Multiple voting location and single voting location on trees

European Journal of Operational Research 181 (2007) 654–667 www.elsevier.com/locate/ejor

Discrete Optimization

Multiple voting location and single voting location on trees H. Noltemeier, J. Spoerhase *, H.-C. Wirth Lehrstuhl fu¨r Informatik I, Universita¨t Wu¨rzburg, Am Hubland, 97074 Wu¨rzburg, Germany Received 20 January 2006; accepted 30 June 2006 Available online 13 October 2006

Abstract We examine voting location problems in which the goal is to place, based on an election amongst the users, a given number of facilities in a graph. The user preference is modeled by shortest path distances in the graph. A Condorcet solution is a set of facilities to which there does not exist an alternative set preferred by a majority of the users. Recent works generalize the model to additive indifference and replaced user majority by c-proportion. We show that for multiple voting location, Condorcet and Simpson decision problems are Rp2 -complete, and investigate the approximability of the Simpson and the Simpson score optimization problem. Further we contribute a result towards lower bounds on the complexity of the single voting location problem. On the positive side we develop algorithms for the optimization problems on tree networks which are substantially faster than the existing algorithms for general graphs. Finally we suggest a generalization of the indifference notion to threshold functions.  2006 Elsevier B.V. All rights reserved. Keywords: Group decisions and negotiations; Facility location; Condorcet; Simpson score; Efficient graph algorithms

1. Introduction Location problems on graphs are characterized as follows: An edge weighted graph models distances in a universe. Weighted nodes of the graph represent customers. The customers can be served by facilities which can be placed at the nodes of the graph. The goal is to find an optimal placement of the facilities. Several objective functions are in common use, e.g. maximum or average distance to the closest facility *

Corresponding author. E-mail addresses: [email protected] (H. Noltemeier), [email protected] (J. Spoerhase), [email protected] (H.-C. Wirth).

(center and median problem), or total sum of distances and costs for opening the selected set of facilities (facility location problem). Voting problems are a means of modeling the process of finding compromises in a group of social individuals. Here, global decisions are often based on individual preferences which can be, more or less explicitely, treated as a formal election between alternative solutions. It is assumed that the resulting solutions are accepted by all participants and hence stable, since they are preferred by a significant majority of the users. Due to the nature of this model voting scenarios are often suitable to be solved by local algorithms, i.e., algorithms which are executed by an individual with a limited amount

0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.06.039

H. Noltemeier et al. / European Journal of Operational Research 181 (2007) 654–667

of information about the total scenario and a limited amount of communication with other individuals. Often local algorithms can be parallelized easily. Voting location problems are a way to combine both lines of research: The static universe is modeled by a weighted graph, while the optimal placement of facilities is the result of an election process performed by the individual users. Here the user preferences are fully determined by the distances in the underlying graph. Single and multiple location problem distinguishes the cases where one or more facilities are to be placed, respectively. The problems under investigation in this paper are as follows: The input graph specifies locations for users and facilities. There is a voting process among the users which aims to find an optimal placement of a set of facilities. The voting preference of individual users is based on the distance to the possible facility sets in the input graph. If a facility set is preferred over a concurrent facility set by more than half of the users, we say that the first set dominates the latter. A Condorcet set is a set of facilities which is not dominated by any alternative. The Simpson score of a facility set is the user preference of the strongest alternative; a low Simpson score can therefore be interpreted as a particular stable voting result. A Simpson solution is the most stable possible result, namely a facility set with minimum Simpson score. We refer the reader to the following chapter for a detailed definition of those notions. Applications of such voting location problems can be found, e.g. in the area of facility location planning. In the classical facility location problem, decisions are performed based solely on a couple of abstract cost functions. These cost functions reflect mainly the view of the manufacturer, hence it is questionable whether the customer would be content enough to accept the implemented solutions. Voting location can be used to make statements about the stability of solutions: we not only seek for solutions minimizing the costs but also minimizing the attractiveness of alternative solutions. When such a solution is implemented then it can be assumed that it is widely accepted by the customers. 2. Problem definition and preliminaries An instance of the multiple voting location problem (MVLP) is given by an undirected graph G = (V, E) with finite node set V and positive edge

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weights d: E ! Rþ inducing a distance function d: V  V ! Rþ 0 [ f1g between pairs of nodes. Two subsets U, L  V denote the set of users and the set of possible locations, respectively. We assume that U = L = V unless otherwise stated. A nonnegative node weight function w: U ! Rþ 0 represents the number of users at nodes in the graph. Additionally the instance specifies a number p 2 N (1 6 p 6 jLj) of facilities to place. An instance of the single voting location problem (SVLP) is defined similarly with the restriction that p = 1. 2.1. Notation We make use of the following standard notation: The eccentricity of a node u 2 V is defined as e(u) :¼ maxv2Vd(u, v). The radius of a graph with node set V is defined by minu2Ve(u). The distance between two node sets V1, V2  V is defined to be dðV 1 ; V 2 Þ :¼ minv1 2V 1 ;v2 2V 2 dðv1 ; v2 Þ. If the input graph T is a tree, and u, v 2 V are two disjoint nodes, then by Tu(v) we denote the subtree with root v hanging from v where the tree T is considered as rooted at u. As said before we are only interested in solutions which consist of exactly p facilities. To this end, let Lp :¼ fX  L j jX j ¼ pg be the family of facility sets of cardinality p. 2.2. Condorcet and Simpson We reflect the notation for Condorcet and Simpson solutions on MVLP instances from [1]: Let u 2 U be a user and X ; Y 2 Lp be two sets of facilities. The user u prefers Y over X if d(u, Y) < d(u, X) and the user is undecided if these distances are equal. By U ðY  X Þ :¼ fu 2 U j dðu; Y Þ < dðu; X Þg;

ð1Þ

we denote the set of users preferring Y over X. The weight of this set is denoted by W ðY  X Þ :¼ wðU ðY  X ÞÞ:

ð2Þ

Definition 2.1 (Dominating solution). Let X ; Y 2 Lp . We say that solution Y dominates X, denoted by Y  X, if W ðY  X Þ > 12 wðU Þ. In other words, Y is preferred over X by a majority of the users. Notice that the dominance relation is not necessarily a transitive relation.

