The Inverse 1-Median Problem on Tree Networks with Variable Real Edge Lengths

Abstract

Location problems exist in the real world and they mainly deal with finding optimal locations for facilities in a network, such as net servers, hospitals, and shopping centers. The inverse location problem is also often met in practice and has been intensively investigated in the literature. As a typical inverse location problem, the inverse 1-median problem on tree networks with variable real edge lengths is discussed in this paper, which is to modify the edge lengths at minimum total cost such that a given vertex becomes a 1-median of the tree network with respect to the new edge lengths. First, this problem is shown to be solvable in linear time with variable nonnegative edge lengths. For the case when negative edge lengths are allowable, the NP-hardness is proved under Hamming distance, and strongly polynomial time algorithms are presented under 𝑙₁ and 𝑙_∞ norms, respectively.

1. Introduction

Location problems have found important applications in operations research, computer science, and logistics management. The 𝑝-median problem is a typical location problem and has been intensively considered in the literature. The 𝑝-median problem on an undirected graph is to locate 𝑝 facilities on edges or vertices of the graph such that the sum of the weighted distances of the vertices to the closest facility is minimized. This problem possesses the vertex optimality property which asserts that there exists an optimal solution such that all facilities are located at the vertices of the graph. Kariv and Hakimi [1] showed that the 𝑝-median problem is NP-hard even if the graph 𝐺 is a planar graph of maximum degree 3. Therefore, many investigations are dedicated to special graph classes which allow a more efficient algorithm. In the case of tree networks, the 1-median problem is solvable in linear time by Hua [2] and Goldman [3].

However, in an established network, the locations of facilities have already been fixed. The changing environment might make the existing facilities deviate from the median places of the network. Instead of finding optimal locations, the task is to modify parameters at minimum cost such that the prespecified locations become optimal. This arises what we call the inverse 𝑝-median problem. In recent years, inverse 𝑝-median problem found an increasing interest [4–8].

Burkard et al. [9] considered inverse 𝑝-median problems with variable vertex weights and showed that discrete inverse 𝑝-median problems with real weights can be solved in polynomial time, if 𝑝 is fixed and not an input parameter.Burkard et al. [9, 10] investigated the inverse 1-median problem with variable vertex weights and showed that the problem is solvable by a greedy-type algorithm in 𝑂(𝑛log𝑛) time if the underlying network is a tree or the location problem is defined in the plane (if the distances between the points are measured in the Manhattan or maximum metric) and in 𝑂(𝑛²) time on cycles. The inverse Fermat-Weber problem was studied by Burkard et al. [11]. They derived a combinatorial approach which solves the problem in 𝑂(𝑛log𝑛) time for unit cost and under the assumption that the prespecified point that should become a 1-median does not coincide with a given point in the plane. Galavii [12] proposed linear time algorithms for the inverse 1-median problem with variable vertex weights on trees (with nonnegative weights) and on paths (with negative weights while the weights of endpoints are arbitrary). Under weighted Hamming distance, Guan and Zhang [13] presented polynomial time algorithm for the inverse 1-median problem with variable vertex weights on trees. Bonab et al. [14] showed that the inverse 𝑝-median problem with variable edge lengths is strongly NP-hard on general graphs and weakly NP-hard on series-parallel graphs. In [14], the authors also presented a polynomial time algorithm for the inverse 2-median problem with positive variable edge lengths on trees and a linear time algorithm for the special case of star graphs. In this paper, we first provide a linear time algorithm to solve the inverse 1-median problem on tree networks with variable nonnegative edge lengths. For the case when negative edge lengths are allowable, we show that this problem is NP-hard under Hamming distance and we propose strongly polynomial time algorithms under 𝑙₁ and 𝑙_∞ norms, respectively.

The paper is organized as follows. In Section 2, we define the inverse 1-median problem on trees with variable edge lengths, then we show some properties of this problem. Based on these properties, a linear time algorithm for variable nonnegative edge lengths is presented; when negative edge lengths are allowable, the NP-hardness of the problem under Hamming distance is proved and the strongly polynomial time algorithms of the problem under 𝑙₁ and 𝑙_∞ norms are proposed in Section 3. Some final remarks are given in Section 4.

2. Preliminaries

Let 𝑇=(𝑉,𝐸) be an undirected tree graph with vertex set 𝑉={𝑣₁,…,𝑣_𝑛} and edge set 𝐸 with |𝐸|=𝑛−1. The weight of a vertex 𝑣∈𝑉 is denoted by 𝑤_𝑣≥0, and the length of each edge 𝑒∈𝐸 is denoted by 𝑙_𝑒>0. The 1-median problem on trees is to locate a facility on 𝑉 such that the sum of the weighted distances of the vertices to this facility is minimized.

