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 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 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 A has the domination ratio at least 1 mn . Analogous results for the standard quadratic assignment problem is an open question. We show that computing a solution whose objective function value is no worse than that of, is NP-hard for any fixed natural numbers a and b such that α = a b > 1. However, a solution with the domination number Ω(m n−1 n m−1 +m n+1 n+mn m+1 ) for BQAP1 and Ω(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.

The Bilinear Assignment Problem: Complexity and polynomially solvable special cases

“…Lastly, we give an upper bound on the domination ratio for any polynomial time heuristic algorithm for the BAP. The result can be shown following the main idea in [10,32], and hence we omit the proof.…”

“…Linearizable instances of QAP have been studied by various authors [8,24,31]. Corresponding properties for the quadratic spanning tree problem (QMST) was investigated in [10]. Before attempting to characterize linearizable instances of BAP, let us first establish some preliminary results.…”

“…If we expand objective value expressions in (10) and cancel out identical parts from the left and right sides, we get q 1111 +q 11kℓ +q ij11 +q ijkℓ +q 1j1ℓ +q 1jk1 +q i11ℓ +q i1k1 = q 111ℓ +q 11k1 +q ij1ℓ +q ijk1 +q 1j11 +q 1jkℓ +q i111 +q i1kℓ , from which it follows that q ijkℓ = q ijk1 + q ij1ℓ + q i1kℓ + q 1jkℓ − q ij11 − q i1k1 − q i11ℓ − q 1jk1 − q 1j1ℓ − q 11kℓ + q i111 + q 1j11 + q 11k1 + q 111ℓ − q 1111 (11) for every i, j, k, ℓ ≥ 2. However, note that (11) also holds true when some of i, j, k, ℓ are equal to 1.…”

“…Since LAP can be solved in cubic running time, we have that the complexity of RxOy and RyOx procedures is O(m 2 n 2 + m 3 + n 3 ). Similar rounding procedures for the unconstrained bipartite boolean quadratic program and bipartite quadratic assignment problem were investigated in [32] and [10], respectively. Theorem 10.…”