Proximity in Concave Integer Quadratic Programming

TL;DR

This paper extends proximity bounds to concave integer quadratic programming, showing that approximate solutions remain close to continuous relaxations, with bounds depending on problem size, subdeterminants, and approximation quality.

Contribution

It introduces the first proximity bounds for concave integer quadratic programming, incorporating approximation quality into the bounds.

Findings

Proximity bounds depend on problem size, subdeterminants, and approximation parameter.

Approximate solutions are close to continuous relaxations in concave integer quadratic problems.

Bounds are applicable even when exact optimality is relaxed.

Abstract

A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of $n \Delta$ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, $n$ is the number of variables and $\Delta$ denotes the maximum of the absolute values of the subdeterminants of the constraint matrix. Hochbaum and Shanthikumar, and Werman and Magagnosc showed that the same upper bound is valid if a more general convex function is minimized, instead of a linear function. No proximity result of this type is known when the objective function is nonconvex. In fact, if we minimize a concave quadratic, no upper bound can be given as a function of $n$ and $\Delta$. Our key observation is that, in this setting, proximity phenomena still occur, but only if we consider also approximate solutions instead of optimal solutions only. In our main result we provide upper bounds on the distance between approximate (resp., optimal) solutions to a Concave Integer Quadratic Programming problem and optimal (resp., approximate) solutions of its continuous relaxation. Our bounds are functions of $n, \Delta$, and a parameter $\epsilon$ that controls the quality of the approximation. Furthermore, we discuss how far from optimal are our proximity bounds.