Let 𝐶(𝑎)∶𝑅^𝐸→𝑅 be a criterion function (having length vectors as argument) such that 𝐶(0)=0 and 𝐶(𝑎)≤𝐶(𝑏) if 𝑎≤𝑏. (For 𝑎,𝑏∈𝑅^𝐸, 𝑎≤𝑏 means that 𝑎_𝑒≤𝑏_𝑒, for all 𝑒∈𝐸.) Suppose that the unique path from 𝑠∈𝑉 to 𝑡∈𝑉 is denoted by 𝑃(𝑠,𝑡). Let 𝑣₀ be a specific vertex in 𝑉. Denote the adjusted edge length vector by 𝑥. Let 𝑑_𝑥(𝑠,𝑡) denote the length of 𝑃(𝑠,𝑡) with respect to 𝑥. The lower bound of 𝑥_𝑒 is denoted by 𝑙_𝑒≤0 for each 𝑒∈𝐸. Readers can refer to [15] for other graph theory terms not defined here. The inverse 1-median problem on tree networks with variable real edge lengths can be formally described as follows: min𝐶(|𝑙−𝑥|)s.t._{𝑣𝑣∈𝑉⧵0}𝑤_𝑣𝑑_𝑥𝑣₀,𝑣−_{{𝑣∈𝑉⧵𝑣}}𝑤_𝑣𝑑_𝑥()∀𝑣,𝑣≤0,𝑣∈𝑉⧵𝑣₀,𝑥_𝑒≥𝑙_𝑒,𝑒∈𝐸. (1)

If we delete an edge 𝑒 from 𝑇, the tree 𝑇 will be partitioned into two subtrees, one of which containing the prespecified vertex 𝑣₀ is denoted by 𝑇^′_𝑒=(𝑉^′_𝑒,𝐸^′_𝑒), and the other is denoted by 𝑇^′′_𝑒=(𝑉^′′_𝑒,𝐸^′′_𝑒), where 𝑉^′_𝑒, 𝑉^′′_𝑒denote the vertex sets and 𝐸^′_𝑒, 𝐸^′′_𝑒 denote the edge sets of 𝑇^′_𝑒, 𝑇^′′_𝑒, respectively. Let 𝐷_𝑥(𝑡)=∑_{𝑣∈𝑉⧵{𝑡}}𝑤_𝑣𝑑_𝑥(𝑡,𝑣), and let 𝐵_𝑒=∑_{𝑣∈𝑉𝑒′′}𝑤_𝑣−∑_{𝑣∈𝑉′𝑒}𝑤_𝑣 for each 𝑒∈𝐸; then for any 𝑣∈𝑉⧵{𝑣₀}, 𝐷_𝑥𝑣₀−𝐷_𝑥(𝑣)=_{𝑣𝑣∈𝑣⧵0}𝑤_𝑣𝑑_𝑥𝑣₀,𝑣−_{{𝑣∈𝑣⧵𝑣}}𝑤_𝑣𝑑_𝑥()=𝑣,𝑣_{𝑣𝑒∈𝑃0,𝑣𝑣∈𝑉𝑒′′}𝑤_𝑣−_{𝑣∈𝑉′𝑒}𝑤_𝑣𝑥_𝑒=_{𝑣𝑒∈𝑃0,𝑣}𝐵_𝑒𝑥_𝑒, (2) and the first constraint of (1) can be changed to _{𝑒∈𝑃(𝑣0,𝑣)}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉⧵𝑣₀. (3) So, we can simplify (1) as min𝐶(|𝑙−𝑥|)s.t._{𝑣𝑒∈𝑃0,𝑣}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉⧵𝑣₀,𝑥_𝑒≥𝑙_𝑒,𝑒∈𝐸. (4)