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Definition 2.2 (Condorcet solution). Solution X 2 Lp is a Condorcet solution if there is no alternative solution Y 2 Lp such that Y  X. Observe that a Condorcet solution does not need to exist in every instance of MVLP. The stability of a solution as a result of a voting process can be measured in terms of the power of the strongest opposition, known as the Simpson score:

defines the set of users preferring Y over X. We say that Y dominates X (notated as Y  X) if the condition in (4) holds. With these new definitions, the notions [a, c]-Condorcet solution, a-Simpson score, and a-Simpson solution are defined as in Definitions 2.2 and 2.3.

Definition 2.3 (Simpson score and Simpson solution). The Simpson score W*(X) of a solution X 2 Lp with respect to user preference (1) is defined to be

The set of all [a, c]-Condorcet solutions of an instance of MVLP is denoted by C(a, c). In [2] it is shown that the family (C(a, c))a,c is inclusion-wise monotonous with respect to both parameters. Moreover the authors show that fixing one of the parameters allows to minimize the remaining parameter, i.e.,

W  ðX Þ :¼ max W ðY  X Þ:

c ðaÞ :¼ minfc 2 Rþ 0 j Cða; cÞ 6¼ ;g

A solution X* minimizing the Simpson score, i.e.,

a ðcÞ :¼ minfa 2 Rþ 0 j Cða; cÞ 6¼ ;g

Y 2Lp

W  ðX  Þ ¼ min W  ðX Þ X 2Lp

is called a Simpson solution. The Simpson score is always well defined, hence a Simpson solution exists in every instance of MVLP. 2.3. Relaxations on Condorcet and Simpson Campos and Moreno have described the notion of indifference or tolerance [2,1]. While in (1) a user u is undecided between two sets X, Y of facilities only if d(u, X)  d(u, Y) = 0, they suggest to extend the state of indifference to the cases jd(u, X)  d(u, Y)j 6 a for a constant a P 0 describing the tolerance of the users. We call this an additive indifference (see Section 7 for a generalization to e.g. multiplicative indifference). This yields the new definition

and

are well defined. Observe that c*(a) = W*(X)/w(U) where X is an a-Simpson solution. Hence the [a, c*(a)]-Condorcet solutions are exactly the a-Simpson solutions. On the other hand, an [a*(c), c]-Condorcet solution is named to be a c-tolerant solution. The sets of all a-Simpson solutions and all c-tolerant solutions are denoted by S(a) and T(c), respectively. A tuple (a, c) such that C(a, c) 5 ; but C(a  , c) = C(a, c  ) = ; for any  > 0 is called an efficient pair (also known as a pareto-optimal solution). An [a, c]-Condorcet solution is called efficient Condorcet solution if (a, c) is an efficient pair. The set of all efficient Condorcet solutions is denoted by EC. 2.4. Decision, optimization, and construction problems

The relaxed Condorcet and Simpson solutions are defined similarly to the previous definitions as follows:

The previous definitions suggest a number of problems to investigate. As usual one distinguishes between decision problems (problems which have a yes/no solution), optimization problems (problems where a numerical goal function is to be minimized or maximized), and construction problems (problems where a witness instance solving the corresponding optimization problem is seeked for). Briefly speaking, decision and optimization problems are polynomially time reducible to each other, while construction problems may or may not be significantly harder, depending on the particular problem (see e.g. [3] for further details). The decision problems (examined in Section 4) are as follows:

Definition 2.4 ([a, c]-Condorcet and a-Simpson solutions). The user set U(Y  X) as defined in (3)

Problem 2.5 (CONDORCET). Given an instance of MVLP together with parameters a; c 2 Rþ 0

U ðY  X Þ :¼ fu 2 U j dðu; Y Þ < dðu; X Þ  ag

ð3Þ

for the set of users preferring Y over X, which is applied to the already stated definitions. Campos and Moreno also relax the conditions on dominating solutions by replacing the term majority (which means more than one half of the users) with the term a proportion c of the users: For solutions X ; Y 2 Lp and c 2 [0, 1] they declare the domination Y X

if and only if W ðY  X Þ > c  wðU Þ:

ð4Þ

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(0 6 c 6 1) and p 2 N (1 6 p 6 jVj), does the instance admit an [a, c]-Condorcet solution? Problem 2.6 (SIMPSON). Given an instance of MVLP together with parameters p 2 N and W 2 Rþ 0 , does the instance admit a Simpson solution X 2 Lp of Simpson score W*(X) 6 W? Unlike in many standard optimization problems, in the case of the Simpson score even evaluating the goal function is already a non-trivial optimization problem since it consists in maximizing the power of an opposition. Problem 2.7 (SIMPSON SCORE). Given an instance of MVLP and a set X 2 Lp , determine the Simpson score W*(X) of X. Moreover, each decision problem induces a corresponding optimization problem. Here Condorcet induces bicriteria optimization problems which are usually solved by fixing one parameter and minimizing the remaining. This yields the following optimization problems (examined in Section 5): Problem 2.8 (MINIMUM SIMPSON). Given an instance of MVLP and a parameter p 2 N (1 6 p 6 jVj), determine the minimum Simpson score, i.e., minX 2Lp W  ðX Þ. Problem 2.9 (c-TOLERANT CONDORCET). Given an instance of MVLP with parameters c 2 [0, 1] and p 2 N (1 6 p 6 jVj), determine a*(c). Problem 2.10 (a-SIMPSON). Given an instance of MVLP with parameters a 2 [0, 1] and p 2 N (1 6 p 6 jVj), determine c*(a). The associated construction problems (which are investigated in Section 6) include computing the sets C(a, c), S(a), and T(c), as well as computing the set of efficient pairs in the family (C(a, c))a,c. 3. Related work and contribution of this paper Hakimi defines in [4] the notions medianoid and centroid as follows: Let p; r 2 N and X 2 Lp be a set of p facilities. If Y  2 Lr such that W ðY   X Þ ¼ maxY 2Lr W ðY  X Þ, then Y* is an (rjX)-medianoid of the graph. A set X  2 Lp minimizing the weight of an (rjX*)-medianoid is called an (rjp)-centroid. Notice that in Hakimi’s setting it is allowed to place points not only at vertices but even on edges. In [4] he proves that the problems of computing the

