Abstract
In the resource constrained shortest path problem we are given a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resources with finite budget limits. A minimum cost source-destination path is sought such that the total consumption of the arcs from each resource does not exceed its budget limit. In the case of constant number of weight functions we give a fully polynomial time multi-criteria approximation scheme for the problem which returns a source-destination path of cost at most the optimum, however, the path may slightly violate the budget limits. On the negative side, we show that there does not exist a polynomial time multi-criteria approximation scheme for the problem if the number of weight functions is not a constant. The latter result applies to a broad class of problems as well, including the multi-dimensional knapsack, the multi-budgeted spanning tree, the multi-budgeted matroid basis and the multi-budgeted bipartite perfect matching problems.
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References
Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)
Ergun, F., Sinha, R., Zhang, L.: An improved FPTAS for restricted shortest path. Information Processing Letters 83(5), 287–291 (2002)
Garroppo, R.G., Giordano, S., Tavanti, L.: A survey on multi-constrained optimal path computation: exact and approximate algorithms. Comput. Netw. 54(17), 3081–3107 (2010)
Gary, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. WH Freeman and Company, New York (1979)
Goel, A., Ramakrishnan, K.G., Kataria, D., Logothetis, D.: Efficient computation of delay-sensitive routes from one source to all destinations. In: INFOCOM 2001. Twentieth Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE, vol 2, pp. 854–858. IEEE (2001)
Grandoni, F., Ravi, R., Singh, M., Zenklusen, R.: New approaches to multi-objective optimization. Math. Program. 146(1–2), 525–554 (2014)
Hassin, R.: Approximation schemes for the restricted shortest path problem. Math. Oper. Res. 17(1), 36–42 (1992)
Hastad, J.: Clique is hard to approximate within \(n^{1-\varepsilon }\). In: Foundations of Computer Science, 1996. Proceedings, 37th Annual Symposium on IEEE, pp. 627–636. IEEE (1996)
Lorenz, D.H., Raz, D.: A simple efficient approximation scheme for the restricted shortest path problem. Oper. Res. Lett. 28(5), 213–219 (2001)
Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on IEEE, pp. 86–92. IEEE (2000)
Song, M., Sahni, S.: Approximation algorithms for multiconstrained quality-of-service routing. IEEE Trans. Comput. 55(5), 603–617 (2006)
Weingartner, H.M., Ness, D.N.: Methods for the solution of the multidimensional 0/1 knapsack problem. Oper. Res. 15(1), 83–103 (1967)
Acknowledgements
The authors are grateful to an anonymous referee for constructive comments that helped to improve the presentation of the paper. This work has been supported by the OTKA Grant K112881, and by the Grant GINOP-2.3.2-15-2016-00002 of the Ministry of National Economy of Hungary.
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Horváth, M., Kis, T. Multi-criteria approximation schemes for the resource constrained shortest path problem. Optim Lett 12, 475–483 (2018). https://doi.org/10.1007/s11590-017-1212-z
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DOI: https://doi.org/10.1007/s11590-017-1212-z