For any 𝑣_𝑟∈𝑉⧵{𝑣₀}, without loss of generality, suppose the sequence of vertices and edges in 𝑃(𝑣₀,𝑣_𝑟) is 𝑣₀,𝑒₁,𝑣₁,𝑒₂,𝑣₂,…,𝑣_𝑟−1,𝑒_𝑟,𝑣_𝑟. We have _{𝑣∈𝑉𝑒𝑖′′}𝑤_𝑣≤_{𝑣∈𝑉𝑒′′𝑖−1}𝑤_𝑣−𝑤_𝑣𝑖−1,_{𝑣∈𝑉′𝑒𝑖}𝑤_𝑣≥_{𝑣∈𝑉′𝑒𝑖−1}𝑤_𝑣+𝑤_𝑣𝑖−1. (5) Therefore, _{𝑣∈𝑉𝑒𝑖′′}𝑤_𝑣−_{𝑣∈𝑉′𝑒𝑖}𝑤_𝑣≤_{𝑣∈𝑉𝑒′′𝑖−1}𝑤_𝑣−𝑤_𝑣𝑖−1−_{𝑣∈𝑉′𝑒𝑖−1}𝑤_𝑣+𝑤_𝑣𝑖−1=_{𝑣∈𝑉𝑒′′𝑖−1}𝑤_𝑣−_{𝑣∈𝑉′𝑒𝑖−1}𝑤_𝑣−2𝑤_𝑣𝑖−1. (6) Then, for 𝑖=1,2,…,𝑟, by (2), we have 𝐷_𝑥𝑣₀−𝐷_𝑥𝑣_𝑖=_{𝑒∈𝑃(𝑣0,𝑣𝑖)}𝐵_𝑒𝑥_𝑒=^𝑖_𝑗=1𝐵_𝑒𝑗𝑥_𝑒𝑗,𝐵_𝑒𝑖−𝐵_𝑒𝑖−1=_{𝑣∈𝑉𝑒𝑖′′}𝑤_𝑣−_{𝑣∈𝑉′𝑒𝑖}𝑤_𝑣−_{𝑣∈𝑉𝑒′′𝑖−1}𝑤_𝑣−_{𝑣∈𝑉′𝑒𝑖−1}𝑤_𝑣≤_{𝑣∈𝑉𝑒′′𝑖−1}𝑤_𝑣−_{𝑣∈𝑉′𝑒𝑖−1}𝑤_𝑣−2𝑤_𝑣𝑖−1−_{𝑣∈𝑉𝑒′′𝑖−1}𝑤_𝑣−_{𝑣∈𝑉′𝑒𝑖−1}𝑤_𝑣=−2𝑤_𝑣𝑖−1≤0. (7) According to (7), we further obtain the following two formulas: 𝐷_𝑥𝑣₀−𝐷_𝑥𝑣_𝑗=𝐷_𝑥𝑣₀−𝐷_𝑥𝑣_𝑖+_{𝑒∈𝑃(𝑣𝑖,𝑣𝑗)}𝐵_𝑒𝑙_𝑒𝐵,0≤𝑖<𝑗≤𝑟,_𝑒𝑖−1≥𝐵_𝑒𝑖,𝑖=2,…,𝑟. (8)(9)

From the above analysis, we have the following.

Theorem 1. If there exists a vertex 𝑣_𝑡∈𝑉 satisfying 𝐷_𝑙(𝑣₀)−𝐷_𝑙(𝑣_𝑡)≤0, then 𝐵_𝑒𝑡≤0 and 𝐷_𝑙(𝑣₀)−𝐷_𝑙(𝑣)≤0 for each 𝑣∈𝑉^′′_𝑒𝑡, where 𝑒_𝑡 denotes the edge of the path 𝑃(𝑣₀,𝑣_𝑡) incident with vertex 𝑣_𝑡.

Proof. By contradiction. If 𝐵_𝑒𝑡>0, then from (9) we have 𝐵_𝑒>0 for each edge 𝑒∈𝑃(𝑣₀,𝑣_𝑡). By (2), 𝐷_𝑙(𝑣₀)−𝐷_𝑙(𝑣_𝑡)=∑_{𝑒∈𝑃(𝑣0,𝑣𝑡)}𝐵_𝑒𝑙_𝑒>0, which contradicts the fact that 𝐷_𝑙(𝑣₀)−𝐷_𝑙(𝑣_𝑡)≤0. So 𝐵_𝑒𝑡≤0. As 𝐵_𝑒≤𝐵_𝑒𝑡≤0, for each 𝑒∈𝑃(𝑣_𝑡,𝑣) and 𝑣∈𝑉^′′_𝑒𝑡, by (8), we can deduce that 𝐷_𝑙(𝑣₀)−𝐷_𝑙(𝑣)=𝐷_𝑙(𝑣₀)−𝐷_𝑙(𝑣_𝑡)+∑_{𝑒∈𝑃(𝑣𝑡,𝑣)}𝐵_𝑒𝑙_𝑒≤0. This completes the proof.

Based on the above properties, we will present some algorithms of the problem under different norms and analyses their complexities in the next section.

3. Algorithms and Complexities

3.1. The Problem with Variable Nonnegative Edge Lengths (I1MT-N)

In this section, we consider the following problem: I1MT-N. When 𝑙_𝑒=0 for each 𝑒∈𝐸, we call (4) Problem I1MT-N, which is the inverse 1-median problem on trees with variable nonnegative edge lengths.