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(rjX)-medianoid and approximating the (rjp)-centroid to any constant factor are NP-hard. If we restrict the allowed positions to nodes only, we get for the special case r = p and X 2 Lp , that the weight of any (pjX)-medianoid is the Simpson score W*(X), and a (pjp)-centroid is a Simpson solution. Hakimi conjectures in [4] the (rjp)-centroid problem for general values of r and p to be very difficult. We contribute in Theorem 4.4 a result which justifies this at least for the case if points can only be placed at vertices. In Theorem 5.1 we provide optimal approximation results for the SIMPSON SCORE problem. Theorem 5.2 raises the constant factor lower bound on the approximability of the minimum Simpson problem shown by Hakimi to jVj1. In Section 6 we investigate the SVLP. As stated before, in this case of p = 1 the Simpson problem becomes equivalent to the (1j1)-centroid problem [4]. Moreover, on tree networks with p = 1 the Simpson solution and Condorcet set both coincide with the 1-median [5] which can be computed in time O(n) [6]. Note that this identity only holds for the case without relaxations, namely a = 0 and c = 1/2. In [2] the authors provide an algorithm to compute the set C(a, c) for p = 1 with running time O(jUj jLj2). We contribute in Theorem 6.1 an observation towards establishing this as a lower bound. In the same paper, Campos and Moreno give algorithms for computing the set S(a) of a-Simpson solutions with running time O(jUj jLj2), for computing the set T(c) of c-tolerant Condorcet solutions with running time O(jUj jLj2(logjUj + logjLj)), and for determining the set EC of efficient Condorcet solutions with running time O(jUj2jLj2 Æ (logjUj + logjLj)) (all algorithms for the case p = 1). We investigate in Section 6 the same problems restricted to tree networks (with U = L = V) and provide faster algorithms with running time O(jVj), O(jVjlogjVj), O(jVj), and O(jVjlogjVj), respectively. Unlike to the model in [5], we allow the use of additive indifference a as well as of c-proportion. In Section 7 we suggest a generalization of the additive indifference introduced in [2] to threshold functions. We can show that most of the essential properties of additive indifference carry over to the new model. We remark for completeness that the Simpson score and solution used in this paper goes back to the work of P. Simpson [7]. In particular this notion should not be confused with the Simpson paradox described by E.H. Simpson [8] about data aggregation

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in complex data systems, which is not subject of this paper. 4. Hardness of decision problems Condorcet and Simpson

xi ^ :xi ^ z where z is any literal distinct from xi and :xi . To enforce m = 2k + n, add yet unused variables to X or to Y respectively, which are nontrivial by definition. h Theorem 4.3. CONDORCET is Rp2 -complete.

In this section we investigate the complexity of the two decision problems CONDORCET and SIMPSON defined in Section 2.4. We will prove that both of the above decision problems are intractable; in fact we can show that they are Rp2 -complete. The class Rp2 is part of the polynomial time hierarchy and contains all decision problems which can be described using a formula of the form $x"y/ where / is a quantifier free formula. It is widely assumed that Rp2 is a proper superset of NP, hence both problems turn out to be even more complex than the well known NP-complete decision problems. The hardness proof uses a reduction from 983SAT decision problem. We briefly recall some preliminaries: A set X denotes the set of logical (i.e., binary) variables. If x 2 X is a variable, then :x is its negation. The set fx; :x j x 2 X g is the set of literals. A term is a conjunction of literals. A formula in 3-DNF is a disjunction of terms where each term consists of exactly three literals. An assignment X0 for a set X of variables is a mapping X0: X ! {0, 1}. If all variables of a formula / are defined this way, the truth value of the whole formula is determined and denoted by X0(/). If X0 is an arbitrary assignment, then X 0 : x 7! 1  X 0 ðxÞ is the assignment generated from X0 where the truth value of each variable has been flipped. The decision problem 983SAT is defined as follows: Given a formula / in 3-DNF over a partitioned set X [ Y of variables, decide whether the formula $x1    $xm"y1    "yn/ is satisfied, where xi 2 X and yj 2 Y. Theorem 4.1 [9]. The problem 983SAT is Rp2 complete. Let m :¼ jXj, n :¼ jYj, and k be the number of terms in /. A variable xi is called trivial in / if / contains either xi or :xi but not both. Corollary 4.2. 983SAT is Rp2 -complete, even if m = 2k + n and / contains no trivial variables. Proof. We show that the known hardness result from above continues to hold even with the additional restrictions. If / contains a trivial variable, say xi or :xi , then add a new term

Proof. Let l be a literal and m(l) denote the number of appearances of l in /. Furthermore P we set M :¼ maxfmðxi Þ; mð:xi Þ j xi 2 X g, N :¼ ðmðy i Þ þ mð:y i ÞÞ and W :¼ m 0 M + 3k + n + N, where m 0 is the number of used variables from X. The graph constructed in the sequel consists of several subgraphs. We use two main types of subgraphs: An s-diamond (for s 2 N) has two terminal nodes t1, t2 and s further nodes v1, . . . , vs, where each vi is connected both to t1 and t2. A term component is a cycle C6 where three pairwise nonadjacent nodes are terminals. The total graph is composed by connecting diamonds and term components at terminal nodes. The weight of each edge in diamonds and term components as well as of each nonterminal node equals 1, while the weight of terminal nodes is set to zero. For each variable xi 2 X, let C(xi) be a (W + 1)diamond, called the variable component, with terminal nodes xi and :xi . Similarly, add an 1-diamond C(yi) for each yi 2 Y. For each term c = l1 ^ l2 ^ l3 in /, add a term component T(c) with terminal set {l1, l2, l3}. Now connect each of those terminal nodes li by a new diamond Lc(i) to the corresponding terminal node in the variable component. The size of these connecting diamonds shall be adjusted such that the degree of each used literal l over X in the variable components is the same, namely M + W + 1; obviously this can be achieved, e.g. by choosing exactly one (M  m(l) + 1)-diamond and all remaining diamonds as 1-diamonds. Finally, connect the variable nodes x1 and :x1 by edges of weight 2 to all non-adjacent nonterminal nodes in the graph. This completes the construction of the graph. Furthermore, we set p :¼ m = 2k + n, c :¼ (W  1)/w(V), and U :¼ L :¼ V. The idea of the proof is to establish a relationship between assignments of the variables of the 983SAT instance on the one hand and node subsets of the CONDORCET instance on the other hand. Throughout the following proof we identify assignments and node subsets in the natural way, namely that a terminal node of a variable component is part of the subset if and only if the corresponding literal is set to 1.

