Complexity of LP in Terms of the Face Lattice

TL;DR

This paper investigates the complexity of linear programming problems based on the face lattice of the associated polytope, showing that algorithms relying solely on combinatorial structure and encoding size can require exponential resources.

Contribution

It demonstrates that for any fixed dimension and size, there exist problems that are computationally hard for algorithms based only on combinatorial data and encoding size.

Findings

Algorithms based solely on combinatorial structure can require exponential time.

Existence of polynomially solvable problems with specific parameters.

Complexity depends critically on the face lattice structure of the polytope.

Abstract

Let $X$ be a finite set in $Z^d$. We consider the problem of optimizing linear function $f(x) = c^T x$ on $X$, where $c\in Z^d$ is an input vector. We call it a problem $X$. A problem $X$ is related with linear program $\max\limits_{x \in P} f(x)$, where polytope $P$ is a convex hull of $X$. The key parameters for evaluating the complexity of a problem $X$ are the dimension $d$, the cardinality $|X|$, and the encoding size $S(X) = \log_2 \left(\max\limits_{x\in X} \|x\|_{\infty}\right)$. We show that if the (time and space) complexity of some algorithm $A$ for solving a problem $X$ is defined only in terms of combinatorial structure of $P$ and the size $S(X)$, then for every $d$ and $n$ there exists polynomially (in $d$, $\log n$, and $S$) solvable problem $Y$ with $\dim Y = d$, $|Y| = n$, such that the algorithm $A$ requires exponential time or space for solving $Y$.