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In this paper we investigate genetic algorithms (GA) as a .heuristic technique for obtaining near optimal solutions to the probabilistic minimum spanning tree (PMST) problem. The PMST problem is a natural generalization of the classical minimum spanning tree (MST) problem and is frequently a more realistic model. The PMST problem addresses the circumstances that arise when not all nodes are deterministically present but, rather, nodes are present with known probabilities. Although there are some special cases that are solvable in polynomial time, it is known that the PMST problem is NP-complete.

Genetic Algorithms(GA) are used to find minimum cost trees with special structures. The desired structure for the trees is M-Star isomorphic, where M = 3. The M-Star isomorphic problem is to find a spanning tree that is a star, and on each branch there are M vertices connected to form a path. The M-Star problem is NP-complete for M > 2. The cost is taken as the sum of the length of the edges forming the spanning tree. The determinant encoding, the Davis encoding, and the Priifer encoding schemes are used to represent the spanning trees. The cost of a greedy algorithm is compared to the three GA encodings and a variety of crossover and mutation operators. Results show that the determinant encoding is better than the Priifer encoding, and is as good as the Davis encoding for the 19 vertex problem. For the 61 vertex problem the Davis encoding produces better results over the determinant encoding in 8 out of 10 test cases.

The genetic algorithm (GA) heuristic is used to find near optimal solutions for the Probabilistic Minimum Spanning Tree problem (PMST), an NP-complete variation of the classical minimum spanning tree problem. Given a connected graph G(V,E), a cost function c:E--+R+, and a probability function P:2" b[O, I], the problem is to find an 0 priori spanning tree of minimum expected length. For the incomplete graph problem, a new encoding scheme for spanning trees that is based on the cycle basis is compared to another new encoding scheme that is based on the factorization of the determinant of the in-degree matrix of the original graph. For edge connectivity probability < 0.4 and problem size of 20 nodes, our results show a significant improvement in using the cycle basis encoding over the use of the determinant encoding, or a greedy algorithm.For the 30 and 40 node problems the determinant encoding performs better than the cycle basis encoding for most edge connectivity probabilities. Recently Bertsimas[4] defined the ProbabilisticMinimum Spanning Tree (PMST), which is a variation of the minimum spanning tree where each vertex is present with some given probability.The PMST model is more realistic and representative of many combinatorial optimization problems than the minimum spanning tree, especially if the MST is NOT the "best" solution for all instances of the problem. Note when the vertex probability is one, for each vertex, the PMST problem reduces to the MST problem. Many applications are natural for the PMST such as VLSI design, communication network design, and organizational structures design.

In recent years several authors have investigated binary search trees with minimal internal path length. In this paper we propose relaxing the requirement of inserting all nodes on one level before going to the next level. This leads to a new class of binary search trees called ISA [k] trees. We investigated the average locate cost per node, average shift cost per node, total insertion cost, and average successful search cost for ISA[k] trees. We also present an insertion algorithm with associated predecessor and successor functions for ISA[k] trees. For large binary search trees (over 160 nodes) our results suggest the use of ISA[2] or ISA[3] trees for best performance.