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A node subset X 0 2 Lp is called a valid X-option if it contains for each variable xi 2 X exactly one of the terminal nodes xi ; :xi of the corresponding variable component; a valid Y-option is defined in a similar way. Note that a valid X-option can be treated as an X-assignment in the aforementioned sense. The converse also holds. It follows from the construction of the graph that each c-Condorcet solution is a valid X-option. We denote all nonterminal nodes v with d(X0, v) > 1 to be opposition nodes. We state the following observation: If X0 is a valid X-option, then there is an opposition Y0 where all nonterminal nodes preferring Y0 over X0 are actually opposition nodes. This can be shown as follows: First observe that since only nonterminal nodes have nonzero weight, terminal nodes do not impact the value W(Y0  X0). Second, let Y0 be an opposition and v a nonterminal node in U(Y0  X0) which is not an opposition node. Then v 2 Y0 and is preferred by no node other than v itself. Thus we can replace v by any yet unused opposition node without decreasing the value W(Y0  X0). Let X0 be an option which is not a c-Condorcet solution. As aforementioned there is an opposition Y0 such that all nonterminals in U(Y0  X0) are opposition nodes. From W(Y0  X0) P W it follows that Y0 must be preferred over X0 by all opposition nodes. Thus each term component must contain at least two nodes from Y0. On the other hand, each variable component C(yi) must contain at least one node from Y0. Since term and variable components do not have common nodes we can deduce the following observation: A valid X-option X0 is not a c-Condorcet solution if and only if there is a valid Y-option Y0 such that:

This means that (as follows by simply counting the available nodes) each node from Y0 is either part of a variable component or a term component, in particular it cannot be a nonterminal node in a literal component. We complete the proof by showing that the instance of 983SAT is satisfied if and only if the constructed graph admits a c-Condorcet solution. To this end, let X0 be an assignment for the variables in X such that the formula "y/ is satisfied. Then we claim that X 0 is a Condorcet solution: For the sake of a contradiction, consider an opposition Y0 satisfying conditions (i)–(iii) stated above. Then ðX 0 [ Y 0 Þð/Þ ¼ 1. Consequently there exists a term c = l1 ^ l2 ^ l3 such that ðX 0 [ Y 0 Þðli Þ ¼ 1 for all i. Hence, the complementary assignment X 0 [ Y 0 does not contain any of l1, l2, l3 in the variable components (cf. Fig. 1 left). To satisfy condition (iii), Y0 must contain all three terminal nodes of T(c), which contradicts condition (ii). Conversely, assume that $x"y/ is not true. Let X0 be an arbitrary valid X-option. Then there exists an Y0 such that ðX 0 [ Y 0 Þð/Þ ¼ 0, hence for all terms c = l1 ^ l2 ^ l3 there is at least one literal li such that ðX 0 [ Y 0 Þðli Þ ¼ 0. Then the set Y 1 :¼ Y 0 can be completed to form an opposition which dominates X0 at the end: Since ðX 0 [ Y 0 Þðli Þ ¼ 1, we can add the two terminals lj, j 5 i, of the term component T(c) to Y1. After this has been performed for all terms, conditions (i)–(iii) from above are satisfied for the set Y1; hence Y1  X0, and X0 is not a Condorcet solution. h

(i) each variable component C(yi) contains exactly one node from Y0,

Observe that the existence of an [a, c]-Condorcet solution X implies W*(X) 6 c Æ w(U). On the other

(ii) each term component contains exactly two nodes from Y0, and (iii) each literal component contains at least one node from X0 [ Y0.

Fig. 1. Illustration of the situation in the proof.

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hand, if the instance does not admit an [a, c]-Condorcet solution, then for all sets X we have W*(X) > c Æ w(U). This reduces CONDORCET to SIMPSON and we can conclude: Theorem 4.4. SIMPSON is Rp2 -complete. Note that CONDORCET can trivially be decided in polynomial time if c = 1 or a = c = 0. We argue that these are the only such pairs [a, c] unless P = NP. Theorem 4.5. The set of pairs (a, c) for which CONDORCET can be decided in polynomial time equals {(a, 1)ja P 0} [ {(0, 0)} unless P = NP. Proof. For the case of c = 0 and a > 0, it can be seen that MINIMUM DOMINATING SET easily reduces to CONDORCET. So consider the case where 0 < c < 1 and a P 0 is fixed. For the sake of simplicity we show only the case a = 0. The proof carries over for arbitrary values of a in a very similar way. We show a reduction from the NP-complete 3-DIMENSIONAL MATCHING problem (3DM for short) which is defined as follows: Let M be a subset of R · S · T where R, S and T are pairwise disjoint sets of cardinality q. Decide whether M has a matching, i.e. a set M 0  M such that jM 0 j = q and the elements of M 0 pairwise do not have common coordinates. Given an instance I of 3DM and positive integers n, k such that n P jMj + 3q we construct an instance I(n, k) of CONDORCET in the following way: The underlying graph G = (V, E) of I(n, k) consists of three sets Vin, Vmid and Vout of nodes.

The cardinalities of these node sets are jVoutj = 3qk and jVinj = jVmidj = n respectively. By construction we guarantee that M [ R [ S [ T  Vmid holds. The elements of Vmid, Vin are indexed in any way. Note that an element u 2 R [ S [ T [ M can always be identified with an xiu 2 V mid , where iu is an appropriate index in {0, . . . , n  1}. The set Vout contains for each u 2 R [ S [ T exactly k elements u0, . . . , uk1. Now let E be the union of Vout · Vmid and Vmid · Vin. Furthermore define the weights of the edges in E as follows: Let u 2 R [ S [ T. Then u corresponds to an xiu 2 V mid and to some {u0, . . . , uk1}  Vout. Let xj be an arbitrary element of Vmid. If xj 2 M and u is a coordinate of xj then c(ul, xj) :¼ n else c(ul,xj) :¼ n + ((j  iu) mod n) for all 0 6 l 6 k  1. For xj 2 Vmid and vi 2 Vin we set c(vi, xj) :¼ n + ((j  i) mod n). While the weights of the nodes in Vout [ Vin are set to 1 all nodes in Vmid are zero weighted. Finally let p :¼ q which completes the construction (see Fig. 2). Let X ; Y 2 Lp . Observe that only the nodes in Vout [ Vin impact the value W(Y  X), since all nodes in Vmid are zero weighted. Hence we call the nodes in Vout [ Vin essential. Now let 0 6 i 6 n  1 and xi 2 Vmid. We define i 0 :¼ (i  1) mod n. Verify that xi0 is preferred over xi by all essential nodes which distance to xi exceeds n. Now let M 0 be a matching of the 3DM instance I and consider X :¼ M 0 as an option in I(n, k). We state that W(Y  X) 6 n for each Y 2 Lp . To this end let Yio be the set of essential nodes in Y and y 2 Yio. Observe that y is the only essential node in

Fig. 2. Illustration of an instance I(n, k). Node xl represents the triple (r, s, t) from the matching. Node xir corresponds to r 2 R. Thick edges are of weight n.