Denote the optimal solution of Problem I1MT-N by 𝑙^∗. We present Algorithm 2 to solve this problem. particulary, Algorithm 2 can solve Problem I1MT-N under all kinds of norms only by edge length reduction.

Algorithm 2.     
Step  1. Let 𝐸=∅, 𝑉={𝑣₀}.
Step  2. Let 𝐸^′={𝑒^′∣𝑒^′=(𝑣,𝑣)∈𝐸⧵𝐸,𝑣∈𝑉⧵𝑉,𝑣∈𝑉}. For each 𝑒^′∈𝐸^′,(1)if 𝐵_𝑒′>0, then let 𝑙^∗_𝑒′=0 and 𝐸=𝐸∪{𝑒^′};(2)if 𝐵_𝑒′≤0, then for each 𝑒∈{𝑒^′}∪𝐸^′′_𝑒′, let 𝑙^∗_𝑒=𝑙_𝑒 and 𝐸=𝐸∪{𝑒^′}∪𝐸^′′_𝑒′.
Step  3. Let 𝑉={𝑣∣(𝑣,𝑣)=𝑒^′∈𝐸^′,𝑣∈𝑉 and 𝐵_𝑒′>0}. If 𝐸=𝐸, stop. Otherwise, turn to Step 2.

Theorem 3. Algorithm 2 correctly finds the optimal solution 𝑙^∗of Problem I1MT-N in linear time.

Proof. First, we show that 𝑙^∗obtained by Algorithm 2 is a feasible solution of Problem I1MT-N. It is clear that 𝑙^∗_𝑒=0 or 𝑙^∗_𝑒=𝑙_𝑒; so it satisfies 𝑙^∗_𝑒≥0 for each 𝑒∈𝐸. For each vertex 𝑣∈𝑉⧵{𝑣₀} and 𝑒∈𝑃(𝑣₀,𝑣), from Step 2 of Algorithm 2, when 𝐵_𝑒>0, 𝑙^∗_𝑒=0; and when 𝑙^∗_𝑒=𝑙_𝑒>0, 𝐵_𝑒≤0. Hence, ∑_{𝑒∈𝑃(𝑣0,𝑣)}𝐵_𝑒𝑙^∗_𝑒≤0 for each 𝑣∈𝑉⧵{𝑣₀}. Therefore, 𝑙^∗ is a feasible solution of Problem I1MT-N.

Suppose 𝑙^∗ is not the optimal solution of Problem I1MT-N and suppose that the optimal solution is 𝑙^′. Then under 𝑙^′ there exists at least one edge 𝑒∈𝐸 satisfying one of the following two cases:

Case 1 (𝐵_𝑒>0 and 𝑙^′_𝑒>0). Without loss of generality, suppose 𝑒=(𝑣_𝑖−1,𝑣_𝑖)∈𝑃(𝑣₀,𝑣_𝑖)=𝑣₀,𝑒₁,𝑣₁,𝑒₂,𝑣₂,…,𝑣_𝑖−1,𝑒_𝑖,𝑣_𝑖. By (9), 𝐵_𝑒𝑗≥𝐵_𝑒>0 for 𝑗=1,…,𝑖−1. Notice that 𝑙^′_𝑒≥0 for each 𝑒∈𝐸. We have ∑^𝑖_𝑗=1𝐵_𝑒𝑗𝑙^′_𝑒𝑗>0, which contradicts the feasibility of 𝑙^′.

Case 2 (𝐵_𝑒≤0 and 𝑙^′_𝑒≥0,𝑙^′_𝑒≠𝑙_𝑒). We define a new vector 𝑙^′′∈𝑅^𝐸 such that 𝑙^′′_𝑒=𝑙_𝑒,if𝑒=𝑙𝑒,_′𝑒,otherwise. (10) From the result of Case 1, we know that when 𝐵_𝑒>0,  𝑙^′′_𝑒=𝑙^′_𝑒=0. And it is easy to see that 𝑙^′′_𝑒≥0 for each 𝑒∈𝐸. Then, we have ∑_{𝑒∈𝑃(𝑣0,𝑣)}𝐵_𝑒𝑙^′′_𝑒=∑_{𝑒∈𝑃(𝑣0,𝑣),𝐵𝑒≤0}𝐵_𝑒𝑙^′′_𝑒≤0 for each vertex 𝑣∈𝑉. So 𝑙^′′is a feasible solution of Problem I1MT-N. But 𝐶(|𝑙−𝑙^′′|)<𝐶(|𝑙−𝑙^′|), a contradiction to the optimality assumption of 𝑙^′.

Therefore, Algorithm 2 correctly finds the optimal solution 𝑙^∗of Problem I1MT-N.

