On Bounded Integer Programming

TL;DR

This paper introduces a new reduction from bounded integer programming to a lattice problem, establishing improved time bounds and complexity results, including #P-hardness and probabilistic solvability under certain assumptions.

Contribution

It presents a novel reduction from BIP to SAP with special properties, leading to the best known upper time bounds and complexity insights for BIP.

Findings

New upper time bound for BIP: $poly() imes n^{n+o(n)}$

Proves #SAP is #P-hard under semi-reductions for some norms

Shows BIP is solvable in probabilistic time $2^{O(n)}$ under reasonable assumptions

Abstract

We present an efficient reduction from the Bounded integer programming (BIP) to the Subspace avoiding problem (SAP) in lattice theory. The reduction has some special properties with some interesting consequences. The first is the new upper time bound for BIP, $poly(\varphi)\cdot n^{n+o(n)}$ (where $n$ and $\varphi$ are the dimension and the input size of the problem, respectively). This is the best bound up to now for BIP. The second consequence is the proof that #SAP, for some norms, is #P-hard under semi-reductions. It follows that the counting version of the Generalized closest vector problem is also #P-hard under semi-reductions. Furthermore, we also show that under some reasonable assumptions, BIP is solvable in probabilistic time $2^{O(n)}$.