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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 156901, 4 pages
The Capacity Expansion Path Problem in Networks
Department of Mathematics, Yunnan University, Kunming 650091, China
Received 8 February 2013; Revised 1 June 2013; Accepted 25 June 2013
Academic Editor: Yuri Sotskov
Copyright © 2013 Jianping Li and Juanping Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper considers the general capacity expansion path problem (GCEP) for the telecommunication operators. We investigate the polynomial equivalence between the GCEP problem and the constrained shortest path problem (CSP) and present a pseudopolynomial algorithm for the GCEP problem, no matter the graph is acyclic or not. Furthermore, we investigate two special versions of the GCEP problem. For the minimum number arc capacity expansion path problem (MN-CEP), we give a strongly polynomial algorithm based on the dynamic programming. For the minimum-cost capacity expansion shortest path problem (MCESP), we give a strongly polynomial algorithm by constructing a shortest paths network.
Currently Chinese telecommunication operators are under the huge pressure of communication capacity and communication quality, followed by the rapidly increasing users. Shortest path problem plays an important role in the telecommunication networks. In this paper, we consider the network capacity expansion strategy for the telecommunication operators, called the general capacity expansion path problem (GCEP), given a weighted digraph with nodes, arcs, and two fixed nodes , . For each arc , nonnegative is the weight, is the capacity, and nonnegative is the cost of expanding the capacity of the arc by one unit. Given two positive and , find a path from to to satisfy two constraints: (1) and (2) for some arc , if the capacity , we must expand the capacity of arc to . The objective is to minimize the total expansion cost on the path , where if and otherwise. In telecommunication network, is the given maximum length of the path , and is the expected maximum number of the users. If the length of the path exceeds , the communication quality would drop quickly. If the capacity (or bandwidth) of the arc is not wide enough, the arc would be congested, and we have to expand this arc to the expected .
Krumke et al.  studied budget constrained network upgrading problems. The goal is how to find a minimum cost set of vertices to be upgraded so that the resulting network has a minimum spanning tree of weight no more than the budget. Their upgrading costs are on the vertices.
Zhang et al.  studied the problem how to increase the capacities of the elements in an edge set efficiently so that the capacity of a given family of subsets of can be increased to the maximum extent, where the total cost for the increment of capacity is within a given budget bound . Yang and Zhang  considered a class of bottleneck capacity expansion problems, which aimed to enhance bottleneck capacity to a certain level with minimum cost. There were two types of expanding models: arc expanding and node expanding. They emphasized the maximum capacity path problem. Our models are different from theirs. For example, when we consider the arc expansion cost of the path in the telecommunication network, on which each signal is sent from the source via some transferring facility to the sink , we must make sure that the transferring length does not exceed the length bound .
Different from our model and method, Seref et al.  studied an incremental optimization problem. They started from a feasible solution and made an incremental change in that would result in the greatest improvement in the objective function. For the incremental shortest path problem, they showed that the arc inclusion version is polynomially solvable whereas the arc exclusion version is -complete.
In this paper, we present a proof of polynomial equivalence between the GCEP problem and the constrained shortest path problem (CSP), which leads to that the GCEP problem remains -hard. We design a pseudopolynomial algorithm and a FPTAS to solve the GCEP problem. Then we study two special versions of the GCEP problem. For the first version, minimum number arc capacity expansion path problem (MNCEP), we present a strongly polynomial algorithm based on dynamic programming. For the second version, the minimum-cost capacity expansion shortest path problem (MCESP), we construct a shortest path network and present a strongly polynomial algorithm.