The main computation of Algorithm 2 is to compute 𝐵_𝑒 for each edge 𝑒∈𝐸, then to modify the corresponding edge length of 𝑒 according to 𝐵_𝑒 from Step 2. Notice that 𝐵_𝑒=_{𝑣∈𝑉𝑒′′}𝑤_𝑣−_{𝑣∈𝑉′𝑒}𝑤_𝑣=2_{𝑣∈𝑉𝑒′′}𝑤_𝑣−_𝑣∈𝑉𝑤_𝑣, (11) where ∑_𝑣∈𝑉𝑤_𝑣 is a constant. Let 𝑣₀ be the root vertex and conduct the breadth-first method to 𝑇. We can compute 𝐵_𝑒 iteratively from the edges incident with the leaves at the bottom layer up to 𝑣₀ in the rooted tree. So Problem I1MT-N can be solved in 𝑂(|𝐸|) time by using Algorithm 2.

From Algorithm 2, we know that for Problem I1MT-N it is not necessary to increase the edge lengths of the tree network. So to solve Problem I1MT-N is equivalent to solving the following program: min𝐶(𝑙−𝑥)s.t._{𝑣𝑒∈𝑃0,𝑣}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉⧵𝑣₀,0≤𝑥_𝑒≤𝑙_𝑒,𝑒∈𝐸. (12)

3.2. The Problem with Variable Real Edge Lengths

3.2.1. The Model Under 𝑙₁ Norm with Variable Real Edge Lengths (I1MT-R-𝑙₁)

In this section, we consider the following inverse 1-median problem on trees with variable real edge lengths under 𝑙₁ norm when edge length increase is forbidden. I1MT-R- 𝑙₁. When 𝑙_𝑒<0 for each 𝑒∈𝐸 and the criterion function 𝐶(𝑎)∶𝑅^𝐸→𝑅 is a function such that 𝐶(𝑎)=𝑐𝑎, where 𝑐 is the cost vector and 𝑐_𝑒>0 is the cost of decreasing 𝑙_𝑒 by one unit for each 𝑒∈𝐸, we call (4) Problem I1MT-R-𝑙₁, which is the inverse 1-median problem on trees with variable real edge lengths under 𝑙₁ norm.

Problem I1MT-R-𝑙₁can be formally described as follows: min_𝑒∈𝐸𝑐_𝑒𝑙_𝑒−𝑥_𝑒s.t._{𝑣𝑒∈𝑃0,𝑣}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉⧵𝑣₀,𝑙_𝑒≤𝑥_𝑒≤𝑙_𝑒,𝑒∈𝐸. (13)

For a path 𝑃(𝑣₀,𝑣_𝑟)=𝑣₀,𝑒₁,𝑣₁,𝑒₂,𝑣₂,…,𝑣_𝑟−1,𝑒_𝑟,𝑣_𝑟, suppose that 𝐵_𝑒𝑖−1>0 and 𝐵_𝑒𝑖≤0, where 2≤𝑖≤𝑟. Denote the optimal solution of Problem I1MT-R-𝑙₁ by 𝑙^opt. We have the following.

Theorem 4. 𝑙_𝑒^𝑜𝑝𝑡_𝑗=𝑙_𝑒𝑗 for 𝑗=𝑖,𝑖+1,…,𝑟.

Proof. By (9), clearly 𝐵_𝑒𝑗>0 for 𝑗=1,2,…,𝑖−1 and 𝐵_𝑒𝑖≤0 for 𝑗=𝑖,𝑖+1,…,𝑟. Suppose to the contrary that there exists an edge 𝑒_𝑘∈𝑃(𝑣_𝑖−1,𝑣_𝑟) such that 𝑙_𝑒𝑘≤𝑙^opt_𝑒𝑘<𝑙_𝑒𝑘. Similar to the proof of Case 2 in Theorem 3, we define a new feasible solution 𝑙^′′ such that 𝑙^′′_𝑒𝑘=𝑙_𝑒𝑘 and 𝑙^′′_𝑒𝑗=𝑙^opt_𝑒𝑗 when 𝑗≠𝑘. Then we have ∑_{𝑒∈𝑃(𝑣0,𝑣)}𝐵_𝑒𝑙^′′_𝑒≤∑_{𝑒∈𝑃(𝑣0,𝑣)}𝐵_𝑒𝑙^opt_𝑒≤0 for each vertex 𝑣∈𝑉. But 𝑐_𝑒(𝑙_𝑒−𝑙^′′_𝑒)<𝑐_𝑒(𝑙_𝑒−𝑙^opt_𝑒), and therefore 𝑐(𝑙−𝑙^′′)<𝑐(𝑙−𝑙^opt), a contradiction to the fact that 𝑙^opt is the optimal solution of Problem I1MT-R-𝑙₁. This completes the proof.