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U(y  X). It follows W(Yio  X) 6 p. Now let Ymid :¼ Y \ Vmid. Since d(u, Vmid) = n for all essential nodes u 2 V, Ymid can only be preferred by essential users u 0 for which d(u 0 , X) > n holds. Since X is a matching, d(uj, X) = n holds for all uj 2 Vout. It follows that U(Ymid  X) and Vout are disjoint. Furthermore, there are exactly p nodes u 2 Yin such that d(u, X) = n. Hence W(Ymid  X) 6 n  p. We conclude W(Y  X) = W(Yio  X) + W(Ymid  X) 6 n  p + p = n. On the other hand, if X is an option but not a matching for I then W*(X) P n  p + k. To see this, let Xmid :¼ X \ Vmid and define an alternative Y such that Y fxi0 j xi 2 X mid g. The case where X and Vmid are disjoint is trivial. We verify that Y is preferred by all essential users u such that d(u, X) > n. First we count the nodes uj 2 Vout such that d(uj, X) 6 n. To this end, let u 2 R [ S [ T. Consider the corresponding xiu 2 V mid and assume that d(uj, X) 6 n for some corresponding uj 2 Vout. This can only be the case if either uj 2 X or xiu 2 X or u is covered by X. Since X is no matching there must be an u not covered by M and such that xiu 62 X . Hence there are at least k  jXoutj corresponding nodes uj 2 Vout for which d(uj, X) > n is satisfied. Here we set Xout :¼ X \ Vout. A similar argument shows that Vin contains at least n  jXmij nodes vj such that d(vj, X) > n and where Xmi :¼ X \ (Vmid [ Vin). We conclude W(Y  X) P k  jXoutj + n  jXmij = n  p + k. Now compute n and k such that n P jMj + 3q, n 6 c Æ w(U) and n  p + k > c Æ w(U) is satisfied. Since w(U) = n + 3pk the value of w(U) itself depends on n, k. It is possible to calculate n, k in polynomial time and such that n, k are polynomially bounded by the input size of I. Thus we can construct I(n, k) with polynomial effort. It follows directly that I(n, k) has a c-Condorcet solution if and only if there is a matching for I. h Theorem 4.5 implies for each  > 0 and fixed a P 0 the existence of a MVLP instance which does not admit an [a, 1  ]-Condorcet solution. 5. Approximability of optimization problems MINIMUM SIMPSON and SIMPSON SCORE In this section we investigate the approximability of the SIMPSON SCORE and the MINIMUM SIMPSON optimization problems as defined in Section 2.4. For the first we give an optimal approximation while for the latter we can improve Hakimi’s result

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and show that it is essentially not approximable at all (unless P = NP). The SIMPSON SCORE turns out to be strongly related to the MAXIMUM COVERAGE problem [10], for which a tight bound of approximability is known [11]. The latter problem is: Given a set M M with weights wC : M ! Rþ 0 and a collection S  2 of subsets of M and a natural number k 6 jSj find a subset S 0  S such that jS 0 j = k and wC([S 0 ) is maximal. Theorem 5.1. The SIMPSON SCORE optimization problem is equivalent to MAXIMUM COVERAGE under approximation preserving reduction and hence e e approximable within e1 but not within e1   unless P = NP. Proof. Consider an instance of MAXIMUM COVERAGE. We construct a graph G = (V, E) in the following way: For all m 2 M and M 0 2 S we introduce nodes in G with weights w(m) :¼ wC(m) and w(M 0 ) :¼ 0 respectively. Furthermore we connect m 2 M and M 0 2 S by an edge if and only if m 2 M 0 holds. The option X consists of p :¼ k nodes such that X and M [ S are disjoint. Finally we choose an arbitrary node v from X which is connected to all M 0 2 S by an edge. The weights of all edges are set to 1. It is easy to see that the weight of an optimal solution S 0 for MAXIMUM COVERAGE equals the Simpson score W*(X). We conclude that SIMPSON SCORE is as least as hard to approximate as MAXIMUM COVERAGE. Conversely, given an instance of MVLP and an option X we construct an equivalent instance of MAXIMUM COVERAGE as follows: First set M :¼ U, wC :¼ w and k :¼ p. Then, let the collection S 0 consist exactly of the sets U(Y  X). Again it follows easily that W*(X) and the optimal weight of the MAXIMUM COVERAGE instance are equal which completes the equivalence. h Theorem 5.2. The MINIMUM SIMPSON problem cannot be approximated within jVj1 unless P = NP. Proof. Let GC = (VC, EC) be an undirected graph. A subset V 0  VC is called a vertex cover if each e 2 EC is incident to a node v 2 V 0 . The NP-complete decision problem VERTEX COVER asks for a given graph GC and a natural number K 6 jVCj, whether GC has a vertex cover containing at most K nodes. Given an instance of VERTEX COVER and an arbitrary positive integer s we construct an instance

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I(s) of MVLP: The underlying undirected graph G = (V, E) of I(s) is obtained by replacing each edge (u, v) 2 EC by an s-diamond D(u, v) with u, v as the terminal nodes. The s-diamond D(u, v) is defined exactly as in the proof of Theorem 4.3. Note that the reduction ensures VC to be a subset of terminal nodes of V. Concluding we set p :¼ K. First assume that GC has a vertex cover V 0 containing at most K nodes. Now consider X :¼ V 0 as a voting candidate in I(s). Since each nonterminal node from G is adjacent to a node from X, it follows easily that each node y 2 V can be preferred over X by at most one user. Hence W(Y  X) cannot exceed p for each Y 2 Lp . Let X 2 Lp be an arbitrary option. Then s(X) denotes a subset of VC obtained by replacing all nonterminal nodes in X by an adjacent terminal node. Let X be an option such that W*(X) < s. Then we claim s(X) to be a vertex cover of GC. To this end assume that GC has no vertex cover with at most K nodes. Then for all options X there is an edge (u, v) which is not covered by s(X). It is clear that each alternative Y containing u is preferred by all s nonterminal nodes of D(u, v) and hence W(Y  X) P s. Note that jVj depends linearly on s. Thus we can determine a suitable s in polynomial time, such that the equation pjVj1 < s is satisfied. Furthermore s is itself polynomially bounded by the input size of the VERTEX COVER instance, which ensures that I(s) can be constructed in polynomial running time. Let A be an algorithm approximating MINIMUM SIMPSON within jVj1 and XA be the result obtained by applying A to I(s). Further assume that GC has a vertex cover with at most K nodes. Then s(XA) is a vertex cover of GC. For the performance guarantee of A yields W*(XA) 6 pjVj1 < s. Hence a polynomial running time for A would imply VERTEX COVER 2 P as follows: First compute s and I(s). Then apply A to I(s). Finally check if s(XA) is a vertex cover for GC. h

