Average value of solutions of the bipartite quadratic assignment problem and linkages to domination analysis

Abstract

In this paper we study the complexity and domination analysis in the context of the \emph{bipartite quadratic assignment problem}. Two variants of the problem, denoted by BQAP1 and BQAP2, are investigated. A formula for calculating the average objective function value $\mathcal{A}$ of all solutions is presented whereas computing the median objective function value is shown to be NP-hard. We show that any heuristic algorithm that produces a solution with objective function value at most $\mathcal{A}$ has the domination ratio at least $\frac{1}{mn}$. Analogous results for the standard \emph{quadratic assignment problem} is an open question. We show that computing a solution whose objective function value is no worse than that of $n^m{m}^n-{\lceil \frac{n}{\alpha }\rceil }^{\lceil \frac{m}{\alpha }\rceil }{\lceil \frac{m}{\alpha }\rceil }^{\lceil \frac{n}{\alpha }\rceil }$ solutions of BQAP1 or $m^m{n}^n-{\lceil \frac{m}{\alpha }\rceil }^{\lceil \frac{m}{\alpha }\rceil }{\lceil \frac{n}{\alpha }\rceil }^{\lceil \frac{n}{\alpha }\rceil }$ solutions of BQAP2, is NP-hard for any fixed natural numbers $a$ and $b$ such that $\alpha =\frac{a}{b}>1$. However, a solution with the domination number $\Omega (m^{n-1}{n}^{m-1}+m^{n+1}n+m{n}^{m+1})$ for BQAP1 and $\Omega (m^{m-1}{n}^{n-1}+m^2{n}^n+m^m{n}^2)$ for BQAP2, can be found in $O(m^3{n}^3)$ time.