof. Suppose the contrary, given a profile 〈p〉 = (p1, p2, . . . , pn) ∈ V n , there is avoter k ∈ N having an incentive to announce p′

k instead of pk , that is,

dT (pk, π (〈p〉)) > dT (pk, π (〈p; p′k, k〉)). (3.1)

Denote by c1 ∈ C(〈p〉) and c2 ∈ C(〈p;p′k ,k〉) the vertices nearest to peak pk , respectively.

(3.1) implies that c2 ∈ C〈p〉. Let P(c1, c2) be the (unique) path connecting c1 and c2 in T ,a1 be the vertex adjacent to c1 on P(c1, c2), T1 and T2 be the two connected componentsof T \ e(c1, a1) concluding c1 and a1, respectively. Then c2 ∈ T2, and pk ∈ T2 by ourassumption. Following the definition of c1, it is easy to see that a1 ∈ C〈p〉. Hence for theprofile 〈p〉, ∑

pi ∈T2ω(i) < ω(N )/2, which implies that

∑c1>i c2

ω(i) ≥∑

c1>i a1

ω(i) >∑

a1>i c1, i =k

ω(i) + ω(k) ≥∑

c2>i c1, i =k

ω(i) + ω(k). (3.2)

On the other hand, by our assumption c2 ∈ C〈p;p′k ,k〉 and pk ∈ T2, we have p′

k ∈ T2 and∑

c1>i c2

ω(i) ≤∑

c2>i c1, i =k

ω(i) + ω(k),

which is contrary to (3.2). Therefore, the majority rule π satisfies the property of strategy-proofness. The proof of group-strategy-proof is similar, which is omitted.

4. Weak quasi-Condorcet winner of a cycle

In this section, we characterize weak quasi-Condorcet winners and discuss the relatedalgorithm for cycles. Throughout this section, we assume that the edge length functionof a cycle is constant (without loss of generality, equal to 1).

For convenience, we denote a cycle of order n by Cn = v1v2 . . . vnv1, whoseconsecutive vertices are adjacent; and denote a path vivi+1 . . . v j of Cn and its length byCn[vi , v j ] and l(Cn[vi , v j ]), respectively. A path of length k is called a k-interval of Cn .For a real number α, denote by �α� the greatest integer no more than α.

Theorem 4.1. Let Cn be a cycle of order n. Then v ∈ V (Cn) is a weak quasi-Condorcetwinner of Cn if and only if the weight of each � n+1

2 �-interval containing v is at least12w(Cn).

Proof. By the definition of weak quasi-Condorcet winner, v is a weak quasi-Condorcetwinner of Cn if and only if for all vertices u ∈ V (Cn) distinct of v,

w(FCn (v, u)) ≥ 1

2w(Cn).

It is easy to verify that FCn (v, u) is an interval containing v of length � n+12 �. As vertex u

runs over all vertices except v, these intervals are exactly all � n+12 �-intervals containing

v, and satisfy the condition that the weight of each interval is at least 12w(Cn).


Corollary 4.2. Let Cn[p, q] be a � n+12 �-interval of Cn . If

∑u∈Cn[p,q] w(u) < 1

2w(Cn),then the path Cn[p, q] contains no weak quasi-Condorcet winner of Cn .

Corollary 4.3. Suppose that vi , v j ∈ V (Cn) are both weak quasi-Condorcet winnersof Cn . If l(Cn[vi , v j ]) ≤ l(Cn[v j , vi ]), then all vertices of path Cn[vi , v j ] are weakquasi-Condorcet winners of Cn .

Corollary 4.4. If v1 and v� n+22 � are both weak quasi-Condorcet winners of Cn , then all

vertices of Cn are weak quasi-Condorcet winners.

Corollary 4.5. Let S be the set of all weak quasi-Condorcet winners of Cn . If S = ∅,then the subgraph induced by S is connected.

