# Robust balanced optimization

- 9 Downloads

## Abstract

An instance of a balanced optimization problem with vector costs consists of a ground set *X*, a cost-vector for every element of *X*, and a system of feasible subsets over *X*. The goal is to find a feasible subset that minimizes the so-called imbalance of values in every coordinate of the underlying vector costs. Balanced optimization problems with vector costs are equivalent to the robust optimization version of balanced optimization problems under the min-max criterion. We regard these problems as a family of optimization problems; one particular member of this family is the known balanced assignment problem. We investigate the complexity and approximability of robust balanced optimization problems in a fairly general setting. We identify a large family of problems that admit a 2-approximation in polynomial time, and we show that for many problems in this family this approximation factor 2 is best-possible (unless P = NP). We pay special attention to the balanced assignment problem with vector costs and show that this problem is NP-hard even in the highly restricted case of sum costs. We conclude by performing computational experiments for this problem.

### Keywords

Balanced optimization Assignment problem Computational complexity Approximation### Mathematics Subject Classification

90C27## Notes

### Acknowledgements

We thank Marc Goerigk for interesting discussions on the relation between balanced optimization problems with vector costs and robust optimization. We are also indebted to the reviewers whose remarks led to a significant speedup of some of the algorithms in this work. This research has been supported by the Netherlands Organisation for Scientific Research (NWO) under Grant 639.033.403, by BSIK Grant 03018 (BRICKS: Basic Research in Informatics for Creating the Knowledge Society), and by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.

### References

- Ahuja R (1997) The balanced linear programming problem. Eur J Oper Res 101(1):29–38CrossRefGoogle Scholar
- Aissi H, Bazgan C, Vanderpooten D (2005) Complexity of the min-max and min-max regret assignment problems. Oper Res Lett 33(6):634–640CrossRefGoogle Scholar
- Aissi H, Bazgan C, Vanderpooten D (2009) Min-max and min-max regret versions of combinatorial optimization problems: a survey. Eur J Oper Res 197(2):427–438CrossRefGoogle Scholar
- Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805CrossRefGoogle Scholar
- Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13CrossRefGoogle Scholar
- Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88(3):411–424CrossRefGoogle Scholar
- Ben-Tal A, Golany B, Nemirovski A, Vial J-P (2005) Retailer-supplier flexible commitments contracts: a robust optimization approach. Manuf Serv Oper Manag 7(3):248–271CrossRefGoogle Scholar
- Ben-Tal A, Boyd S, Nemirovski A (2006) Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math Program 107(1–2):63–89CrossRefGoogle Scholar
- Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, PrincetonCrossRefGoogle Scholar
- Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98(1):49–71CrossRefGoogle Scholar
- Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501CrossRefGoogle Scholar
- Camerini P, Maffioli F, Martello S, Toth P (1986) Most and least uniform spanning trees. Discrete Appl Math 15(2–3):181–197CrossRefGoogle Scholar
- Cappanera P, Scutellà M (2005) Balanced paths in acyclic networks: tractable cases and related approaches. Networks 45(2):104–111CrossRefGoogle Scholar
- Deineko V, Woeginger G (2006) On the robust assignment problem under a fixed number of cost scenarios. Oper Res Lett 34:175–179CrossRefGoogle Scholar
- Dokka T, Crama Y, Spieksma F (2014) Multi-dimensional vector assignment problems. Discrete Optim 14:111–125CrossRefGoogle Scholar
- Dunning I, Huchette J, Lubin M (2017) Jump: a modeling language for mathematical optimization. SIAM Rev 59(2):295–320CrossRefGoogle Scholar
- Ficker AMC, Spieksma FCR, Woeginger GJ (2018) Robust balanced optimization, KU Leuven, FEB Research report KBI_1802Google Scholar
- Gabrel V, Murat C, Thiele A (2014) Recent advances in robust optimization: an overview. Eur J Oper Res 235(3):471–483CrossRefGoogle Scholar
- Galil Z, Schieber B (1988) On finding most uniform spanning trees. Discrete Appl Math 20(2):173–175CrossRefGoogle Scholar
- Gorissen BL, Yanıkoğlu İ, den Hertog D (2015) A practical guide to robust optimization. OMEGA 53:124–137CrossRefGoogle Scholar
- Kamura Y, Nakamori M (2014) Modified balanced assignment problem in vector case: System construction problem, In: 2014 international conference on computational science and computational intelligence (CSCI), vol 2. IEEE, pp 52–56Google Scholar
- Katoh N, Iwano K (1994) Efficient algorithms for minimum range cut problems. Networks 24(7):395–407CrossRefGoogle Scholar
- Kinable J, Smeulders B, Delcour E, Spieksma F (2017) Exact algorithms for the equitable traveling salesman problem. Eur J Oper Res 261(2):475–485CrossRefGoogle Scholar
- Koster AM, Kutschka M, Raack C (2013) Robust network design: formulations, valid inequalities, and computations. Networks 61(2):128–149CrossRefGoogle Scholar
- Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, NorwellCrossRefGoogle Scholar
- Larusic J, Punnen A (2011) The balanced traveling salesman problem. Comput Oper Res 38(5):868–875CrossRefGoogle Scholar
- Lee C, Lee K, Park K, Park S (2012) Branch-and-price-and-cut approach to the robust network design problem without flow bifurcations. Oper Res 60(3):604–610CrossRefGoogle Scholar
- Martello S, Pulleyblank W, Toth P, De Werra D (1984) Balanced optimization problems. Oper Res Lett 3(5):275–278CrossRefGoogle Scholar
- Poss M (2014) Robust combinatorial optimization with variable cost uncertainty. Eur J Oper Res 237:836–845CrossRefGoogle Scholar
- Punnen A, Nair K (1999) Constrained balanced optimization problems. Comput Math Appl 37(9):157–163CrossRefGoogle Scholar
- Turner L (2012) Variants of shortest path problems. Algorithmic Oper Res 6(2):91–104Google Scholar
- Wiesemann W, Kuhn D, Sim M (2014) Distributionally robust convex optimization. Oper Res 62(6):1358–1376CrossRefGoogle Scholar