Abstract
We study a 0–1 knapsack problem, in which the objective value is forbidden to take some values. We call gaps related forbidden intervals. The problem is NP-hard and pseudo-polynomially solvable independently on the measure of gaps. If the gaps are large, then the problem is polynomially non-approximable. A non-trivial special case with respect to the approximate solution appears when the gaps are small and polynomially close to zero. For this case, two fully polynomial time approximation schemes are proposed. The results can be extended for the constrained longest path problem and other combinatorial problems.
Similar content being viewed by others
References
Abboud, A., Lewi, K.: Exact weight subgraphs and the k-sum conjecture. In: Fomin, F.V., et al. (eds.) ICALP 2013. Part I, LNCS 7965, pp. 1–12. Springer-Verlag, Berlin Heidelberg (2013)
Barahona, F., Pulleyblank, W.R.: Exact aborescences, matchings and cycles. Discrete Appl. Math. 16, 91–99 (1987)
Halman, N., Nannicini, G., Orlin, J.: A computationally efficient FPTAS for convex stochastic dynamic programs. SIAM J. Optim. 25(1), 317–350 (2015)
Hassin, R.: Approximation schemes for the restricted shortest path problem. Math. Oper. Res. 17, 36–42 (1992)
Joksch, H.C.: The shortest route problem with constraints. J. Math. Anal. Appl. 14, 191–197 (1966)
Karzanov, A.V.: Maximum matching of given weight in complete and complete bipartite graphs. Cybernetics 23, 8–13 (1987). (Translation from Kibernetika 1, 7–11 (1987), in Russian)
Kellerer, H., Pferschy, U.: A new fully polynomial time approximation scheme for the knapsack problem. J. Comb. Optim. 3, 59–71 (1999)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)
Leclerc, M.: Polynomial time algorithms for exact matching problems, Masters Thesis, University of Waterloo, Waterloo (1986)
Li, W., Li, J.: Approximation algorithms for k-partitioning problems with partition matroid constraint. Optim. Lett. 8(3), 1093–1099 (2014)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York (1990)
Milanic, M., Monnot, J.: The exact weighted independent set problem in perfect graphs and related graph classes. Electron. Notes Discrete Math. 35, 317–322 (2009)
Nguyen, T.H.C., Richard, P., Grolleau, E.: An FPTAS for response time analysis of fixed priority real-time tasks with resource augmentation. IEEE Trans. Comput. 64(7), 1805–1818 (2015)
Papadimitriou, C.H., Yannakakis, M.: The complexity of restricted spanning tree problems. J. ACM 29, 285–309 (1982)
Sahni, S.: General techniques for combinatorial approximation. Oper. Res. 25(6), 920–936 (1977)
Sahni, S., Horowitz, E.: Combinatorial problems: reducibility and approximation. Oper. Res. 26, 718–759 (1978)
Samanta, R., Erzin, A.I., Raha, S., Shamardin, Y.V., Takhonov, I.I., Zalyubovskiy, V.V.: A provably tight delay-driven concurrently congestion mitigating global routing algorithm. Appl. Math. Comput. 255, 92–104 (2015)
Schemeleva, K., Delorme, X., Dolgui, A., Grimaud, F., Kovalyov, M.Y.: Lot-sizing on a single imperfect machine: ILP models and FPTAS extensions. Comput. Ind. Eng. 65(4), 561–569 (2013)
Vassilevska, V., Williams, R.: Finding, minimizing, and counting weighted subgraphs. In: Mitzenmacher, M. (ed.), STOC, pp. 455–464. Association for Computing Machinery (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dolgui, A., Kovalyov, M.Y. & Quilliot, A. Knapsack problem with objective value gaps. Optim Lett 11, 31–39 (2017). https://doi.org/10.1007/s11590-016-1043-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-016-1043-3