Statistical bound of genetic solutions to quadratic assignment problems

Quadratic assignment problems (QAPs) are commonly solved by heuristic methods, where the optimum is sought iteratively. Heuristics are known to provide good solutions but the quality of the solutions, i.e., the confidence interval of the solution is unknown. This paper uses statistical optimum estimation techniques (SOETs) to assess the quality of Genetic algorithm solutions for QAPs. We examine the functioning of different SOETs regarding biasness, coverage rate and length of interval, and then we compare the SOET lower bound with deterministic ones. The commonly used deterministic bounds are confined to only a few algorithms. We show that, the Jackknife estimators have better performance than Weibull estimators, and when the number of heuristic solutions is as large as 100, higher order JK-estimators perform better than lower order ones. Compared with the deterministic bounds, the SOET lower bound performs significantly better than most deterministic lower bounds and is comparable with the best deterministic ones. 

The determination of all syzygies for the dependent polynomial invariants of the Riemann tensor. I. Pure Ricci and pure Weyl invariants

In this paper, we shall consider all pure Ricci and pure Weyl scalar invariants of any degree, in a four-dimensional Lorentzian space. We present a general graph-theoretic based reduction algorithm which decomposes, using syzygies, any pure invariant in terms of the independent base invariants {r₁,r₂,r₃} or {w₁,w₂}

The determination of all syzygies for the dependent polynomial invariants of the Riemann Tensor. II. mixed invariants of Eeven degree in the Ricci Spinor

We continue our analysis of the polynomial invariants of the Riemann tensor in a four-dimensional Lorentzian space. We concentrate on the mixed invariants of even degree in the Ricci spinor Φ_ABȦḂ and show how, using constructive graph-theoretic methods, arbitrary scalar contractions between copies of the Weyl spinor ψ_ABCD, its conjugate ψ_ȦḂĊḊ and an even number of Ricci spinors can be expressed in terms of paired contractions between these spinors. This leads to an algorithm for the explicit expression of dependent invariants as polynomials of members of the complete set. Finally, we rigorously prove that the complete set as given by Sneddon [J. Math. Phys. 39, 1659-1679 (1998)] for this case is both complete and minimal.

Polyhedral combinatorics of the cardinality constrained quadratic knapsack problem and the quadratic selective travelling salesman problem

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods. Good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper studies the polyhedral combinatorics of the QSTSP and the QKP, i.e. amongst others we identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets.

Determination of all syzygies for the dependent polynomial invariants of the Riemann Tensor. III. mixed invariants of arbitrary degree in the Ricci Spinor

In this paper, we rigorously prove that the complete set of Riemann tensor invariants given by Sneddon [J. Math. Phys. 40, 5905 (1999)] is both minimal and complete. Furthermore, we provide a two-stage algorithm for the explicit construction of polynomial syzygies relating any dependent Riemann tensor invariant to members of the complete set.