Based on Theorem 4.1, we propose the following linear algorithm for determiningwhether a vertex is a weak quasi-Condorcet winner of a cycle. Denote Cn = v1v2 . . . vnv1

and v = v� n+12 �.

Step 1. Calculate W1 = w(v1) + w(v2) + · · · + w(v� n+12 �) and W (Cn) = w(v1) + w(v2)

+ · · · + w(vn);Step 2. For i = 1, 2, . . . , � n+1

2 � − 1, calculate Wi+1 = Wi − w(vi ) + w(v� n+12 �+i );

Step 3. If Wi ≥ 12w(Cn) for each i = 1, 2, . . . , � n+1

2 �, then v = v� n+12 � is a weak

quasi-Condorcet winner of Cn; otherwise, v = v� n+12 � is not a weak quasi-Condorcet

winner.

Furthermore, we have

Theorem 4.6. The problem of finding a weak quasi-Condorcet winner of a cycle withvertex-weight function is solvable in linear time.

Let N be a set of voters with weight function ω : N → R+, and C be a cycle. Ifω(i) < ω(N )/2 for each i ∈ N , then there always exists a profile of peaks 〈p〉 of Non C such that the corresponding weak quasi-Condorcet winners do not exist. If thereis a voter i∗ ∈ N with a power weight ω(i∗) ≥ ω(N )/2, then the peak of voter i∗ mustbe a weak quasi-Condorcet winner for any profile 〈p〉 on C . That is, the voter i∗ has adecisive power on the majority rule, which is called the “cycle dictator”. In this case, wehave the following result.

Theorem 4.7. Let C = (V, E) be a cycle, and N = {1, 2, . . . , n} be a set of voters withweight function ω : N → R+. If there is a voter i∗ ∈ N such that ω(i∗) ≥ ω(N )/2, thenfor any profile of peaks 〈p〉 = (p1, . . . , pn) on C , i∗ is always a weak quasi-Condorcetwinner, and the majority rule π : V n → V of choosing weak quasi-Condorcet winnersof C satisfies the property of (group) strategy-proofness.

238 CHEN ET AL.

5. Weak quasi-Condorcet winner of a cactus graph

In this section we discuss the problem of finding weak quasi-Condorcet winners ofa cactus graph. We also assume that the edge length function of a graph is constantthroughout this section.

Given a graph G = (V, E), a vertex v of G is a cut vertex if E(G) can be partitionedinto two nonempty subsets E1 and E2 such that the induced graphs G[E1] and G[E2]have just the vertex v in common. A block of G is a connected subgraph of G that hasno cut vertices and is maximal with respect to this property. Every graph is the union ofits blocks. A block of G is said to be pendant if it contains exactly one cut vertex.

Definition 2. A graph G is called a cactus graph, if G is a connected graph in whicheach block is an edge or a cycle.

Note that if a cactus graph G of order n ≥ 3 is not a cycle, then G contains at leasttwo pendant blocks (edges and/or cycles). Based on the algorithms given for trees andof cycles, we discuss the algorithm of finding weak quasi-Condorcet winners of a cactusgraph G as follows.

Case 1. G is a block. Then G is an edge or a cycle. We can find the weak quasi-Condorcetwinners of G by the algorithms given in Section 3 and Section 4.

Case 2. G is not a block. Let B1, B2, . . . , Bk (k ≥ 2) be the pendant blocks of G,and bi be the cut vertex contained in Bi (i = 1, 2, . . . , k). Assume without loss ofgenerality, that w(B1 − b1) ≤ w(B2 − b2) ≤ · · · ≤ w(Bk − bk). Since 2w(B1 − b1) +w(b1) ≤ w(B1 − b1) + w(b1) + w(B2 − b2) ≤ w(G), w(B1 − b1) < 1

2w(G). LetG∗ = G − (V (B1) − b1) and define the weight function w∗ on G∗ as follows:

w∗(v) ={w(B1), if v = b1;

w(v), if v = b1.

