On the flow cost lowering problem
Introduction
The minimum cost flow problem (MCFP) is well studied in the literature (cf. [1] for a collection of algorithms). A minimum cost flow can be used to model various applications in areas such as transportation, logistics, and telecommunication network design problems.
While in the classical minimum cost flow the cost per unit of flow is fixed, we investigate a scenario where the unit flow cost can be lowered by upgrading arcs. A typical application of this model is a telecommunication network with given transmission time or cost and given amount of data to be transmitted. The network is modeled by a directed graph with arc capacities and cost. By investing money on selected arcs one can lower the transmission time or cost on this arc. The goal is: Given an upgrade budget and an amount of data, find an optimum upgrade strategy and an optimum routing such that the flow cost in the upgraded network is as small as possible.
In [4] a related problem is investigated, where upgrading means increasing the capacity of arcs, and the goal is to find an upgrade strategy of minimum cost which admits a flow of given value.
The paper is organized as follows. In Section 2 we state the problem explicitly and introduce some notations. Section 3 gives hardness results for the problem even on series–parallel graphs. In Section 4 we use the ideas from [4] to develop an approximation algorithm for series–parallel graphs. This algorithm produces for arbitrary a solution which satisfies the flow constraint, exceeds the given budget constraint by a factor of at most (1+δ) and has flow cost within (1+ε) times the cost of an optimum solution. The running time is polynomial both in the input size and 1/δε.
Section snippets
Preliminaries and problem formulation
Definition 1 Minimum cost flow problem Given a graph G=(V,R) (parallel arcs allowed) with terminals s,t∈V, arc capacities , a cost function , and a flow value , find a feasible flow of value F from s to t of minimum cost Definition 2 k-Criteria minimization problem A k-criteria minimization problem on weighted graphs is specified by a tuple (f1,…,fk) of polynomial time computable objectives, a tuple (F2,…,Fk) of (upper or lower) bounds, and a graph class Γ. The set of feasible solutions is given by
Hardness results
By a reduction from Continous Multiple Choice Knapsack [3, Problem MP11] one can show that FCLP is NP-hard to solve even if the restriction to integer numbers is dropped. We omit this proof and proceed with some stronger non-approximability results. Theorem 6 Non-approximability of FCLP For any polynomial time computable function α(n), the existence of an (α(n),1,1)-approximation algorithm for FCLP on series–parallel graphs with n nodes implies P=NP. Proof We show the hardness by a reduction from Knapsack [3, Problem MP9]. An instance of K
Approximation on series–parallel graphs
This section provides an exact algorithm on series–parallel graphs with pseudopolynomial running time which uses a dynamic programming scheme. In the following section we use a scaling technique to evolve a polynomial running time approximation algorithm.
Acknowledgements
This work is supported by the Deutsche Forschungsgemeinschaft (DFG), Grant No. 88/15–3.
References (6)
- et al.
Networks Flows
(1993) - U. Feige, A threshold of lnn for approximating set cover, in: Proceedings of the 28th Annual ACM Symposium on the...
- et al.
Computers and Intractability (A Guide to the Theory of NP-Completeness)
(1979)
Cited by (11)
The p-median problem with upgrading of transportation costs and minimum travel time allocation
2023, Computers and Operations ResearchUpgrading edges in the Graphical TSP
2023, Computers and Operations ResearchUpgrading edges in the maximal covering location problem
2022, European Journal of Operational ResearchCitation Excerpt :Next, we briefly review the literature of upgrading problems. The upgrading version of many classical problems has been studied during the last decades, e.g. for the spanning tree problem (Álvarez-Miranda & Sinnl, 2017), for the min-max spanning tree problem (Sepasian & Monabbati, 2017), for bottleneck problems (Burkard, Lin, & Zhang, 2004a), for minimum flow cost problems (Demgensky, Noltemeier, & Wirth, 2002), for the shortest path problem (Dilkina, Lai, & Gomes, 2011), for the maximal shortest path interdiction problem (Zhang, Guan, & Pardalos, 2021), or for communication and signal flow problems (Paik & Sahni, 1995). For sake of clarity, we summarise the cited literature of upgrading problems in Table 1.
Erratum: Minimum cost flows with minimum quantities (Information Processing Letters (2011) 111: 11 (533-537))
2012, Information Processing LettersMinimum cost flows with minimum quantities
2011, Information Processing Letters