Approximability results for the p -centdian and the converse centdian problems

Given an undirected graph G = ( V, E ) with a nonnegative edge length function and an integer p , 0 < p < | V | , the p -centdian problem is to ﬁnd p vertices (called the centdian set ) of V such that the eccentricity plus median-distance is minimized, in which the eccentricity is the maximum (length) distance of all vertices to their nearest centdian set and the median-distance is the total (length) distance of all vertices to their nearest centdian set . The eccentricity plus median-distance is called the centdian-distance . The purpose of the p -centdian problem is to ﬁnd p open facilities (servers) which satisfy the quality-of-service of the minimum total distance ( median-distance ) and the maximum distance ( eccentricity ) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the centdian-distance and the objective function is to minimize the cardinality of the centdian set , this problem is called the converse centdian problem. In this paper, we prove the p -centdian problem is NP-Complete. Then we design the ﬁrst non-trivial brute force exact algorithms for the p -centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems.

Given a set of customers on the network, the network location theory is concerned with the optimal locations of new facilities (servers) to minimize transportation distances (costs) of serving these customers and consider the population density area.The most fundamental problems of the network location theory are the pCP and the pMP, respectively.The pCP is suitable for emergency services where the objective is to have the farthest customers as close as possible to their facility centers.But this solution of the pCP may cause a substantial increase in total distance (cost), thus this result takes a huge loss of the spatial efficiency.The pMP is suitable for locating facilities providing a routine service, by minimizing the average distances from customers to these selected facilities.The solution of the pMP is beneficial in serving centrally located and high-population density areas but sacrifices the remote and low-population density areas [41,42,50].Motivated by the application of finding p open facilities (servers) which satisfy the quality-of-service of the minimum total distance (median-distance) and the maximum distance (eccentricity) to their service customers, simultaneously [21,22,25,41,42,50], Halpern [21,22] introduced a convex combination of the 1CP and the 1MP, which he called the 1-centdian problem.Hooker et al. [25] studied the generalization of the 1-centdian problem, called the p-centdian problem.Given an undirected complete graph G = (V, E, ℓ) with a nonnegative edge length function ℓ, a real number λ, 0 ≤ λ ≤ 1, and an integer p, If the centdian set can be the continuum set of points on the edges of G, Hooker et al. [25] proposed the possible centdian set for the pDP.Perez-Brito et al. [41] fixed the flaw of Hooker et al. [25] theorem for the pDP.Tamir et al. [50] presented a polynomial time exact algorithm for the pDP on trees.Ben-Moshe et al. [6] gave O(|V |log|V |) time exact algorithms for the 1DP on cycle graphs and cactus graphs, respectively.If the induced subgraph by the centdian set is connected, Nguyen et al. [39] proposed a linear time algorithm for the pDP on unweighted block graphs and proved the problem is NP-Complete on weighted block graphs.If λ = 0, the pDP is equal to the pMP, and however λ = 1 the pDP is equal to the pCP.Hence, it is not hard to see that the pDP is NP-hard.However, it is still unclear whether there exists a polynomial time deterministic approximation algorithm for the pDP.Given an undirected graph G = (V, E) and two independent minimization criteria with a bound on the first criterion, a generic bicriteria network design problem involves the minimization of the second criterion but satisfies the bound on the first criterion among all possible subgraphs from G [37].Many multiple criteria problems had been studied [9,17,29,34,37].Clearly, the pCP, pMP, and pDP are one kind of bicriteria network design problems.The first criterion is the cardinality of the vertex set V ′ and the second is the eccentricity, median-distance, and centdian-distance, respectively.Hence, if we converse the two criteria, that is given the bound of the eccentricity from each vertex to V ′ and the objective function is to minimize the cardinality of the V ′ , this problem is called the converse p-center problem (also called the balanced p-center problem) [4,5,16].Given a graph G = (V, E, ℓ) with a nonnegative edge length function ℓ and an integer U , U > 0, the converse p-center problem is to find a vertex set V ′ in V with minimum cardinality such that the eccentricity of V ′ is at most U [4,5,16].This problem had been shown to be NP-Complete [16] and a (log U + 1)-approximation algorithm had been proposed [4,5].However, the converse version of the p-centdian problem is undefined.Hence, we present the converse version of the p-centdian problem, called the converse centdian problem.Given a graph G = (V, E, ℓ) with a nonnegative edge length function ℓ and two integers λ and U , 0 ≤ λ ≤ 1, U > 0, the converse centdian problem (CDP) is to find a vertex set V ′ in V with minimum cardinality such that λ£ In this paper, we focus on a special case of the centdiandistance for the pDP (respectively, CDP) : £ C (V ′ ) + £ M (V ′ ) and discuss the complexity, the non-trivial brute force exact algorithms, and the approximation algorithms for the pDP and CDP, respectively.First, we prove that the pDP is NP-Complete even when the centdian-distance is £ C (V ′ ) + £ M (V ′ ).Then we present the first non-trivial brute force exact algorithms for the pDP and CDP, respectively.Finally, we design a (1 + ǫ)-approximation algorithm for the pDP satisfying the cardinality of the centdian set is less than or equal to (1 + 1/ǫ)(ln|V | + 1)p and a (1 + 1/ǫ)(ln|V | + 1)-approximation algorithm for the CDP satisfying the centdian-distance is less than or equal to (1 + ǫ)U , in which ǫ > 0, respectively.
The rest of this paper is organized as follows.In Section 2, some definitions and notations are given.In Section 3, we prove that the pDP is NP-Complete even when the centdian-distance is In Section 4, we present non-trivial brute force exact algorithms for the pDP and CDP, respectively.In Section 5, we design a (1 + ǫ)-approximation algorithm for the pDP satisfying the cardinality of the centdian set is less than or equal to (1 + 1/ǫ)(ln|V | + 1)p.In Section 6, we design a (1 + 1/ǫ)(ln|V | + 1)approximation algorithm for the CDP satisfying the centdian-distance is less than or equal to (1 + ǫ)U , in which ǫ > 0. Finally, we make a conclusion in Section 7.