Thus, from a similar argument as in the proof of Theorem 1, the set of weak quasi-Condorcet winners of G∗ is the same as that of G.

Note that |V (G∗)| < |V (G)|, we will get Case 1 after repeating applications of thealgorithm in Case 2 in no more than |V (G)| times. Thus we have

Theorem 5.1. The problem of finding a weak quasi-Condorcet winner of a cactus graphwith vertex-weight function is solvable in linear time.

Moreover, we can see that if weak quasi-Condorcet winners exist for a cactus graphG, then all weak quasi-Condorcet winners lie in a block (an edge or a cycle) of G, and as ageneralization of Corollary 4.5, the subgraph induced by weak quasi-Condorcet winnersof G is connected.


6. Complexity issues for public location in general networks

In general, the problem can have various extensions. There may be a number of publicfacilities to be allocated during one voting process. The community network may be ofgeneral graphs. The public facilities may be of the same type, or they may be of differenttypes. The utility functions of the voters may be of different forms. Our discussion inthis section will take such variations into consideration. However, in our discrete model,we keep the restriction that the public facilities will be located at vertices of the graph.We have the following general results for any of the four types of Condorcet winners.

Theorem 6.1. If there are a bounded constant number of public facilities to be locatedat one voting process under the majority rule, then a Condorcet winner (of any of thefour types) can be computed in polynomial time.

Proof. We present the proof for the strong Condorcet solution. (Other cases can beproven similarly.) Let k be the bounded constant for the number of public facilities to belocated in the network G = (V, E, w), |V | = n. The total number of possible choicesfor the locations of k publics facilities is nk . We construct an auxiliary directed graphD = (U, A) as follows: The vertex set U consists of nk points each representing a set oflocations for the k public facilities. An arc (u, v) ∈ A if and only if the total weight ofcommunities that prefers the set of locations u to the set of locations v is more than onehalf of the total weight. Therefore, a vertex in D with in-degree zero, that is, without anyincoming arc, is a Condorcet solution.

In the construction of the auxiliary graph D, every vertex in D has to be comparedwith every other vertex, and each comparison can be done by an algorithm of timecomplexity O(n) by passing through all n vertices in the original network G. So the totaltime complexity of computing a strong Condorcet winner is O(n2k+1).

Theorem 6.2. If the number of public facilities to be located is not a constant butconsidered as the input size, the problem of computing a Condorcet winner is NP-hard;and the corresponding decision problem: deciding whether a candidate set of publicfacilities is a Condorcet winner, is co-NP-complete.

Proof. We present the proof also for the strong Condorcet solution. We obtain an NP-hard proof by reduction from EXACT COVER BY 3-SETS. We are given a family F ={S1, . . . , Sm} of subsets of Q = {1, 2, . . . , 3n}, where every set in F has three elements,and we are asking whether a subfamily F ′ ⊆ F with n sets exists that covers Q withoutoverlap. We construct a network G = (V, E) with edge length function l and vertexweight function ω as follows.

The vertex set V of G consists of 3 types:

(1) X of m elements (one for each 3-set in F), and ∀v ∈ X , ω(v) = 0; (2) n sets ofvertices Yi , i = 1, 2, . . . , n, each Yi consists of 3n vertices (one for each element in

240 CHEN ET AL.

Q), and ∀v ∈ Yi (i = 1, 2, . . . , n), ω(v) = 1; (3) Z of n vertices z1, z2, . . . , zn , and∀v ∈ Z , ω(v) = 0.

The edge set E of G consists of 2 parts:

(1) There is an edge between a vertex u in X and a vertex v in Yi (i = 1, 2, . . . , n), ifthe element corresponding to v is in the 3-set corresponding to u. Denote the set ofthese edges by E0, and all edges in E0 have the same length 1.

