Approximation schemes for r-weighted Minimization Knapsack problems
- 22 Downloads
Stimulated by salient applications arising from power systems, this paper studies a class of non-linear Knapsack problems with non-separable quadratic constrains, formulated in either binary or integer form. These problems resemble the duals of the corresponding variants of 2-weighted Knapsack problem (a.k.a., complex-demand Knapsack problem) which has been studied in the extant literature under the paradigm of smart grids. Nevertheless, the employed techniques resulting in a polynomial-time approximation scheme (PTAS) for the 2-weighted Knapsack problem are not amenable to its minimization version. We instead propose a greedy geometry-based approach that arrives at a quasi PTAS (QPTAS) for the minimization variant with boolean variables. As for the integer formulation, a linear programming-based method is developed that obtains a PTAS. In view of the curse of dimensionality, fast greedy heuristic algorithms are presented, additionally to QPTAS. Their performance is corroborated extensively by empirical simulations under diverse settings and scenarios.
KeywordsWeighted Minimization Knapsack Quasi polynomial-time approximation scheme Polynomial-time approximation scheme Power generation planning Smart grid Economic dispatch control
We would like to thank the Editor and anonymous reviewers for their careful reading of our manuscript and their helpful comments that improved the presentation of the paper.
- Chau, S. C., Elbassioni, K. M., & Khonji, M. (2014). Truthful mechanisms for combinatorial AC electric power allocation. In International conference on autonomous agents and multi-agent systems, AAMAS ’14 (pp. 1005–1012), Paris, France, 5–9 May 2014.Google Scholar
- Csirik, J., Frenk, J. B. G., Labbé, M., & Zhang, S. (1991). Heuristic for the 0–1 min-knapsack problem. Acta Cybernetica, 10(1–2), 15–20.Google Scholar
- Garey, M., & Johnson, D. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: W.H. Freeman.Google Scholar
- Hedengren, J. D. (2014). APMonitor Modeling Language. http://APMonitor.com. Accessed 18 Aug 2017.
- Ibaraki, T., & Katoh, N. (1988). Resource allocation problems. Cambridge, MA: MIT Press.Google Scholar
- Khonji, M., Karapetyan, A., Elbassioni, K., & Chau, C. K. (2016). Complex-demand scheduling problem with application in smart grid. In Computing and combinatorics (pp. 496–509). Berlin: Springer.Google Scholar
- Nemhauser, G. L., & Wolsey, L. A. (1999). Integer and combinatorial optimization. New York, NY: Wiley-Interscience.Google Scholar
- Pferschy, U., & Schauer, J. (2013). Approximating the quadratic knapsack problem on special graph classes. In Approximation and online algorithms—11th international workshop, WAOA 2013 (pp. 61–72), Sophia Antipolis, France, September 5–6, 2013, Revised selected papers.Google Scholar
- Schrijver, A. (1986). Theory of linear and integer programming. New York: Wiley.Google Scholar
- Wood, A. J., & Wollenberg, B. F. (2012). Power generation, operation, and control. London: Wiley.Google Scholar
- Yu, L., & Chau, C. (2013). Complex-demand knapsack problems and incentives in AC power systems. In International conference on autonomous agents and multi-agent systems, AAMAS ’13 (pp. 973–980), Saint Paul, MN, USA, May 6–10, 2013.Google Scholar