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# The Alternating Stock Size Problem and the Gasoline Puzzle

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## Cite As

Alantha Newman, Heiko Röglin, and Johanna Seif. The Alternating Stock Size Problem and the Gasoline Puzzle. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.71

## Abstract

Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that: (i) all prefixes of the ordering are non-negative, and (ii) the maximum value of a prefix sum is minimized. Kellerer et al. referred to this problem as the stock size problem and showed that it can be approximated to within 3/2. They also showed that an approximation ratio of 2 can be achieved via several simple algorithms. We consider a related problem, which we call the alternating stock size problem, where the number of positive and negative integers in the input set S are equal. The problem is the same as above, but we are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations. We show that it can be approximated to within 1.79. Then we show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovász, in which we want to minimize the size of the gas tank necessary to go around the track. We present a 2-approximation for this problem, using a natural linear programming relaxation whose feasible solutions are doubly stochastic matrices. Our novel rounding algorithm is based on a transformation that yields another doubly stochastic matrix with special properties, from which we can extract a suitable permutation.
##### Keywords
• approximation algorithms
• stock size problem
• scheduling with non-renewable resources

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## References

1. H. M. Abdel-Wahab and T. Kameda. Scheduling to minimize maximum cumulative cost subject to series-parallel precedence constraints. Operations Research, 26(1):141-158, 1978.
2. Wojciech Banaszczyk. The steinitz constant of the plane. Journal für die Reine und Angewandte Mathematik, 373:218-220, 1987.
3. Imre Bárány. On the power of linear dependencies. In Building bridges, pages 31-45. Springer, 2008.
4. Jacek Blazewicz, Jan Karel Lenstra, and AHG Rinnooy Kan. Scheduling subject to resource constraints: classification and complexity. Discrete Applied Mathematics, 5(1):11-24, 1983.
5. Dirk Briskorn, Byung-Cheon Choi, Kangbok Lee, Joseph Leung, and Michael Pinedo. Complexity of single machine scheduling subject to nonnegative inventory constraints. European Journal of Operational Research, 207(2):605-619, 2010.
6. J Carlier and AHG Rinnooy Kan. Scheduling subject to nonrenewable-resource constraints. Operations Research Letters, 1(2):52-55, 1982.
7. V. S. Grinberg and S. V. Sevastyanov. The value of the Steinitz constant. Functional Analysis and its Applications, 14:56-57, 1980.
8. Péter Györgyi and Tamás Kis. Approximation schemes for single machine scheduling with non-renewable resource constraints. Journal of Scheduling, 17(2):135-144, 2014.
9. Péter Györgyi and Tamás Kis. Approximability of scheduling problems with resource consuming jobs. Annals of Operations Research, 235(1):319-336, 2015.
10. Nick Harvey and Samira Samadi. Near-optimal herding. In COLT, pages 1165-1182, 2014.
11. Hans Kellerer, Vladimir Kotov, Franz Rendl, and Gerhard J. Woeginger. The stock size problem. Operations Research, 46(3):S1-S12, 1998.
12. L. Lovász. Combinatorial Problems and Exercises. North-Holland, 1979.
13. Clyde L. Monma. Sequencing to minimize the maximum job cost. Operations Research, 28(4):942-951, 1980.
14. Ehab Morsy and Erwin Pesch. Approximation algorithms for inventory constrained scheduling on a single machine. Journal of Scheduling, 18(6):645-653, 2015.
15. Klaus Neumann and Christoph Schwindt. Project scheduling with inventory constraints. Mathematical Methods of Operations Research, 56(3):513-533, 2003.
16. Alantha Newman, Heiko Röglin, and Johanna Seif. The alternating stock size problem and the gasoline puzzle. arXiv:1511.09259, 2015.
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