1-T(c), and 1-EC on a tree which are faster than the algorithms described in [2] for general graphs. 6.1. Towards lower bounds for computing Condorcet set 1-C(a, c) We can show that computing 1-C(a, c) on a graph is as least as hard as solving the vector maximization problem. The vector maximization problem determines for a given set S of n vectors in Rk the set of maximal vectors. A vector u = (u1, . . . , uk) is called maximal, if there is no v = (v1, . . . , vk) 2 S such that u 5 v and ui 6 vi for all 1 6 i 6 k. This problem was introduced in [12] and is well investigated in literature so far (see [13] for a comprehensive survey). From our result it follows that an algorithm for 1-C(a, c) with running time faster than O(jUj jLj2) as stated in [2] would imply an algorithm for the vector maximization problem which is also faster than the currently known algorithms. Theorem 6.1. If t(Æ,Æ) is a function such that there is an algorithm computing 1-C(a, c) in time t(jUj, jLj), then this implies an algorithm for the vector maximization problem with running time O(kn log n + t(k, n)). Proof. In the sequel we describe an algorithm for the vector maximization problem which calls a subroutine for 1-C(a, c). To this end, let S = {x1, . . . , xn} be a set of n k-dimensional vectors. For each 1 6 i 6 k let vi denote a lexicographic order of S such that the ith coordinates of the vectors have the highest priority. For 1 6 i 6 k and 1 6 j 6 n let m(i, j) :¼ j{xr 2 S : xjvixr}j. The values of m can be computed in O(kn log n). Now we introduce a set U = {u1, . . . , uk} of users. Furthermore we set L :¼ S, w :¼ 1 and c :¼ n1 . We connect each pair n ui, xj by an edge of length c(ui, xj) :¼ n + m(i, j). Observe that the set 1-C(a, c) is exactly the set of maximal vectors of S. By introducing an additional user uk+1 with an appropriate weight we can even assume that 0 < c < 1 is fixed to some value. h

6. Single voting location problems

6.2. Computing Condorcet set 1-C(a, c) on trees

In this section we examine the SVLP (i.e., the voting location problem where only one facility is to be placed) under an additive indifference as described in Section 2.3. All problem names throughout this section are prepended by ‘‘1-’’ to remind that p = 1 is assumed. In detail we provide algorithms for computing the sets 1-C(a, c), 1-S(a),

We first investigate the problem 1-SIMPSON SCORE under additive indifference relation (see (3) in Section 3) on a tree. To this end, let Puv denote the unique path between nodes u and v in the tree. Let N a ðuÞ :¼ fv 2 V j dðu; vÞ > a but dðu; v0 Þ 6 a for all inner nodes v0 on P uv g

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denote the a-neighborhood of node u 2 V. Observe that for a = 0 we obtain the usual graph neighborhood N(u) = N0(u). We can prove the following lemmas: Lemma 6.2. Let u be a node and v 2 Na(u). Then U(x  u)  Tu(v) holds for each x 2 Tu(v). Proof. Let x 2 Tu(v) and y 62 Tu(v). We show y 62 U(x  u) which implies the assertion (cf. Fig. 3). There is a z 2 N(v) visited by Puv, Pyv and hence by Pyx. Clearly d(y, x) = d(y, z) + d(z, x). Furthermore d(z, u) 6 a since v is an a-neighbor of u. It follows dðy; uÞ  dðy; xÞ 6 ðdðy; zÞ þ dðz; uÞÞ  ðdðy; zÞ þ dðz; xÞÞ ¼ dðz; uÞ  dðz; xÞ 6 a which shows the claim.

h

Observation 6.3. Let u, v be nodes. If d(u, v) 6 a then U(v  u) is empty, otherwise it contains the nodes in Tu(v). Lemma 6.4. Let u 2 V and v 2 Na(u). Then U(v  u) = Tu(v). Moreover Na(u) contains an opposition of u. Proof. The equation U(v  u) = Tu(v) follows immediately from the preceding lemmas. If W*(u) = 0 then v is an opposition of u. Otherwise Observation 6.3 shows d(u, y 0 ) > a for an opposition y 0 of u. The path P uy 0 visits an a-neighbor y of u. Lemma 6.2 shows U(y 0  u)  Tu(y) = U(y  u) and thus y is an opposition of u too. h

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Lemma 6.5. For additive indifference relation (3), problem 1-SIMPSON SCORE on a tree can be computed in O(n). By investigating the properties of an [a, c]-Condorcet solution on a tree we can derive an algorithm for computing 1-C(a, c) on a tree. To this end, we call a subtree Tu(v) to be qualified if w(Tu(v)) > c Æ w(V). If we define for any node u 2 V and v 2 N(u) du ðvÞ :¼ supfdðu;xÞ j x 2 T u ðvÞ ^ wðT u ðxÞÞ > c  wðV Þg; we can characterize the situation where a node belongs to this Condorcet set as follows: Lemma 6.6. The following statements are equivalent: (i) A node u is in C(a, c). (ii) d(u, v) 6 a for all nodes v 5 u with qualified Tu(v). (iii) du(v) 6 a for each v 2 N(u). The algorithm for computing 1-C(a, c) is based on a routine for propagating the values of the du(Æ) functions (cf. Fig. 4 for details). Let (u, v) be an edge. Then propagate(u, v) computes du(v) if the values of dv(u 0 ) are known for each u 0 2 N(v) different from u. This can be done as follows: If Tu(v) is not qualified then du(v) = 1. Otherwise the following equation holds and can be derived by simple considerations: du ðvÞ ¼ dðu; vÞ þ maxfdv ðu0 Þ j u0 2 N ðvÞ ^ u0 6¼ ug [ f0g: ð5Þ

Since all a-neighbors v and weights w(Tu(v)) can be determined in a single tree traversal, the desired value W*(u) for a node u 2 V can be computed in linear time:

Fig. 3. Illustration of the proof of Lemma 6.2.

