On the Shadow Simplex Method for Curved Polyhedra

TL;DR

This paper introduces a new dual analysis of the shadow simplex method for polyhedra with discrete curvature bounds, leading to improved diameter bounds and efficient algorithms for linear optimization in such settings.

Contribution

It develops a novel dual analysis technique for the shadow simplex method, providing new diameter bounds and optimization algorithms for polyhedra with curvature constraints.

Findings

Diameter bound of O(n^2/δ log(n/δ)) for curved polyhedra.

Polyhedra from totally unimodular matrices have diameter O(n^3 log n).

Expected O(n^3/δ log(n/δ)) pivots to find optimal vertices.

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 (SOCG 2012), Brunsch and R\"oglin (ICALP 2013), and Eisenbrand and Vempala (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 (SODA 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 $O(m n)$ time to compute. An initial feasible solution can be found using $O(\frac{m n^3}{\delta} \ln \frac{n}{\delta})$ pivot steps.