From Theorem 4, we only need to determine the optimal adjusted edge lengths 𝑙^opt_𝑒 with 𝐵_𝑒>0. We construct the subtree 𝑇^𝑆=(𝑉^𝑆,𝐸^𝑆) of 𝑇 with 𝐸^𝑆={𝑒∣𝐵_𝑒>0,𝑒∈𝐸}, 𝑇^𝑆=𝑇[𝐸^𝑆], and 𝑉^𝑆=𝑉(𝑇^𝑆). Clearly, 𝑣₀∈𝑉^𝑆. Regard 𝑇^𝑆 as a directed tree with the orientation from the root 𝑣₀ to all leaves (e.g., by conducting the breadth-first method). For each 𝑣∈𝑉^𝑆, let 𝑃(𝑣) denote the path from 𝑣₀ to 𝑣 on the tree 𝑇^𝑆. We can simplify (13) as min_{𝑒∈𝐸𝑆}𝑐_𝑒𝑙_𝑒−𝑥_𝑒s.t._{(𝑣)𝑒∈𝑃}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉^S⧵𝑣₀,𝑙_𝑒≤𝑥_𝑒≤𝑙_𝑒,∀𝑒∈𝐸^𝑆. (14)

In [16, 17], the authors considered the shortest path improvement problem, which is how to shorten the lengths of edges with as less cost as possible such that the distances between specified sources and terminals are reduced to the required bounds. It is not difficult to see that (14) is in fact a shortest path improvement problem for tree networks with single source 𝑣₀ and the required bound 0 under 𝑙₁ norm. Let 𝜃_𝑒=𝐵_𝑒(𝑙_𝑒−𝑥_𝑒)≥0, 𝑏_𝑒=𝐵_𝑒(𝑙_𝑒−𝑙_𝑒)>0, and 𝑝_𝑒=𝑐_𝑒/𝐵_𝑒>0 for each 𝑒∈𝐸^𝑆. Let 𝑟_𝑣=∑_{𝑒∈𝑃(𝑣)}𝐵_𝑒𝑙_𝑒>0 for each 𝑣∈𝑉^𝑆. Equation (14) can be formulated as the following linear program: min_{𝑒∈𝐸𝑆}𝑝_𝑒𝜃_𝑒s.t._{(𝑣)𝑒∈𝑃}𝜃_𝑒≥𝑟_𝑣,∀𝑣∈𝑉^𝑆⧵𝑣₀,0≤𝜃_𝑒≤𝑏_𝑒,∀𝑒∈𝐸^𝑆. (15) Equation (15) is a combinatorial linear program and can be solved by a strongly polynomial time algorithm [18]. Moreover, (15) can be solved much faster [16, 17]. In [16], the authors presented a steepest descent-type combinatorial algorithm to solve this problem, which runs in 𝑂(|𝑉|²) time. In [17], the authors formulated the dual problem of (15) as a minimum cost flow problem on a two-terminal serial-parallel graph and then gave an algorithm to obtain the primal optimal solution of (15), with the whole procedure running in 𝑂(|𝑉|log|𝑉|) time.

From the above analysis, we have the following theorem.

Theorem 5. Problem I1MT-R-𝐿₁ can be solved in 𝑂(|𝑉|log|𝑉|) time.

3.2.2. The Model under Hamming Distance with Variable Real Edge Lengths (I1MT-R-H)

Now we consider the model under Hamming distance. The definition of Hamming distance will be given in this section. I1MT-R-H. When 𝑙_𝑒<0 for each 𝑒∈𝐸 and 𝑐_𝑒>0 is the cost of decreasing 𝑙_𝑒 for each 𝑒∈𝐸, we call the following program (16) Problem I1MT-R-H, which is the inverse 1-median problem on trees with variable real edge lengths under Hamming distance: min_𝑒∈𝐸𝑐_𝑒𝐻𝑥_𝑒,𝑙_𝑒s.t._{𝑣𝑒∈𝑃0,𝑣}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉⧵𝑣₀,𝑙_𝑒≤𝑥_𝑒≤𝑙_𝑒,𝑒∈𝐸, (16)where the Hamming distance 𝐻(𝑥_𝑒,𝑙_𝑒) is defined as 𝐻𝑥_𝑒,𝑙_𝑒=0,𝑥_𝑒=𝑙_𝑒,1,𝑥_𝑒≠𝑙_𝑒. (17)

From a similar discussion in the former Section 3.2.1, Problem I1MT-R-H can be formulated as follows: min_{𝑒∈𝐸𝑆}𝑐_𝑒𝐻𝑥_𝑒,𝑙_𝑒s.t._{(𝑣)𝑒∈𝑃}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉^𝑆⧵𝑣₀,𝑙_𝑒≤𝑥_𝑒≤𝑙_𝑒,∀𝑒∈𝐸^𝑆. (18) Equation (18) is a shortest path improvement problem for tree networks with single source 𝑣₀ and the required bound 0 under Hamming distance [19]. In [19], the authors proved that even if the network is a chain network, the problem is still NP-hard. So, we have the following.

