Discrete & Computational Geometry

, Volume 56, Issue 4, pp 882–909

# On the Shadow Simplex Method for Curved Polyhedra

• Nicolai Hähnle
Article

## Abstract

We study the simplex method over polyhedra satisfying certain “discrete curvature” lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al. (Discrete Comput. Geom. 52(1):102–115, 2014), Brunsch and Röglin (Automata, languages, and programming. Part I, pp. 279–290, Springer, Heidelberg, 2013), and Eisenbrand and Vempala (http://arxiv.org/abs/1404.1568, 2014) have improved our understanding of such polyhedra. We develop a new type of dual analysis of the shadow simplex method which provides a clean and powerful tool for improving all previously mentioned results. Our methods are inspired by the recent work of Bonifas and the first named author (in: Indyk P (ed) Proceedings of the Twenty-Sixth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 295–314, SIAM, 2015), who analyzed a remarkably similar process as part of an algorithm for the Closest Vector Problem with Preprocessing. For our first result, we obtain a constructive diameter bound of $$O(\frac{n^2}{\delta } \ln \frac{n}{\delta })$$ for n-dimensional polyhedra with curvature parameter $$\delta \in (0,1]$$. For the class of polyhedra arising from totally unimodular constraint matrices, this implies a bound of $$O(n^3 \ln n)$$. For linear optimization, given an initial feasible vertex, we show that an optimal vertex can be found using an expected $$O(\frac{n^3}{\delta } \ln \frac{n}{\delta })$$ simplex pivots, each requiring O(mn) time to compute, where m is the number of constraints. An initial feasible solution can be found using $$O(\frac{m n^3}{\delta } \ln \frac{n}{\delta })$$ pivot steps.

## Keywords

Optimization Linear programming Simplex method Diameter of polyhedra

## Notes

### Acknowledgments

We would like to thank Friedrich Eisenbrand and Santosh Vempala for useful discussions. We also thank the anonymous reviewers for their useful comments.

## References

1. 1.
Adiprasito, K.A., Benedetti, B.: The Hirsch conjecture holds for normal flag complexes. Math. Oper. Res. 39(4), 1340–1348 (2014)
2. 2.
Balinski, M.L.: The Hirsch conjecture for dual transportation polyhedra. Math. Oper. Res. 9(4), 629–633 (1984)
3. 3.
Barnette, David: An upper bound for the diameter of a polytope. Discrete Math. 10, 9–13 (1974)
4. 4.
Bonifas, N., Di Summa, M., Eisenbrand, F., Hähnle, N., Niemeier, M.: On sub-determinants and the diameter of polyhedra. Discrete Comput. Geom. 52(1), 102–115 (2014). Preliminary version in SOCG 12
5. 5.
Borgwardt, K.-H.: The Simplex Method: A Probabilistic Analysis. Algorithms and Combinatorics: Study and Research Texts. Springer, Berlin (1987)Google Scholar
6. 6.
Brightwell, G., van den Heuvel, J., Stougie, J.: A linear bound on the diameter of the transportation polytope. Combinatorica 26(2), 133–139 (2006)
7. 7.
Brunsch, T., Röglin, H.: Finding short paths on polytopes by the shadow vertex algorithm. Automata, Languages, And Programming, Part I. Lecture Notes in Computer Science, pp. 279–290. Springer, Heidelberg (2013)Google Scholar
8. 8.
Dadush, D., Bonifas, N.: Short paths on the Voronoi graph and closest vector problem with preprocessing. In: Indyk, P. (eds) Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, 4–6 Jan 2015, pp. 295–314. SIAM, 2015Google Scholar
9. 9.
De Loera, J.A., Kim, E.D., Onn, S., Santos, F.: Graphs of transportation polytopes. J. Combin. Theory Ser. A 116(8), 1306–1325 (2009)
10. 10.
Dyer, M., Frieze, A.: Random walks, totally unimodular matrices, and a randomised dual simplex algorithm. Math. Program. 64(1, Ser. A), 1–16 (1994)
11. 11.
Eisenbrand, F., Vempala, S.: Geometric random edge. arXiv:1404.1568 (2014)
12. 12.
Kalai, G.: The diameter of graphs of convex polytopes and $$f$$-vector theory. Applied Geometry and Discrete Mathematics. DIMACS Series Discrete Mathematics and Theoretical Computer Science, pp. 387–411. American Mathematical Society, Providence, RI (1991)Google Scholar
13. 13.
Kalai, G., Kleitman, D.J.: A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Am. Math. Soc. (N.S.) 26(2), 315–316 (1992)
14. 14.
Larman, D.G.: Paths of polytopes. Proc. Lond. Math. Soc. 3(20), 161–178 (1970)
15. 15.
Matschke, B., Santos, F., Weibel, C.: The width of 5-dimensional prismatoids. arXiv:1202.4701 (2013)
16. 16.
Naddef, D.: The Hirsch conjecture is true for (0,1)-polytopes. Math. Program. 45(1, Ser. B), 109–110 (1989)
17. 17.
Santos, F.: A counterexample to the Hirsch conjecture. Ann. Math. 176(1), 383–412 (2012)
18. 18.
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
19. 19.
Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)
20. 20.
Todd, M.J.: An improved Kalai-Kleitman bound for the diameter of a polyhedron. SIAM J. Discrete Math. 28(4), 1944–1947 (2014)
21. 21.
Vershynin, R.: Beyond Hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. SIAM J. Comput. 39(2), 646–678 (2009)