Subdeterminants and Concave Integer Quadratic Programming

TL;DR

This paper presents algorithms for approximating solutions to NP-hard concave quadratic integer programming problems, with complexity depending on subdeterminants of the constraint matrix and the number of nonlinear variables.

Contribution

It introduces a polynomial-time approximation algorithm for separable concave integer quadratic programming based on subdeterminant bounds, including the first for D at most two.

Findings

Algorithm finds epsilon-approximate solutions efficiently.

First polynomial-time approximation for D ≤ 2 with fixed nonlinear variables.

Improved algorithm for totally unimodular case D=1, independent of dimension n.

Abstract

We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by D the largest absolute value of the subdeterminants of the constraint matrix. In this paper we give an algorithm that finds an epsilon-approximate solution for this problem by solving a number of integer linear programs whose constraint matrices have subdeterminants bounded by D in absolute value. The number of these integer linear programs is polynomial in the dimension n, in D and in 1/epsilon, provided that the number k of variables that appear nonlinearly in the objective is fixed. As a corollary, we obtain the first polynomial-time approximation algorithm for separable concave integer quadratic programming with D at most two and k fixed. In the totally unimodular case D=1, we give an improved algorithm that only needs to solve a number of linear programs that is polynomial in 1/epsilon and is independent on n, provided that k is fixed.