Abstract
Motivated by the power allocation problem in AC (alternating current) electrical systems, we study the multi-objective (combinatorial) optimization problem where a constant number of (nonnegative) linear functions are simultaneously optimized over a given feasible set of 0–1 points defined by quadratic constraints. Such a problem is very hard to solve if no specific assumptions are made on the structure of the constraint matrices. We focus on the case when the constraint matrices are completely positive and have fixed cp-rank. We propose a polynomial-time algorithm which computes an \(\epsilon \)-Pareto curve for the studied multi-objective problem when both the number of objectives and the number of constraints are fixed, for any constant \(\epsilon >0\). This result is then applied to obtain polynomial-time approximation schemes (PTASes) for two NP-hard problems: multi-criteria power allocation and sum-of-ratios optimization.
Keywords
- Power Allocation
- Knapsack Problem
- Pareto Optimal Solution
- Quadratic Constraint
- Linear Objective Function
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Notes
- 1.
Actually, this combinatorial optimization problem was first studied by Woeginger [35] under the name 2-weighted Knapsack, in terms of inapproximability.
- 2.
The Equi-Partition is defined as follows: Given \((a_1,a_2,\ldots ,a_{2n})\in \mathbb {Z}^{2n}\) with \(\sum _{i=1}^{2n}a_i=2k\), does exist a subset \(S\subset \{1,2,\ldots ,{2n}\}\), \(|S|=n\), such that \(\sum _{i\in S}a_i=\sum _{i\not \in S}a_i=k?\).
References
Anagnostopoulos, A., Grandoni, F., Leonardi, S., Wiese, A.: A mazing \(2+\epsilon \) approximation for unsplittable flow on a path. In: SODA, pp. 26–41 (2014)
Bansal, A., Chakrabarti, N., Epstein, A., Schieber, B.: A quasi-ptas for unsplittable flow on line graphs. In: STOC, pp. 721–729 (2006)
Bansal, N., Friggstad, Z., Khandekar, R., Salavatipour, M.R.: A logarithmic approximation for unsplittable flow on line graphs. In: SODA, pp. 702–709 (2009)
Bansal, N., Korula, N., Nagarajan, V., Srinivasan, A.: Solving packing integer programs via randomized rounding with alterations. Theory of Computing 8(1), 533–565 (2012)
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. In STOC, pp. 735–744 (2000)
Bartlett, M., Frisch, A.M., Hamadi, Y., Miguel, I., Tarim, S.A., Unsworth, C.: The temporal knapsack problem and its solution. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 34–48. Springer, Heidelberg (2005)
Bazgan, C., Gourvès, L., Monnot, J.: Approximation with a fixed number of solutions of some multiobjective maximization problems. J. Discrete Algorithms 22, 19–29 (2013)
Bazgan, C., Gourvès, L., Monnot, J., Pascual, F.: Single approximation for the biobjective max TSP. Theor. Comput. Sci. 478, 41–50 (2013)
Bazgan, C., Hugot, H., Vanderpooten, D.: Solving efficiently the 0–1 multi-objective knapsack problem. Comput. & OR 36(1), 260–279 (2009)
Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, Singapore (2003)
Calinescu, G., Chakrabarti, A., Karloff, H., Rabani, Y.: Improved approximation algorithms for resource allocation. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 401–414. Springer, Heidelberg (2002)
Chakrabarti, A., Chekuri, C., Gupta, A., Kumar, A.: Approximation algorithms for the unsplittable flow problem. Algorithmica 47(1), 53–78 (2007)
Chau, C., Elbassioni, K., Khonji, M.: Truthful mechanisms for combinatorial ac electric power allocation. In: AAMAS, pp. 1005–1012 (2014)
Cheng, T., Janiak, A., Kovalyov, M.: Bicriterion single machine scheduling with resource dependent processing times. SIAM J. Optim. 8(2), 617–630 (1998)
Diakonikolas, I.: Approximation of Multiobjective Optimization Problems. PhD thesis, Deptartment of Computer Science, Columbia University, May 2011
Diakonikolas, I., Yannakakis, M.: Small approximate pareto sets for biobjective shortest paths and other problems. SIAM J. Comput. 39(4), 1340–1371 (2009)
Dickinson, P., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57(2), 403–415 (2014)
Ehrgott, M.: Multicriteria Optimization. Springer, Heidelberg (2005)
Ehrgott, M., Gandibleux, X.: Multiple Criteria Optimization: State of the Art Annotated Bibliographical Surveys. Kluwer, Boston (2002)
Elbassioni, K., Nguyen,T.T.: Approximation schemes for binary quadratic programming problems with low cp-rank decompositions. CoRR (2014). abs/1411.5050
Erlebach, T., Kellerer, H., Pferschy, U.: Approximating multiobjective knapsack problems. Manage. Sci. 48(12), 1603–1612 (2002)
Escoffier, B., Gourvès, L., Monnot, J.: Fair solutions for some multiagent optimization problems. Auton. Agent. Multi-Agent Syst. 26(2), 184–201 (2013)
Frieze, A., Clarke, M.: Approximation algorithms for the \(m\)-dimensional 0–1 knapsack problem: worst-case and probabilistic analyses. Eur. J. Oper. Res. 15, 100–109 (1984)
Gandibleux, X., Freville, A.: Tabu search based procedure for solving the 0–1 multiobjective knapsack problem: the two objective case. J. Heuristics 6, 361–383 (2000)
Grainger, J., Stevenson, W.: Power System Analysis. McGraw-Hill, New York (1994)
Hong, S.-P., Chung, S.-J., Park, B.H.: A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. Oper. Res. Lett. 32(3), 233–239 (2004)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Heidelberg (2004)
Leonardi, S., Marchetti-Spaccamela, A., Vitaletti, A.: Approximation algorithms for bandwidth and storage allocation problems under real time constraints. In: FSTTCS, pp. 409–420 (2000)
Mittal, S., Schulz, A.: A general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one. Oper. Res. 61(2), 386–397 (2013)
Nemirovski, A., Todd, M.: Interior-point methods for optimization. Acta Numerica 17, 191–234 (2008)
Papadimitriou, C., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: FOCS, pp. 86–92 (2000)
Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18, 219–229 (2003)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)
Tsaggouris, G., Zaroliagis, C.D.: Multiobjective optimization: improved FPTAS for shortest paths and non-linear objectives with applications. Theory Comput. Syst. 45(1), 162–186 (2009)
Woeginger, G.J.: When does a dynamic programming formulation guarantee the existence of an FPTAS?. In: SODA, pp. 820–829 (1999)
Woeginger, G.J.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS J. Comput. 12(1), 57–74 (2000)
Yu, L., Chau, C.: Complex-demand knapsack problems and incentives in ac power systems. In: AAMAS, pp. 973–980 (2013)
Acknowledgments
We would like to thank Gerhard Woeginger for helpful discussions, especially for pointing us the papers [35, 36]. We thank the ADT-15 reviewers for their helpful comments, suggestions and insights that have helped us improve our manuscript. This work was supported by the MI-MIT Flagship project 13CAMA1.
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Elbassioni, K., Nguyen, T.T. (2015). Approximation Schemes for Multi-objective Optimization with Quadratic Constraints of Fixed CP-Rank. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_17
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DOI: https://doi.org/10.1007/978-3-319-23114-3_17
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