Preliminaries
In this paper, a graph is simple, connected and undirected.By G = (V, E, ℓ), we denote a graph G with vertex set V , edge set E, and edge length function ℓ.The edge length function is assumed to be nonnegative.We use |V | to denote the cardinality of vertex set V .Let (v, v ′ ) denote an edge connecting two vertices v and v ′ .For any vertex v ∈ V is said to be adjacent to a vertex v ′ ∈ V if vertices v and v ′ share a common edge (v, v ′ ).
Definition 1: For u, v ∈ V , SP (u, v) denotes a shortest path between u and v on G.The shortest path length is denoted by d(u, v) = e∈SP (u,v) ℓ(e).
Definition 2: Let H be a vertex set of V .For a vertex v ∈ V , we let d(v, H) denote the shortest distance from v to H, i.e., d(v, H) = min h∈H {d(v, h)}.
For the pDP, we have two criteria.The first criterion is the cardinality of the vertex set V ′ and the second is the is called the centdian-distance.Hence, we can converse the two criteria, that is given the bound of the centdian-distance of the centdian set and the objective function is to minimize the cardinality of the centdian set.
The following examples illustrate the pDP and the CDP.Consider the instance shown in Fig. 1, in which the graph G = (V, E, ℓ) and integers p = 2 and U = 117.An optimal solution of G for the pDP is shown in Fig. 2, in which the centdian-distance is 252 An optimal solution of G for the CDP is shown in Fig. 3, in which the centdian set is {A, B, D}.
In this paper, we will prove that the pDP is NP-Complete by a reduction from the dominating set problem [7,11,44,52] to the pDP.Hence, we review the definition of the dominating set problem.A dominating set of G, denoted by Z, is a subset of V such that each vertex in V \ Z is adjacent to a vertex in Z [7,11,44,52].DSP (dominating set problem) [7,11,44,52] Instance: A connected, undirected graph G = (V, E).
Question: Find a dominating set Z ′ with minimum cardinality.
Note that the DSP had been shown to be NP-Complete [16].Since our approximation algorithm for the pDP is based on the set cover problem [10,28,36].We also review the definition of the set cover problem.Given a finite set U of elements and a collection S of (non-empty) subsets of U. A set cover [10,28,36] is to find a subset S ′ ⊆ S such that every element in U belongs to at least one element of S ′ .SCP (Set cover problem) [10,28,36] Instance: A finite set U of elements, a collection S of (non-empty) subsets of U.
Question: Find a set cover S ′′ such that the number of sets in S ′′ is minimized.

