Parameterized Integer Quadratic Programming: Variables and Coefficients

TL;DR

This paper introduces a fixed parameter tractable algorithm for Integer Quadratic Programming based on the combined parameters of variable count and maximum coefficient size, with applications to graph problems.

Contribution

It presents a novel fixed parameter tractable algorithm for Integer Quadratic Programming parameterized by n+a, addressing an open problem in graph theory.

Findings

Algorithm is fixed parameter tractable for n+a.

Solves the open problem for Optimal Linear Arrangement.

Provides a new approach for parameterized quadratic programming.

Abstract

In the Integer Quadratic Programming problem input is an n*n integer matrix Q, an m*n integer matrix A and an m-dimensional integer vector b. The task is to find a vector x in Z^n, minimizing x^TQx, subject to Ax <= b. We give a fixed parameter tractable algorithm for Integer Quadratic Programming parameterized by n+a. Here a is the largest absolute value of an entry of Q and A. As an application of our main result we show that Optimal Linear Arrangement is fixed parameter tractable parameterized by the size of the smallest vertex cover of the input graph. This resolves an open problem from the recent monograph by Downey and Fellows.