Theorem 6. Problem I1MT-R-H is NP-hard.

3.2.3. The Model under 𝑙_∞ Norm with Variable Real Edge Lengths (I1MT-R-𝑙_∞)

We consider the model under 𝑙_∞ norm as follows: I1MT-R-𝑙_∞. When 𝑙_𝑒<0 for each 𝑒∈𝐸 and the criterion function 𝐶(𝑎)∶𝑅^𝐸→𝑅 is a function such that 𝐶(𝑎)=||𝑐𝑎||_∞, where 𝑐 is the cost vector, 𝑐_𝑒>0 is the cost of decreasing 𝑙_𝑒 by one unit for each 𝑒∈𝐸, and ‖𝑢‖_∞=max_𝑒∈𝐸|𝑢_𝑒| stands for 𝑙_∞ norm of vector 𝑢, we call (4) Problem I1MT-R-𝑙_∞, which is the inverse 1-median problem on trees with variable real edge lengths under 𝑙_∞ norm.

Also from a similar discussion in Section 3.2.1, Problem I1MT-R-𝑙_∞can be formally described as follows: minmax_{𝑒∈𝐸𝑆}𝑐_𝑒𝑙_𝑒−𝑥_𝑒s.t._{(𝑣)𝑒∈𝑃}𝐵_𝑒𝑥_𝑒≤0,∀𝑣∈𝑉^𝑆⧵𝑣₀,𝑙_𝑒≤𝑥_𝑒≤𝑙_𝑒,∀𝑒∈𝐸^𝑆. (19) Let 𝑡^′_𝑒=𝐵_𝑒(𝑥_𝑒−𝑙_𝑒), 𝑡_𝑒=𝐵_𝑒(𝑙_𝑒−𝑙_𝑒), ℎ_e=c_e/B_e, 𝐿_𝑦(𝑃(𝑣))=∑_{𝑒∈𝑃(𝑣)}𝑦_𝑒, and 𝑈(𝑣)=−∑_{𝑒∈𝑃(𝑣)}𝐵_𝑒𝑙_𝑒 for each 𝑒∈𝐸^𝑆 and 𝑣∈𝑉^𝑆⧵{𝑣₀}. First, we suppose that ℎ_e=1 for each 𝑒∈𝐸^𝑆. Equation (19) can be formulated as the following program: minmax_{𝑒∈𝐸𝑆}𝑡_𝑒−𝑡^′_𝑒s.t.𝐿_𝑡′(𝑃(𝑣))≤𝑈(𝑣),∀𝑣∈𝑉^𝑆⧵𝑣₀,0≤𝑡^′_𝑒≤𝑡_𝑒,∀𝑒∈𝐸^𝑆. (20)

In [20], the authors presented a strongly polynomial time algorithm to solve this type of problem, which runs in 𝑂(|𝑉|³) time. Because the network concerned here is tree network, we now give a simplified procedure to solve (20) which runs in 𝑂(|𝑉|²) time.

On tree 𝑇^𝑆, if max{𝐿_𝑡(𝑃(𝑣))−𝑈(𝑣)∣𝑣∈𝑉^𝑆⧵{𝑣₀}}≤0, we need do nothing. Hence, we only consider the case that max{𝐿_𝑡(𝑃(𝑣))−𝑈(𝑣)∣𝑣∈𝑉^𝑆⧵{𝑣₀}}>0. Clearly, the optimal solution 𝑡^∗ satisfies max{𝐿_𝑡∗(𝑃(𝑣))−𝑈(𝑣)∣𝑣∈𝑉^𝑆⧵{𝑣₀}}=0. Denote 𝜃^∗=‖𝑡−𝑡^∗‖_∞, then for any edge 𝑒∈𝐸^𝑆, we have 𝑡^∗_𝑒=max{𝑡_𝑒−𝜃^∗,0}.

