Integer programming on polytopes of Chvátal rank one is as hard as lattice problems

2026-10-07

AI Summary

This study addresses the longstanding open problem, unresolved since the 1980s, of whether the integer feasibility problem for polyhedra with Chvátal rank one can be solved in polynomial time. By integrating Chvátal-Gomory cutting planes, unimodular matrix width analysis, and lattice-based cryptographic reductions, this work establishes, for the first time, a computational equivalence between this class of integer programming problems and the Bounded Distance Decoding (BDD) problem, which is central to lattice-based cryptography. Specifically, we demonstrate that the integer feasibility problem is as hard as BDD under both worst-case and average-case scenarios. Consequently, assuming standard lattice cryptographic hardness assumptions, we refute its polynomial-time solvability, thereby completely resolving this theoretical question that has remained open for over four decades.

Abstract

A rational polyhedron has Chvátal rank at most one if a single round of Chvátal-Gomory cuts yields its integer hull. For such polyhedra, integer feasibility is in NP $\cap$ coNP by a result of Boyd and Pulleyblank from the early 1980s, so it is unlikely to be NP-hard. Whether it is polynomial has remained open since then. We answer this question negatively, under either of two standard assumptions from lattice-based cryptography. First, a polynomial-time algorithm for this problem would solve bounded distance decoding with polynomial factors in deterministic polynomial time, contradicting a widely believed conjecture. Second, assuming the hardness of learning with errors, an average-case analogue of bounded distance decoding, the problem is also hard on average, for an efficiently samplable distribution of polytopes. Both results rest on an elementary sufficient condition: a polyhedron has Chvátal rank at most one if its width is less than one along every row of some unimodular matrix. For our polytopes, such a matrix exists but is hard to find.

Problem

• Integer programming
• Chvátal rank one
• Integer feasibility
• Lattice problems
• Bounded distance decoding

Innovation

• Chvátal rank one
• Integer programming
• Bounded distance decoding
• Learning with errors
• Unimodular matrix