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.
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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 \(\sum _{i\in S}a_i=\sum _{i\not \in S}a_i=k?\).
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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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