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A novel approach to subgraph selection with multiple weights on arcs

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Abstract

In this paper, an extension of the minimum cost flow problem is considered in which multiple incommensurate weights are associated with each arc. In the minimum cost flow problem, flow is sent over the arcs of a graph from source nodes to sink nodes. The goal is to select a subgraph with minimum associated costs for routing the flow. The problem is tractable when a single weight is given on each arc. However, in many real-world applications, several weights are needed to describe the features of arcs, including transit cost, arrival time, delay, profit, security, reliability, deterioration, and safety. In this case, finding an optimal solution becomes difficult. We propose a heuristic algorithm for this purpose. First, we compute the relative efficiency of the arcs by using data envelopment analysis techniques. We then determine a subgraph with efficient arcs using a linear programming model, where the objective function is based on the relative efficiency of the arcs. The flow obtained satisfies the arc capacity constraints and the integrality property. Our proposed algorithm has polynomial runtime and is evaluated in rigorous experiments.

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Notes

  1. This sentence is a direct quote from Cooper et al. (2006).

  2. See GAMS—A User’s Guide, Frechen, Germany: GAMS Development Corporation, 2021.

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Funding

We acknowledge funding support by the National Natural Science Foundation of China under Grants 61673359 and 71520107002. P.M. Pardalos was conducted within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE).

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Dr. Raayatpanah has conceptualized and conducted the research. Dr. Khodayifar has helped developing the mathematical modules. Dr. Weise and Dr. Pardalaos both have helped writing, rewriting, and revising the manuscript.

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Correspondence to Mohammad Ali Raayatpanah.

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Raayatpanah, M.A., Khodayifar, S., Weise, T. et al. A novel approach to subgraph selection with multiple weights on arcs. J Comb Optim 44, 242–268 (2022). https://doi.org/10.1007/s10878-021-00823-0

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  • DOI: https://doi.org/10.1007/s10878-021-00823-0

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