Better and Simpler Reducibility Bounds over the Integers
Asaf Levin
TL;DR
The paper addresses how to replace a function $f:\mathbb{Z}^n\to\mathbb{Z}$ defined on a box $[-N,N]^n$ with an equivalent function $g$ that has a smaller domain and bounded gap, enabling strongly polynomial-time reductions of weakly polynomial algorithms. It develops a scalable linear-programming framework and leverages the Frank–Tardos results to obtain explicit existence bounds—$\rho_{lin}$ for linear, $\rho_{sep}$ for separable, $\rho_{sep-quad}$ for separable quadratic, and $\rho_{quad}$ for general quadratic—and, in several cases, computable bounds with polynomial-time construction of $g$ (e.g., for separable quadratic and quadratic cases). The main contributions are improved upper bounds on the gap relative to prior work (notably surpassing prior bounds in Eisen23 for several function classes) and new constructive reductions with polynomial-time algorithms in key regimes. The significance lies in enabling efficient transformation of weakly polynomial-time integer programs into strongly polynomial-time ones, expanding the practical impact of reducibility techniques in optimization and algorithms.
Abstract
We study the settings where we are given a function of n variables defined in a given box of integers. We show that in many cases we can replace the given objective function by a new function with a much smaller domain. Our approach allows us to transform a family of weakly polynomial time algorithms into strongly polynomial time algorithms.