(2) For each i = 1, 2, . . . , n, there is an edge between vertex zi and every vertex in Yi ,and denote the set of these edges by Ei . For each set Ei , only one edge in Ei has thelength 2 and the other edges have the the same length 1.

The voters’ utility function is the minimum distance to one of n public facilitiesto be located in the network, and the strong Condorcet winner is required to be a subsetof n vertices of V under the majority rule. It is not hard to see that all n vertices of thestrong Condorcet winner will belong to X if there is an exact cover F ′ ⊆ F for theoriginal EXACT COVER BY 3-SETS problem, and vice versa. For the decision problem,we set the vertex set Z to be the candidate set for the strong Condorcet winner. It canbe shown that Z is not a strong Condorcet winner if and only if there is an exact coverF ′ ⊆ F of Q.

7. Remarks and further discussion

The study on public decision making has been an important area in political economicsand majority rule has had fundamental influence in the study of this area (Black, 1958;Buchanan and Tullock, 1962; Plott, 1967; Hansen and Thisse, 1981; Romero, 1978;Demange, 1983; Schummer and Vohra, 2002). In this work, we apply a computationalcomplexity approach to the study of public facility location problem decided via a votingprocess under the majority rule. Our study follows the network model that has beenapplied to the study of similar problems in economics (Romero, 1978; Hansen and Thisse,1981; Schummer and Vohra, 2002). We prove that the general problem is NP-hard andestablish efficient algorithms to interesting networks as used in the study of strategy-proof model for public facility location problem (Schummer and Vohra, 2002). Ourmathematical results depend on understanding of combinatorial structures of underlyingnetworks.

Our approach can be applied to more general public decision making processes.As a demonstration, in the following, we discuss majority winners for a public roadrepair problem, pioneered by Tullock (1959) to study redistribution of tax revenue undera majority rule system. In the public road repair problem, an edge weighted graph G =(V, E, w) is used to represent a network of local roads. There is a distinguished vertexS ∈ V representing the entry point to the highway system, and the weight of each edgerepresenting the cost of repairing. The majority decision problem involves a set of agentsA ⊆ V situated at vertices of the network who would choose a subset F of edges (F


is called a solution of this problem). The cost of repairing F , which is the sum of theweights of edges in F , will be shared by all n agents, each an n-th of the total.

An agent benefits more under solution H than under solution F if the shortest path(if exists) from its location to the entry point in H is no longer than in F and it paysthe cost of repairing in solution H no more than in solution F , and at least one of theinequalities is strict. Solution H dominates solution F , if there is a majority of agentswho would benefit more under H than under F . A solution not dominated by any othersolution is called a majority stable solution under the majority rule (or a majority winner).More formally, a majority stable solution under the majority rule is a collection F ofedges that connects to S a subset A1 ⊂ A of agents with |A1| > |A|/2 such that no othersolution H connecting to S a subset of agents A2 ⊂ A with |A2| > |A|/2 satisfies theconditions that w(H ) ≤ w(F), and that for each agent in A2, its shortest path to S insolution H is no longer than that in solution F , and at least one of the inequalities isstrict. We are able to show that the public road repair problem is in general NP-hard. Onthe other hand, we obtain a polynomial time algorithm when the network is a tree.

Theorem 7.1. It is strongly NP-hard to compute a majority stable solution under themajority rule of the public road repair problem.

Theorem 7.2. When the network is a tree, there is a polynomial time algorithm thatfinds a majority stable solution under the majority rule of the public road repair problem.

Many problems open up from our study. The complexity study for other rules forpublic facility location is very interesting and deserves further study. And it would beinteresting to extend our study to other areas and problems of public decision makingprocess.

Acknowledgment

The results reported in this work are supported by a RGC CERG grant (CityU 1081/02E)and a SRG grant (7001514) of City University of Hong Kong. The authors would like tothank the anonymous referees for their careful reviews of the manuscript and constructivesuggestions.

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