Advances on Strictly $\Delta$-Modular IPs

TL;DR

This paper advances the understanding of strictly $ ext{Delta}$-modular integer programs by developing a randomized strongly polynomial-time feasibility algorithm for $ ext{Delta}\

Contribution

It introduces new techniques that go beyond prime $ ext{Delta}$ cases, enabling feasibility checks for $ ext{Delta}\

Findings

Feasibility testing for strictly $ ext{Delta}$-modular IPs is possible in strongly polynomial time for $ ext{Delta}\

The approach works for $ ext{Delta}\

The work extends previous results beyond prime $ ext{Delta}$.

Abstract

There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $\Delta\in \mathbb{Z}_{>0}$, $\Delta$-modular IPs are efficiently solvable, which are IPs where the constraint matrix $A\in \mathbb{Z}^{m\times n}$ has full column rank and all $n\times n$ minors of $A$ are within $\{-\Delta, \dots, \Delta\}$. Previous progress on this question, in particular for $\Delta=2$, relies on algorithms that solve an important special case, namely strictly $\Delta$-modular IPs, which further restrict the $n\times n$ minors of $A$ to be within $\{-\Delta, 0, \Delta\}$. Even for $\Delta=2$, such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $\Delta$-modular IPs. Prior advances were restricted to prime $\Delta$, which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly $\Delta$-modular IPs in strongly polynomial time if $\Delta\leq4$.