Computational complexity of complementary pivot methods

Abstract

The class of orthants in R ⁿ contains 2ⁿ orthants. A straight line L in R ⁿ is said to cut across an orthant C of R ⁿ if L has a nonempty intersection with the interior of C. It is well known that the maximum number of orthants of R ⁿ that a straight line in R ⁿ can cut across is n+1.

A generalization of the class of orthants of R ⁿ, known as a class of complementary cones, appears in the study of the linear complementarity problem (LCP). In a class of complementary cones there are always 2ⁿ cones. As before, a straight line L in R ⁿ is said to cut across a complementary cone C, if the interior of C has a nonempty intersection with L. We study the following geometrical question: what is the maximum number of complementary cones in a class of complementary cones for R ⁿ that a straight line in R ⁿ can cut across? We show that this number is 2ⁿ, for each n≧1.

This geometrical problem arises in the study of the computational complexity of complementary pivot methods for solving LCPs. We show that the number of pivot steps required for solving an LCP of order n by complementary pivot methods is not bounded above by a polynomial in n. For each n≧2, we construct a simple LCP of order n, for solving which the complementary pivot methods are shown to require exactly 2ⁿ pivot steps.

Klee and Minty have previously shown that the computational effort required to solve a linear program by the simplex method is not bounded above by a polynomial in the size of the linear program. Our study shows that complementary pivot methods for solving LCPs exhibit exactly similar behavior in terms of computational requirements.