2. The Equivalence between the GCEP Problem and the CSP Problem
Motivated by the polynomial equivalence between the Hitchcock problem and the minimum-cost flow problem , we prove that the GCEP problem is polynomially equivalent to the constrained shortest path problem, which leads to Theorem 1. Hassin  studied the constrained (or restricted) shortest path problem (CSP) as follows. Let be a weighted digraph with nodes, arcs, and two fixed nodes , . Each arc has a length and a transition time . These numbers are assumed to be positive integers. For any path from to in , the length and transition time are defined as the sum of the lengths and transition times of all arcs on , respectively. The CSP problem is to find the minimum length path in such that transition time along this path does not exceed a given bound . For convenience, we denote a notation, by , to represent an instance of the CSP problem. The CSP problem is a classical -hard problem . To obtain Theorem 1, we use to represent an instance of the GCEP problem.
Theorem 1. The GCEP problem is polynomially equivalent to the CSP problem.
Proof. It is sufficient to prove that the GCEP problem can be transformed to the CSP problem in polynomial operations, and vice versa.
For any instance , say , of the GCEP problem, we construct an instance of the CSP problem: a digraph consisting of the same structure as the digraph with nodes, arcs, two fixed nodes and , and two positive and . For each arc , define two nonnegative and , where if and otherwise. The objective is to find a path from to in such that and the value of is minimized.
On the converse direction, for any instance , say , of the CSP problem, we construct an instance of the GCEP problem: a digraph consisting of the same structure as the digraph with nodes, arcs, a fixed positive integer , and the bound . For each arc , define nonnegative , , and , which implies that and . The objective is to find a path from to in such that and the value of is minimized.
It is easy to prove the claim: there is an optimal solution to the instance of the CSP problem with the optimal value if and only if there is an optimal solution to the instance of the GCEP problem with the optimal value . Moreover, the transformation is executed in polynomial operations.
Since the CSP problem is -hard, the proof of Theorem 1 implies that the GCEP problem is -hard, even if . And if , the GCEP problem is reduced to regular CSP problem.
3. A Pseudopolynomial Algorithm for the GCEP Problem
According to Theorem 1, for any instance of the GCEP problem in the acyclic digraphs, by constructing an instance of the CSP problem and utilizing the FPTAS in [6, 8] for the CSP problem, we can get a FPTAS to solve the GCEP problem. Now for the GCEP problem, no matter acyclic digraphs or not, by applying the following dynamic programming method which is similar to the Bellman-Ford algorithm [9, 10], we give a pseudopolynomial algorithm. To describe our algorithm, for each node and two integers and , define to be the minimum length of an optimal path from to with respect to the weight , such that at most arcs are traversed and the expansion cost along is at most . Set if no such path exists. For convenience, denote as the expansion cost for each arc and -path as the directed path if its weight is less than or equal to . Set OPT to be the minimum capacity expansion cost of - -path.
Algorithm 2 (EXACT()). Comment. Return minimum capacity expansion path from to with constraint .
Step 1 (initialization). Set for each and set if .
Step 2 For , do For , do For each , do , ; If , output and the path , and exit.
End of Algorithm EXACT().
Actually algorithm EXACT() is a dynamic programming algorithm, which generalizes the Bellman-Ford method [9, 10]. Note that OPT is not known but it satisfies . is the maximum number of traversing arcs on the path from node to , and the algorithm enumerates each and each for . To get , the algorithm computes first for , , and each , then for ,, and each and so on, until the first value of for which . According to the definition of , is the minimum length of an optimal path from to with respect to the weight , such that at most arcs are traversed and the expansion cost along is at most . indicates the length with respect to weight on this optimal path is less than and equal to (optimal condition), and . Hence OPT is set to this value of . So the complexity of algorithm EXACT() is , and we obtain Theorem 3.
Theorem 3. For the GCEP problem, the algorithm EXACT() produces an optimal solution, and its time complexity is .
Moreover, the technique of rounding and scaling for acyclic digraphs in  is applied to our pseudopolynomial algorithm EXACT(), and we can present an FPTAS in time for the GCEP problem even in general digraphs, where . The construction of our FPTAS for general digraphs is similar to the one due to  for acyclic digraphs. The reader can find the algorithm in details from ; thus we omit it here.
