Pivoting in Linear Complementarity: Two Polynomial-Time Cases

TL;DR

This paper analyzes the efficiency of principal pivoting methods for P-matrix linear complementarity problems, demonstrating quadratic complexity for some rules and linear for others, using combinatorial cube orientations.

Contribution

It resolves an open problem by proving quadratic bounds for Murty's pivot rule and establishes linear bounds for K-matrix instances, advancing understanding of pivoting algorithms.

Findings

Murty's rule can require quadratic iterations on cyclic P-LCPs.

All pivot rules need only linear iterations on K-matrix LCPs.

Unique-sink orientations of cubes are effective tools for analyzing LCP algorithms.

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.