Where the Really Hard Quadratic Assignment Problems Are: the QAP-SAT instances

TL;DR

A new QAP-SAT design of the initial problem based on submodularity is introduced to capture its difficulty with new features and the critical parameter of phase transition satisfaction and that of the solving effort are shown to be highly correlated for tabu search, thus allowing the prediction of difficult instances.

Abstract

The Quadratic Assignment Problem (QAP) is one of the major domains in the field of evolutionary computation, and more widely in combinatorial optimization. This paper studies the phase transition of the QAP, which can be described as a dramatic change in the problem's computational complexity and satisfiability, within a narrow range of the problem parameters. To approach this phenomenon, we introduce a new QAP-SAT design of the initial problem based on submodularity to capture its difficulty with new features. This decomposition is studied experimentally using branch-and-bound and tabu search solvers. A phase transition parameter is then proposed. The critical parameter of phase transition satisfaction and that of the solving effort are shown to be highly correlated for tabu search, thus allowing the prediction of difficult instances.

Figures (7)

• Figure 1: For problem dimension $n=5$, examples of a-clause and b-clause of size $k=3$ with $V_A = \{ 2, 3, 5 \}$, and $V_B = \{ 1, 2, 5 \}$. $A^{(3)}$, and $B^{(3)}$ are sub-matrix with variables of $V_A$, and $V_B$. Distance matrix $B$ composed of $m_1=1$b-clause complementary to matrix $B_3$.
• Figure 2: Proportion of satisfied instances, for which the minimal values is the lower bound, according to the number of a-clauses $m$. Facet: problem dimension $n$.
• Figure 3: Critical parameter $m_c$ according to the number of b-clauses $m_1$ (left), and problem dimension $n$ (right).
• Figure 4: Probability of satisfied instances according to the phase transition parameter $\frac{m}{n^{\alpha_1} m^{\alpha_2}_1}$ across all instances. $k=\exp(1.65453) \approx 5.23$.
• Figure 5: Average computation time of B&B algorithm to find the global minimum. Problem dimensions $n$ = 10-17.