The Bilinear Assignment Problem: Complexity and polynomially solvable special cases

TL;DR

This paper investigates the computational complexity of the bilinear assignment problem (BAP), identifying NP-hardness results, special polynomially solvable cases, and providing heuristics and exact conditions for problem equivalences.

Contribution

It establishes complexity bounds for BAP, characterizes special cases with polynomial solutions, and introduces heuristics and formulas for solution evaluation.

Findings

BAP cannot be approximated within a constant factor unless P=NP.

BAP is polynomially solvable when m=O(log n).

Heuristic algorithms can find solutions within certain bounds, but some solution quality problems are NP-hard.

Abstract

In this paper we study the {\it bilinear assignment problem} (BAP) with size parameters $m$ and $n$, $m\leq n$. BAP is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence NP-hard. We show that BAP cannot be approximated within a constant factor unless P=NP even if the associated quadratic cost matrix $Q$ is diagonal. Further, we show that BAP remains NP-hard if $m = O(\sqrt[r]{n})$, for some fixed $r$, but is solvable in polynomial time if $m = O(\sqrt{\log n})$. When the rank of $Q$ is fixed, BAP is observed to admit FPTAS and when this rank is one, it is solvable in polynomial time under some additional restrictions. We then provide a necessary and sufficient condition for BAP to be equivalent to two linear assignment problems. A closed form expression to compute the average of the objective function values of all solutions is presented, whereas the median of the solution values cannot be identified in polynomial time, unless P=NP. We then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of $(m-1)!(n-1)!$ solutions. However, computing a solution whose objective function value is no worse than that of $m!n!-\lceil\frac{m}{\beta}\rceil !\lceil\frac{n}{\beta}\rceil !$ solutions is NP-hard for any fixed rational number $\beta>1$.