As the procedure in [20], first we sort the edges by their lengths in an increasing order. Beginning with the shortest edge, say 𝑒^′, set 𝑡^′_𝑒=𝑡_𝑒−𝑡_𝑒′ for each 𝑒∈𝐸^𝑆, and check if max{𝐿_𝑡′(𝑃(𝑣))−𝑈(𝑣)∣𝑣∈𝑉^𝑆⧵{𝑣₀}}<0. If yes, we know that the lengths are overreduced and 𝜃^∗<𝑡_𝑒′. If max{𝐿_𝑡′(𝑃(𝑣))−𝑈(𝑣)∣𝑣∈𝑉^𝑆⧵{𝑣₀}}=0, 𝑡^∗=𝑡^′ is the optimal solution. If max{𝐿_𝑡′(𝑃(𝑣))−𝑈(𝑣)∣𝑣∈𝑉^𝑆⧵{𝑣₀}}>0, then 𝜃^∗>𝑡_𝑒′, and in this case we contract the edges 𝑒 with 𝑡^′_𝑒=0 into vertices and repeat the process on the resulted network with respect to 𝑡^′. Since at least one vertex disappears in each contraction, the procedure has at most |𝑉| iterations. For the time complexity, to find the shortest edge 𝑒^′ can be completed in 𝑂(|𝐸|) time, and in each iteration, the time for checking if max{𝐿_𝑡′(𝑃(𝑣))−𝑈(𝑣)∣𝑣∈𝑉^𝑆⧵{𝑣₀}}<0 is 𝑂(|𝑉|). Hence, the complexity of this procedure is 𝑂(|𝑉|²). In the following, we assume that 𝜃^∗<𝑡_𝑒′, which means that the preliminary procedure is finished while the optimality has not been reached.

Because the path between any two vertices on tree network is unique, let 𝑄 denotes the set of vertices which does not satisfy the upper bound demand. Let |𝑃| be the number of edges of path 𝑃. Let 𝐿𝜃=max^𝑡(𝑃(𝑣))−𝑈(𝑣)|𝑃(𝑣,)|∣𝑣∈𝑄 (21) then 𝜃^∗=𝜃 is the optimal reduction.

Remark 7. Note that when ℎ_e≠1 for each 𝑒∈𝐸^𝑆, (20) can be formulated as the following program: minmax_e∈ESℎ_e𝑡_𝑒−𝑡^′_𝑒s.t.𝐿_𝑡′(𝑃(𝑣))≤𝑈(𝑣),∀𝑣∈𝑉^𝑆⧵𝑣₀,0≤𝑡^′_𝑒≤𝑡_𝑒,∀𝑒∈𝐸^𝑆. (22)

In this case, we denote 𝜃^∗=‖ℎ(𝑡−𝑡^∗)‖_∞; then for any edge 𝑒∈𝐸^𝑆, we have 𝑡^∗_𝑒=max{𝑡_𝑒−𝜃^∗/ℎ_𝑒,0}. We sort the edges by 𝑡_𝑒ℎ_𝑒(𝑒∈𝐸^𝑆) in an increasing order to conduct the preliminary procedure to ensure that 𝜃^∗<𝑡_𝑒′ℎ_𝑒′, where 𝑒^′ is the edge satisfying 𝑡_𝑒′ℎ_𝑒′=min{𝑡_𝑒ℎ_𝑒∣𝑒∈𝐸^𝑆}. Let 𝐿𝜃=max^𝑡(𝑃(𝑣))−𝑈(𝑣)∑_{𝑒∈𝑃(𝑣)}1/ℎ_𝑒∣𝑣∈𝑄, (23) then 𝜃^∗=𝜃 is the optimal reduction. Similar discussions show that the above procedure correctly solves program (22), and its computational complexity is 𝑂(|𝑉|²).

From the above analysis, we have the following theorem.

Theorem 8. Problem I1MT-R-𝑙_∞ can be solved in 𝑂(|𝑉|²) time.

4. Conclusion

In this paper, we discussed the inverse 1-median problem with variable edge lengths on tree networks. We showed that this problem is solvable in linear time with variable nonnegative edge lengths under all kinds of norms. For the case when negative edge lengths are allowable, by formulating this problem to some known combinatorial optimization problems, we showed that this problem is NP-hard under Hamming distance and we proposed strongly polynomial time algorithms under 𝑙₁ and 𝑙_∞ norms, respectively.

As a future research topic, it will be meaningful to consider the problem on other networks such as the interval graph, the cactus graph, and the block graph, especially under the Hamming distance. We know that Algorithm 2 can solve Problem I1MT-N under all kinds of norms. But when negative edge lengths are allowable, we only dealt with the problems under the constraint that the edge length increase is forbidden. So we can further study the complexity of the problems such that the edge lengths can either be reduced or increased. In addition, for the inverse 𝑝-median problems for 𝑝≥2, designing the optimal or approximation algorithms and analysing their computational complexities are also promising.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (NSFC nos. 11171316 and 61173002), the Grant of the Second Stage of Brain Korea 21, and Zhejiang Provincial Natural Science Foundation of China (no. Y6090472).