Hardness Result for the pDP
In this section, we prove that the pDP is NP-Complete.We transform the DSP to the pDP by the reduction.Hence we need to define pDP and DSP decision problems.

pDP Decision Problem
Instance: A connected, undirected complete graph G = (V, E, ℓ) and two integers p > 0 and U > 0.
Question: Does there exist a vertex set

DSP Decision Problem
Instance: A connected, undirected graph G = (V, E), and a positive integer κ.
Question: Does there exist a dominated set Z such that |Z| is less than or equal to κ? Theorem 1: The pDP decision problem is NP-Complete.
Proof: First, it is easy to see that the pDP decision problem is in NP.Then we show the reduction: the transformation from the DSP decision problem to the pDP decision problem.
Let a graph G = (V, E) and a positive integer κ be an instance of the DSP decision problem.We transform it into an instance of the pDP decision problem, say G = (V , E, ℓ) and two positive integers p and U , as follows. (1) Now, we show that there is a dominating set Z such that |Z| is κ if and only if there is a vertex set (Only if) If there exists a dominating set Z in G and the cardinality of Z is at most κ.Then we choice the corresponding vertex set V ′ in G of the dominating set Z in G. Hence, we have Hence, we choice the corresponding vertex set Z in G of the vertex set V ′ in G and Z is a dominating set in G with |Z| = p.✷

Exact Algorithms for the pDP and CDP
In this section, we show integer programmings to solve the pDP and CDP, respectively.We combine the integer programmings for the pMP and pCP by [13].Given an undirected complete graph G = (V, E, ℓ) with a nonnegative edge length function ℓ, the pDP can be formulated as an integer programming (I) as follows. minimize subject to j∈V x i,j ≤ y j , ∀i, j ∈ V (5) x i,j , y j ∈ {0, 1} where the variable y j = 1 if and only if vertex j is chosen as a centdian, and the variable x i,j = 1 if and only if y j = 1 and vertex i is assigned to vertex j, and C is a feasible eccentricity.For completeness, we list the exact algorithm for the pDP as follows.

Algorithm OPT-pDP
Input: A connected, undirected complete graph G = (V, E, ℓ) with a nonnegative length function ℓ on edges and an integer p > 0.
Output: A vertex set P opt with |P opt | = p.
1. Use the integer programming (I) to find all y j = 1 and put the corresponding vertex j of y j to P opt .
It is easy to show that Algorithm OPT-pDP is an exact algorithm for the pDP.However, to solve an integer programming is NP-hard [12,16].Hence, next section we show (1 + ǫ)-approximation algorithm for the pDP satisfying the cardinality of centdian set is less than or equal to (1 + 1/ǫ)(ln|V | + 1)p, ǫ > 0.
Next, we modify integer programming (I) to design another integer programming (II) for the CDP with an integer U as follows.minimize j∈V y j (9) subject to i∈V j∈V d(i, j)x i,j + C ≤ U (11) x i,j , y j ∈ {0, 1} For completeness, we list the exact algorithm for the CDP as follows.
Algorithm OPT-CDP Input: A connected, undirected complete graph G = (V, E, ℓ) with a nonnegative length function ℓ on edges and an integer U > 0.
Output: A vertex set P opt with £ C (P opt ) + £ M (P opt ) ≤ U .
1. Use the integer programming (II) to find all y j = 1 and put the corresponding vertex j of y j to P opt .

