Smallest defining sets of designs associated with

For d 2:: 2, let Dd be the symmetric block design formed from the points and hyperplanes of the projective space PG(d, 2). Let 3d equal the number of blocks in a smallest defining set of D d. The known results 32 = 3 and S3 = 9 are reviewed and it is shown that 34 = 24 and 52:::; 35:::; 55. If J-Ld = 3d / (2d+ 1-1) is the proportion of blocks in a smallest defining set of Dd, then J-L2 = 3/7, J-L3 = 9/15 and J-L4 = 24/31. The main result of this paper is that J-Ld-+ 1 as d-+ 00.

AWOCA'96 -- Seventh Australasian Workshop on Combinatorial Algorithms

Computing spanning trees with specific properties and constraints lies at the heart of many real-life network optimization problems. Here, a compilation of 35 constrained spanning tree problems is presented. Since most of these problems are NP-complete, good approximate heuristics are needed to solve them on parallel machines. Our method of iterative refinement is one such generic technique to compute good suboptimal solutions for large instances of these problems in a reasonable time. It is suited for problems in which the goal is to find a spanning tree which satisfies a specified constraint while minimizing its total weight. First, a minimum spanning tree (MST) of the given edge-weighted graph is computed without considering the specified constraint. Then, the constraint is used to increase the weights of selected edges (in this tree) in order that when the next MST is constructed (in the graph with modified edge weights) it has fewer violations of the constraint. Next, an MST of th..