Fig. 4. Subroutines used for computing 1-C(a, c) on trees.

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Note that propagate(u, v) takes time O(g(v)) where g(v) denotes the node degree of v. Let s be an arbitrary node and consider T as a s-rooted tree. The algorithm 1-C(a, c) starts with computing the values du(v) for all edges (u, v) by two depth first search traversals: The first traversal invokes propagate_up(s). This subroutine traverses the tree bottom up and calculates the value du(v) for all edges (u, v) where u is the father of v. The running time is linear in the number of nodes: this follows since propagate(Æ, v) takes time O(g(v)) and is called exactly one time for each node v. The second traversal invokes propagate_down(s) and is a top down traversal computing the missing values dv(u). A naive implementation as in Fig. 4 results in a quadratic running time since propagate(Æ, u) would be called at least g(u)  1 times per node u. This can be avoided as follows: When node u is reached, first traverse the set of all neighbors of u (i.e., the sons and the father) and determine v1 :¼ arg maxv2N(u)du(v) and v2 :¼ arg maxv2N ðuÞv1 du ðvÞ. With that modification we can evaluate the values dv(u) for all sons of u in total running time O(g(u)), which guarantees an overall linear running time. When all values du(v) are known, the algorithm outputs the set {u 2 V j du(v) 6 a for all neighbors v of u} (see Lemma 6.6). This yields the following result: Theorem 6.7. The set 1-C(a, c) of Condorcet nodes on a tree can be computed in O(n).

 1-T ðcÞ ¼

    0 max du ðvÞ u 2 V  max du ðvÞ ¼ min u0 2V v2N ðu0 Þ v2N ðuÞ

stated in Lemma 6.6. However in the sequel we present a more general view on the problem and a relation to centers on trees to derive algorithms both for computing c-tolerant solutions and efficient solutions. We define a digraph GT from T as follows: For each (undirected) edge (u, v) in T we add the (directed) arc (u, v) to G if Tu(v) is qualified, i.e., w(Tu(v)) > c Æ w(V). Note that this definition allows also adding pairs of anti-parallel arcs. We call a strongly connected component of GT nontrivial if it contains more than one node. Lemma 6.9. The digraph GT contains at most one nontrivial strongly connected component.

Proof. Assume we have two such strongly connected components. Let (u, v), (v, u) be a pair of antiparallel arcs in the first component, x be a node in the second component. Without loss of generality, x 2 Tu(v). Since by definition, Tv(u) is qualified, this holds also for each super-tree, hence u can be reached from x in graph GT. By interchanging the role of the two components we show also that x is reachable from u which completes the proof. h Lemma 6.10. Let GT contain a nontrivial strongly connected component CT. Then a node u is in 1-C(a, c) if and only if its eccentricity in CT satisfies e(u) 6 a.

6.3. Computing a-Simpson set 1-S(c) on trees From Lemma 6.4 we can deduce that c*(a) is always in the set    wðT v ðuÞÞ wðT u ðvÞÞ  ; ðu; vÞ 2 E [ f0g: C¼ wðV Þ wðV Þ  Performing a binary search on C and using the algorithm of Theorem 6.7, we conclude Corollary 6.8. The set 1-S(a) of a-Simpson nodes on a tree can be computed in O(n log n). 6.4. Computing c-tolerant Condorcet set 1-T(c) on trees It is easy to observe that the set 1-T(c) of c-tolerant nodes can be computed in O(n) by calculating the values of d and using the property

Proof. If e(u) > a for some u, then the claim follows immediately from Lemma 6.6. Let e(u) 6 a. Since all arcs outside the component CT are directed towards CT itself, tree Tu(v) is not qualified for any node v 62 CT. Hence u must be part of the Condorcet set. h Lemma 6.11. If GT contains a nontrivial strongly connected component CT, then a*(c) equals the radius of CT and 1-T(c) coincides with the center of CT. Otherwise, a*(c) = 0 and 1-T(c) contains a median of T. Proof. The first statement has been shown in the previous considerations. Assume now that GT has no nontrivial components. From Lemma 6.2 we conclude that a node u is a Condorcet solution if and only if the outdegree g+(u) = 0. Since GT does not contain a nontrivial strongly connected compo-

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nent, there are at most jVj  1 arcs and hence there exists at least one node with this property. h Theorem 6.12. The set 1-T(c) of c-tolerant nodes on a tree can be computed in O(n). Proof. The algorithm starts by computing GT and CT in linear time. If there is a nontrivial component CT, it is actually a cactus and the center of the underlying tree can be computed in linear time as described in [14]. Otherwise, we refer to Theorem 6.7. h

Fig. 5. Algorithm for computing Efficient 1-Condorcet solutions on a tree.

6.5. Efficient 1-Condorcet solutions on trees 7. Generalization to threshold functions In this section we develop an algorithm on a tree for determining the set 1-EC, i.e., all nodes u and pairs (a, c) such that u 2 1-C(a, c) is an [a, c]-Condorcet node and (a, c) is an efficient pair. Recall that a pair is efficient if 1-C(a  , c) = 1-C(a, c  ) = ; for all  > 0. We use a similar construction as before. To this end we replace the cactus CT by the underlying tree now. Consider the scenario where we have c = 0: Then CT actually equals the whole tree, if we assume without loss of generality that there are no zero weighted leaves in the input instance T. With continuously increasing c, CT shrinks in general. Let a be the radius of CT. Obviously, a is a step-wise monotonously decreasing function of c. At each point where a decreases, (a, c) is an efficient pair. Observe first that the changes in CT appear only at those values of c where c attains the value   wðT u ðvÞÞ wðT v ðuÞÞ ~cðeÞ :¼ min ; wðV Þ wðV Þ for some edge e :¼ (u, v). In other words, ~cðeÞ is the smallest value of c for which edge e is not contained in CT. Hence we do not need to compute CT each time from scratch: Instead, we can update CT by removing those edges with smallest value of ~c. To this end, we need to compute ~c in a preprocessing step (which can be performed in linear time), and then sort the edges according to ~c. The center of the subtree CT can be maintained in time O(log n) per edge removal using the algorithms from [15]. This yields the algorithm depicted in Fig. 5 with the following result: Theorem 6.13. The set of efficient 1-Condorcet solutions on a tree can be computed in O(n log n).