An Approximation Algorithm for the pDP
In this section, we show (1+ǫ)-approximation algorithm for the pDP satisfying the cardinality of centdian set is less than or equal to (1 + 1/ǫ)(ln|V | + 1)p, ǫ > 0. First, we relax the integer programming (I) for the pDP to the linear programming (I L ) to solve the pDP called the fractional pDP as follows.
subject to j∈V x i,j ≤ y j , ∀i, j ∈ V j∈V The main difference between I L and I is that y j and x i,j can take rational values between 0 and 1 for I L .Let ỹ and x be the output values of the linear programming I L .Then it is clear that the centdian-distance of the optimal solution for the fractional pDP is a lower bound on the centdian-distance of the optimal solution for the pDP.Moreover, the linear programming can be solved in polynomial time [32,33].
Proof: Similar with [2], for each i ∈ V , we sort d(i, j), j ∈ V , so that d(i, j and let s be a value such that , and otherwise xi,j = 0. ✷ Given a fractional solution xi,j , for each i ∈ V , let D(i) = j∈V d(i, j)x i,j be the distance of assigning vertex i to its fractional centdian.Given ǫ > 0, we also let the neighborhood set Lemma 3: [35] For each i ∈ V and ǫ > 0, we have j∈N (i) ỹj ≥ j∈N (i) xi,j > ǫ/(1 + ǫ).
Then we transform the pDP to the SCP.An instance of SCP contains a finite set U of elements, a collection S of (non-empty) subsets of U. We let each vertex i ∈ V correspond to each element in U, and each vertex j ∈ V with ỹj > 0 correspond to each set in S, respectively.Then for each vertex i ∈ V , if j ∈ N (i), then the corresponding element of i in U belongs to the corresponding set of j in S.
Then we use the greedy approximation algorithm for the SCP whose approximation ratio is (ln |U| + 1) [10,28,36] to find a set cover of U and S. Let A SCP be the greedy approximation algorithm for the SCP.Finally, output the corresponding vertex set for the output set by A SCP .Given a graph G = (V, E, ℓ), let P AP X be a vertex set in G. Initially, P AP X is empty.Now, for clarification, we describe the (1 + ǫ)-approximation algorithm for the pDP as follows.
2. Use linear programming (I L ) to solve the fractional pDP and find the fractional solutions ỹ and x.

3.
For each i ∈ V , compute D(i) and find its neighborhood set 4. For each i ∈ V do create an element u i in U.
end for 5.For each j ∈ V with ỹj > 0 do create a subset end for 6.Use the greedy approximation algorithm A SCP for the SCP to find a set cover S ′ of the instance U and S. Let y j = 1 if S j ∈ S ′ , and then x i,j = 1 if set S j ∈ S ′ and u i is covered by S j , and otherwise is 0.

7.
Let P AP X be the corresponding vertex set of S ′ .
The result of this section is summarized in the following theorem.
Proof: Let P OP T be the optimal solution for the pDP.Clearly, by Step 5 and Step 6, a subset S j contains the element u i in U if d(i, j) ≤ (1 + ǫ) D(i), where i is the corresponding vertex of u i and j is the corresponding vertex of S j , and each i ∈ V , j∈V d(i, j)x i,j ≤ (1 + ǫ) D(i) Hence, we have since the centdian-distance of the fractional pDP is a lower bound on the centdian-distance of the optimal solution for the pDP.

An Approximation Algorithm for the CDP
In this section, we show a (1 + 1/ǫ)(ln|V | + 1)-approximation algorithm for the CDP satisfying the centdian-distance is less than or equal to (1 + ǫ)U , ǫ > 0. We only run Algorithm APX-pDP for the pDP, for p = 1 to |V | and find the first centdian set such its centdian-distance is less than or equal to (1 + ǫ)U .
For the completeness, we describe the approximation algorithm for the CDP and obtain the centdian set P γ as follows.

2.
Use Algorithm APX-pDP to find a vertex set P p that satisfies Theorem 4.

✷ 7 Conclusion
In this paper, we have investigated the pDP and the CDP and prove that these problems are NP-Complete even when the centdian-distance is £ C (V ′ ) + £ M (V ′ ).Then we have presented non-trivial brute force exact algorithms for the pDP and the CDP, respectively.Moreover, we have designed a (1 + ǫ)approximation algorithm for the pDP satisfying the cardinality of the centdian set is less than or equal to (1 + 1/ǫ)(ln|V | + 1)p and a (1 + 1/ǫ)(ln|V | + 1)-approximation algorithm for the CDP satisfying the centdian-distance is less than or equal to (1 + ǫ)U , in which ǫ > 0. It would be interesting to find approximation complexities for the pDP and the CDP.Another direction for future research is whether the pDP has a polynomial time exact algorithm for some special graphs.