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# Combinatorial Redundancy Detection

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LIPIcs.SOCG.2015.315.pdf
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## Cite As

Komei Fukuda, Bernd Gärtner, and May Szedlák. Combinatorial Redundancy Detection. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 315-328, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.SOCG.2015.315

## Abstract

The problem of detecting and removing redundant constraints is fundamental in optimization. We focus on the case of linear programs (LPs) in dictionary form, given by n equality constraints in n+d variables, where the variables are constrained to be nonnegative. A variable x_r is called redundant, if after removing its nonnegativity constraint the LP still has the same feasible region. The time needed to solve such an LP is denoted by LP(n,d). It is easy to see that solving n+d LPs of the above size is sufficient to detect all redundancies. The currently fastest practical method is the one by Clarkson: it solves n+d linear programs, but each of them has at most s variables, where s is the number of nonredundant constraints. In the first part we show that knowing all of the finitely many dictionaries of the LP is sufficient for the purpose of redundancy detection. A dictionary is a matrix that can be thought of as an enriched encoding of a vertex in the LP. Moreover - and this is the combinatorial aspect - it is enough to know only the signs of the entries, the actual values do not matter. Concretely we show that for any variable x_r one can find a dictionary, such that its sign pattern is either a redundancy or nonredundancy certificate for x_r. In the second part we show that considering only the sign patterns of the dictionary, there is an output sensitive algorithm of running time of order d (n+d) s^{d-1} LP(s,d) + d s^{d} LP(n,d) to detect all redundancies. In the case where all constraints are in general position, the running time is of order s LP(n,d) + (n+d) LP(s,d), which is essentially the running time of the Clarkson method. Our algorithm extends naturally to a more general setting of arrangements of oriented topological hyperplane arrangements.
##### Keywords
• system of linear inequalities
• redundancy removal
• linear programming
• output sensitive algorithm
• Clarkson’s method

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## References

1. A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler. Oriented Matroids. Cambridge University Press, 1993.
2. T. M. Chan. Output-sensitive results on convex hulls, extreme points, and related problems. Discrete & Computational Geometry, 16(4):369-387, 1996.
3. V. Chvatal. Linear Programming. W. H. Freeman, 1983.
4. K. L. Clarkson. More output-sensitive geometric algorithms. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., pages 695-702, 1994.
5. G. B. Dantzig. Linear Programming and Extensions. Princeton University Press, Princeton, NJ, 1963.
6. J. H. Dulá, R. V. Helgason, and N. Venugopal. An algorithm for identifying the frame of a pointed finite conical hull. INFORMS J. Comput., 10(3):323-330, 1998.
7. K. Fukuda. Introduction to optimization. http://www.ifor.math.ethz.ch/teaching/Courses/Fall_2011/intro_fall_11, 2011.
8. K. Fukuda. Walking on the arrangement, not on the feasible region. Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?, IPAM, UCLA, 2011. presentation slides available as http://helper.ipam.ucla.edu/publications/sm2011/sm2011_9630.pdf. .
9. K. Fukuda. Lecture: Polyhedral computation. http://www-oldurls.inf.ethz.ch/personal/fukudak/lect/pclect/notes2015/, 2015.
10. K. Fukuda, B. Gärtner, and M. Szedlák. Combinatorial redundancy removal. Preprint: arXiv:1412.1241, 2014.
11. K. Fukuda and T. Terlaky. Criss-cross methods: A fresh view on pivot algorithms. Mathematical Programming, 79:369-395, 1997.
12. V. Klee and G. J. Minty. How good is the simplex algorithm? In O. Shisha, editor, Inequalities III, pages 159-175. Academic Press, 1972.
13. J. Matoušek, M. Sharir, and E. Welzl. A subexponential bound for linear programming. Algorithmica, 16:498-516, 1996.
14. Th. Ottmann, S. Schuierer, and S. Soundaralakshmi. Enumerating extreme points in higher dimensions. In E.W. Mayer and C. Puech, editors, STACS 95: 12th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 900, pages 562-570. Springer-Verlag, 1995.
15. C. Roos. An exponential example for Terlaky’s pivoting rule for the criss-cross simplex method. Mathematical Programming, 46:79-84, 1990.
16. T. Terlaky. A finite criss-cross method for the oriented matroids. Journal of Combinatorial Theory Series B, 42:319-327, 1987.
17. Z. Wang. A finite conformal-elimination free algorithm over oriented matroid programming. Chinese Annals of Math., 8B:120-125, 1987.