4. Dynamic Programming Algorithm for the MNCEP Problem
In this section, we study the first special version of the GCEP problem, minimum number arc capacity expansion path problem (MNCEP), where for each arc , , if and otherwise. Equivalently, the objective is to find a path of , on which the number of arcs to be expanded is minimized.
Like the GCEP problem, no matter acyclic digraphs or not, by applying the method similar to Bellman-Ford algorithm [9, 10], we can design a strongly polynomial algorithm to solve the MNCEP problem. For each node , given two integers and , define to be the minimum length of the path from to with respect to the weight , which traverses at most arcs with exact arcs to be expanded. Set if no such path exists.
Algorithm 4 (MNCEP). Comment. Return the minimum number of arcs to be expanded on the path from to with constraint .
Step 1 (initialization). Set and set if .
Step 2 For , do For , do For each node , do , If , then output and the path , and exit.
End of Algorithm MNCEP.
The algorithm MNCEP is also based on dynamic programming. Different from algorithm EXACT(), no unknown OPT appears in the algorithm. is the maximum number of traversing arcs on the path from node to , and is the exact number of arcs to be expanded on this path. So . The algorithm enumerates each , each , and each till . So the algorithm computes first for , , and each , then for , , and each , and so on, until the first value for which . Hence, output . Thus, the algorithm runs in time. We obtain Theorem 5.
Theorem 5. For the MNCEP problem, the algorithm MNCEP produces an optimal solution, and its time complexity is .
5. Strongly Polynomial Algorithm for the MCESP Problem
The second special version of GCEP is the minimum-cost capacity expansion shortest path problem (MCESP). In MCESP problem, the bound is set to be the shortest distance from to in , with respect to weight . So the objective of MCESP problem is to find a shortest path from to with respect to the weight , such that the total expansion cost on the path is minimized, where if and otherwise. To solve this MCESP problem optimally, the shortest paths network plays an important role in the algorithm.
Property 1 (see ). Let the vector dist represent the shortest paths distances. Then a directed path from the source node to the node is a shortest path if and only if for every arc .
For convenience, set as the capacity expansion cost for each arc .
Algorithm 6 (MCESP). Comment. Return the minimum-cost capacity expansion shortest path.
Step 1. Perform the breadth-first search in the digraph , using the arcs to satisfy the equality . Construct the shortest paths network of , where . The values of and addcost in remain the same as in .
Step 2. Find the shortest path from to in digraph , with respect to the arc capacity expansion cost addcost. Output and the path .
End of Algorithm MCESP.
Theorem 7. The algorithm MCESP optimally solves the MCESP problem, and it runs in the time .
Proof. Property 1 implies the fact: if is a shortest path from the source to some node , then for every arc . The converse is also true: if for every arc in a directed path from the source to the node , then must be a shortest path. The algorithm MCESP searches all arcs for in . Hence, the new digraph includes all minimum distance path from to in , with respect to the weight . The Step 2 of the algorithm selects the minimum cost capacity expansion path in . Hence, the algorithm MCESP solves the MCESP problem optimally.
Step 1 of the algorithm needs steps to compute all by using Dijkstra algorithm for shortest paths. Step 2 needs steps to perform breadth-first search. Step 2 needs steps as Step 1. Hence, the time complexity of the algorithm MCESP is .
In this paper, we focus on the general capacity expansion path problem for telecommunication operators, a new practical capacity expansion model. The polynomial equivalence between the GCEP problem and the CSP problem is proved. Then by utilizing dynamic programming technique and Bellman-Ford method [9, 10], we design a pseudopolynomial algorithm for the GCEP problem. We also present two special versions of the GCEP problem: the MNCEP problem and the MCESP problem. For the MNCEP problem, by using dynamic programming method and Bellman-Ford method, we design a strongly polynomial algorithm to solve it. For the MCESP problem, by constructing a shortest paths network, we design a strongly polynomial algorithm.
For further work, we would investigate some other special versions of the GCEP problem and related algorithms, which exist in the current telecommunication networks.
The work is fully supported by the National Natural Science Foundation of China [no. 11126355, 61063011].
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