Analysis of Probabilistic Combinatorial Optimization Problems in Euclidean Spaces

Probabilistic combinatorial optimization problems are generalized versions of deterministic combinatorial optimization problems with explicit inclusion of probabilistic elements in the problem definitions. Based on the probabilistic traveling salesman problem (PTSP) and on the probabilistic minimum spanning tree problem (PMSTP), the objective of this paper is to give a rigorous treatment of the probabilistic analysis of these problems in the plane. More specifically we present general finite-size bounds and limit theorems for the objective functions of the PTSP and PMSTP. We also discuss the practical implications of these results and indicate some open problems.