On the flow cost lowering problem

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Abstract

This paper presents the flow cost lowering problem (FCLP), which is an extension to the integral version of the well-known minimum cost flow problem (MCFP). While in the MCFP the flow costs are fixed, the FCLP admits lowering the flow cost on each arc by upgrading the arc. Given a flow value and a bound on the total budget which can be used for upgrading the arcs, the goal is to find an upgrade strategy and a flow of minimum cost. The FCLP is shown to be NP-hard even on series–parallel graphs. On the other hand the paper provides a polynomial time approximation algorithm on series–parallel graphs.

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 δ,ε>0 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,tV, arc capacities u:R→N, a cost function c:R→N, and a flow value F∈N, find a feasible flow x:R→N0 of value F from s to t of minimum costr∈Rx(r)c(r).

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 byx∈Γ:∀i=2,…,k:fi(x)⩽[⩾]FiifFiisupper[lower]bound.

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.

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