Abstract
We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty’s least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris’s highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Balogh, J., Pemantle, R.: The Klee–Minty random edge chain moves with linear speed. Random Struct. Algorithms 30(4), 464–483 (2007)
Bard, Y.: An eclectic approach to nonlinear programming. In: Optimization, Proc. Sem. Austral. Nat. Univ., Canberra, 1971, pp. 116–128. University of Queensland Press, St. Lucia (1972)
Borici, A., Lüthi, H.-J.: Pricing American put options by fast solutions of the linear complementarity problem. In: Kontoghiorghes, E.J., Rustem, B., Siokos, S. (eds.) Computational Methods in Decision-Making, Economics and Finance. Applied Optimization, vol. 74. Kluwer Academic, Dordrecht (2002)
Borici, A., Lüthi, H.-J.: Fast solutions of complementarity formulations in American put pricing. J. Comput. Finance 9, 63–82 (2005)
Chandrasekaran, R.: A special case of the complementary pivot problem. Opsearch 7, 263–268 (1970)
Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. Technical report TR05-140. Electronic colloquium on computational complexity (2005)
Chung, S.-J.: NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60(3), 393–399 (1989)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Computer Science and Scientific Computing. Academic Press, New York (1992)
Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)
Develin, M.: LP-orientations of cubes and crosspolytopes. Adv. Geom. 4(4), 459–468 (2004)
Fukuda, K., Namiki, M.: On extremal behaviors of Murty’s least index method. Math. Program. Ser. A 64(3), 365–370 (1994)
Fukuda, K., Namiki, M., Tamura, A.: EP theorems and linear complementarity problems. Discrete Appl. Math. 84(1–3), 107–119 (1998)
Fukuda, K., Terlaky, T.: Linear complementarity and oriented matroids. J. Oper. Res. Soc. Japan 35(1), 45–61 (1992)
Gärtner, B.: The random-facet simplex algorithm on combinatorial cubes. Random Struct. Algorithms 20(3), 353–381 (2002)
Gärtner, B., Morris, W.D., Rüst, L.: Unique sink orientations of grids. Algorithmica 51(2), 200–235 (2008)
Holt, F., Klee, V.: A proof of the strict monotone 4-step conjecture. In: Advances in Discrete and Computational Geometry. Contemporary Mathematics, vol. 223, pp. 201–216. Am. Math. Soc., Providence (1999)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)
Khachiyan, L.G.: Polynomial algorithms in linear programming. USSR Comput. Math. Math. Phys. 20, 53–72 (1980)
Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Shisha, O. (ed.) Inequalities, vol. III, pp. 159–175. Academic Press, New York (1972)
Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. Lecture Notes in Computer Science, vol. 538. Springer, Berlin (1991)
Kojima, M., Megiddo, N., Ye, Y.: An interior point potential reduction algorithm for the linear complementarity problem. Math. Program. Ser. A 54(3), 267–279 (1992)
Lemke, C.E.: Recent results on complementarity problems. In: Nonlinear Programming, pp. 349–384. Academic Press, New York (1970)
Megiddo, N.: A note on the complexity of P-matrix LCP and computing an equilibrium. RJ 6439, IBM Research, Almaden Research Center, San Jose, CA (1988)
Morris, W.D. Jr.: Randomized pivot algorithms for P-matrix linear complementarity problems. Math. Program. Ser. A 92(2), 285–296 (2002)
Murty, K.G.: Note on a Bard-type scheme for solving the complementarity problem. Opsearch 11(2–3), 123–130 (1974)
Murty, K.G.: Computational complexity of complementary pivot methods. Math. Program. Stud. 7, 61–73 (1978)
Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Sigma Series in Applied Mathematics, vol. 3. Heldermann, Berlin (1988)
Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)
Saigal, R.: A note on a special linear complementarity problem. Opsearch 7, 175–183 (1970)
Samelson, H., Thrall, R.M., Wesler, O.: A partition theorem for Euclidean n-space. Proc. Am. Math. Soc. 9, 805–807 (1958)
Stickney, A., Watson, L.: Digraph models of Bard-type algorithms for the linear complementarity problem. Math. Oper. Res. 3(4), 322–333 (1978)
Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS’01), pp. 547–555 (2001)
Wessendorp, F.: Notes on Morris’ cube orientation. Diploma thesis, ETH, Zürich (March 2001)
Zoutendijk, G.: Methods of Feasible Directions: A Study in Linear and Non-Linear Programming. Elsevier, Amsterdam (1960)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Foniok, J., Fukuda, K., Gärtner, B. et al. Pivoting in Linear Complementarity: Two Polynomial-Time Cases. Discrete Comput Geom 42, 187–205 (2009). https://doi.org/10.1007/s00454-009-9182-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-009-9182-2