Exploiting Problem Structure in Combinatorial Landscapes: A Case Study on Pure Mathematics Application

In this paper, we present a method using AI techniques to solve a case of pure mathematics applications for finding narrow admissible tuples. The original problem is formulated into a combinatorial optimization problem. In particular, we show how to exploit the local search structure to formulate the problem landscape for dramatic reductions in search space and for non-trivial elimination in search barriers, and then to realize intelligent search strategies for effectively escaping from local minima. Experimental results demonstrate that the proposed method is able to efficiently find best known solutions. This research sheds light on exploiting the local problem structure for an efficient search in combinatorial landscapes as an application of AI to a new problem domain.Comment: 7 pages, 2 figures, conferenc

Article electronically published on March 22, 2001 DENSE ADMISSIBLE SEQUENCES

Abstract. A sequence of integers in an interval of length x is called admissible if for each prime there is a residue class modulo the prime which contains no elements of the sequence. The maximum number of elements in an admissible sequence in an interval of length x is denoted by ϱ ∗ (x). Hensley and Richards showed that ϱ ∗ (x)>π(x) for large enough x. We increase the known bounds on the set of x satisfying ϱ ∗ (x) ≤ π(x) and find smaller values of x such that ϱ ∗ (x)>π(x). We also find values of x satisfying ϱ ∗ (x)> 2π(x/2). This shows the incompatibility of the conjecture π(x+y)−π(y) ≤ 2π(x/2) with the prime k-tuples conjecture. 1

Article electronically published on March 22, 2001 DENSE ADMISSIBLE SEQUENCES

Abstract. A sequence of integers in an interval of length x is called admissible if for each prime there is a residue class modulo the prime which contains no elements of the sequence. The maximum number of elements in an admissible sequence in an interval of length x is denoted by ϱ ∗ (x). Hensley and Richards showed that ϱ ∗ (x)>π(x) for large enough x. We increase the known bounds on the set of x satisfying ϱ ∗ (x) ≤ π(x) and find smaller values of x such that ϱ ∗ (x)>π(x). We also find values of x satisfying ϱ ∗ (x)> 2π(x/2). This shows the incompatibility of the conjecture π(x+y)−π(y) ≤ 2π(x/2) with the prime k-tuples conjecture. 1