Redundancies in Linear Systems with two Variables per Inequality

Abstract

The problem of detecting and removing redundant constraints is fundamental in optimization. We focus on the case of linear programs (LPs), given by $d$ variables with $n$ inequality constraints. A constraint is called \emph{redundant}, if after its removal, the LP still has the same feasible region. The currently fastest method to detect all redundancies is due to Clarkson: it solves $n$ linear programs, but each of them has at most $s$ constraints, where $s$ is the number of nonredundant constraints. In this paper, we study the special case where every constraint has at most two variables with nonzero coefficients. This family, denoted by $LI(2)$, has some nice properties. Namely, as shown by Aspvall and Shiloach, given a variable $x_i$ and a value $\lambda$, we can test in time $O(nd)$ whether there is a feasible solution with $x_i = \lambda$. Hochbaum and Naor present an $O(d^2 n \log n)$ algorithm for solving the feasibility problem in $LI(2)$. Their technique makes use of the Fourier-Motzkin elimination method and the earlier mentioned result by Aspvall and Shiloach. We present a strongly polynomial algorithm that solves redundancy detection in time $O(n d^2 s \log s)$. It uses a modification of Clarkson's algorithm, together with a revised version of Hochbaum and Naor's technique. Finally we show that dimensionality testing can be done with the same running time as solving feasibility.