In this section we introduce threshold functions as a concept which covers both the additive indifference suggested by Campos and Moreno (see Section 2.3) as well as obvious generalizations, e.g. multiplicative indifference. We can show that most of the basic properties found in [2] continue to hold in the generalized model. þ þ A function d: Rþ 0  R0 ! R0 is called a threshold function if d(0, y) = 0 for all y 2 Rþ 0 and d is (weakly) monotonously increasing in both parameters. Given a threshold function d and a parameter b P 0, we replace the definition of user preference stated in (1) in Section 1 to the general form U ðY  X Þ :¼ fu 2 U j dðu; Y Þ < dðu; X Þ  dðb; dðu; X ÞÞg: ð6Þ Examples: The additive a-indifference introduced in [2] is modeled by the threshold function d: (b, d) # b with parameter b :¼ a. We can also model a multiplicative indifference where a user is undecided if the ratio of the distances to two facilities does not exceed a factor a by choosing the threshold function d: (b, d) # b Æ d and b :¼ a  1. Definition 7.1 ([b, c]-Condorcet solution). Let b; c 2 Rþ , d be a threshold function determining 0 user preference according to (6); further let X ; Y 2 Lp . Solution Y is said to dominate X, denoted by Y  X, if W ðY  X Þ > c  wðU Þ: Solution X is called a [b, c]-Condorcet solution if there is no alternative solution Y 2 Lp such that Y  X.

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As before, the Condorcet set is denoted by C(b, c). In contrast to Condorcet solutions, [b, c]-Condorcet solutions always do exist. In more detail, for each instance of MVLP there are parameters b, c such that there is a [b, c]-Condorcet solution, e.g. if you set c :¼ 1. It is the goal to keep both parameters low to approximate the classical concept of a Condorcet solution as much as possible.

We remark that while the multiplicative threshold function is not invertible, for each  > 0 the function dðb; dÞ :¼ b  maxfd; g is invertible and comes closest to the multiplicative threshold function. 8. Conclusions and further remarks

7.1. Basic properties In [2] the authors investigate monotonicity of the family of Condorcet sets with respect to both arguments and state results about a-minimal solutions and c-minimal solutions. In the sequel we generalize their results to our model in almost all aspects. Lemma 7.2 (Monotonicity). The family (C(b, c))b,c of Condorcet sets is inclusion-wise monotonous with respect to b and c. Lemma 7.3 (c-Minimality). For each b 2 Rþ 0 the value c ðbÞ :¼ fc 2 Rþ 0 j Cðb; cÞ 6¼ ;g is well defined. While minimizing the second parameter, c, is possible for all threshold functions, this does not hold in general for the first parameter, b. To see this, consider e.g. the multiplicative threshold function d: (b, d) # b Æ d. Since the function value vanishes for small distance values y, the same instances which show that a classical Condorcet solution does not exist also prove the non-existence of [b, c]-Condorcet solutions. In a certain sense we need to require threshold functions to attain a threshold value large enough for creating indifferent solutions in every situation. To this end, we call a threshold function d invertible if the value minfb P 0 j dðb; dÞ P cg is well defined for all c; d 2 Rþ 0 . (We remark that this definition is not literally applicable for the case where d(u, l) = 1 for some pair (u, l). This situation can be avoided by adding an edge of large but finite weight between this pair of nodes to the graph.) Then we have the following result: Lemma 7.4 (b-Minimality). For all invertible threshold functions and each c 2 [0, 1], the value b ðcÞ :¼ minfb 2 Rþ 0 j Cðb; cÞ 6¼ ;g is well defined.

The MVLP was conjectured in [4] to be exceedingly difficult. Our results on the Rp2 -completeness of MVLP on general graphs (see Theorems 4.3 and 4.4) confirm this, as we can show that the problems are substantially harder to solve than NP-complete problems. Under standard assumptions, namely that the polynomial time hierarchy does not collapse below Rp2 , this has the following impact on the complexity of any possible algorithm: While for NP-complete problems we can always search an exponential size solution space and verify each solution in polynomial time, this fast verification time is no longer possible for Rp2 -complete problems. Moreover, it also prevents to directly apply approximation techniques which are known to work well for NP-complete problems. While the MVLP on trees is known to be in NP (as follows from the results in [16]), the question stated in [4] whether the centroid problem can be solved efficiently on trees remains open. We thank the anonymous referees for useful suggestions which helped to improve the presentation. References [1] C.M. Campos Rodrı´guez, J.A. Moreno Pere´z, Multiple voting location problems, unpublished manuscript, 2002. [2] C.M. Campos Rodrı´guez, J.A. Moreno Pere´z, Relaxation of the Condorcet and Simpson conditions in voting location, European Journal of Operations Research 145 (2003) 673– 683. [3] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, M. Protasi, Complexity and Approximation, Springer, Berlin, Heidelberg, New York, 1999. [4] S.L. Hakimi, Locations with spatial interactions: Competitive locations and games, in: in [17], 1990, pp. 439–478. [5] P. Hansen, J.-F. Thisse, R.E. Wendell, Equilibrium analysis for voting and competitive location problems, in: in [17], 1990, pp. 479–501. [6] O. Kariv, S.L. Hakimi, An algorithmic approach to network location problems, part II: p-medians, SIAM Journal of Applied Mathematics 37 (1979) 539–560. [7] P. Simpson, On defining areas of voter choice, Quarterly Journal of Economics 83 (1969) 478–490.

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[13] P. Godfrey, R. Shipley, R. Gryz, Maximal vector computation in large data sets, Technical Report, Technical Report CS-2004-06, University of York, 2004. [14] G.Y. Handler, Minimax location of a facility in an undirected tree graph, Transportation Science 7 (1973) 287–293. [15] S. Alstrup, J. Holm, M. Thorup, Maintaining center and median in dynamic trees, in: Proceedings of the 7th Scandinavian Workshop on Algorithm Theory (SWAT 2000), Springer-Verlag, London, UK, 2000, pp. 46–56. [16] N. Megiddo, E. Zemel, S. Hakimi, The maximum coverage location problem, SIAM Journal on Algebraic and Discrete Methods 4 (2) (1983) 253–261. [17] P.B. Mirchandani, R.L. Francis, Discrete Location Theory, Series in Discrete Mathematics and Optimization, WileyInterscience, 1990.