An Approximation Algorithm for Indefinite Mixed Integer Quadratic Programming

TL;DR

This paper presents a polynomial-time approximation algorithm for mixed integer quadratic programming problems with fixed rank and integer variables, introducing new concepts like spherical form MIQP and aligned vectors.

Contribution

It introduces a novel approximation algorithm for MIQP problems, utilizing new concepts such as spherical form MIQP and aligned vectors, and provides related matrix decomposition results.

Findings

Algorithm finds epsilon-approximate solutions efficiently

Provides a strongly polynomial algorithm for matrix symmetric decomposition

Shows a method for simultaneous diagonalization of matrices

Abstract

In this paper, we give an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P=NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices.