On properties of geometric random problems in the plane

In Beardwood et al. [3], the authors prove that for any bounded uniform i.i.d. random variables {X i : 1 < i < oo} with values in R 2, the length of the shortest tour through {X~ ..... X,} is asymptotic to a constant times ~ with probability one (the same being true in expectation). In fact, this result is valid for any uniform i.i.d. random variables with compact support of measure one in R a, d > 2, provided is replaced by n (dl)ld, the constant depending only on the dimension of the space and not on the shape of the compact support. This theoretical result has become widely recognized to be at the heart of the probabilistic evaluation of the performance of heuristic algorithms for vehicle routing problems. It is used as the main argument in the probabilistic analysis of partitioning algorithms for the traveling salesman problem (TSP) in Karp [18]. It also plays a crucial role in Haimovich and Rinnooy Kan [9] in which a probabilistic analysis of a class of heuristics is performed for the capacitated vehicle routing problems. For an overview of these algorithms and related ones, the reader is referred to Karp and Steele [19] and Haimovich et al. [10], respectively. Another analysis of partitioning algorithms for the Euclidean traveling salesman problem is contained in Halton and Terada [11 ]. More recent probabilistic analyses of heuristics for routing problems include the works of Bramel et al. [5] and Bramel and Simchi-Levi [4]. In the non-routing context, one of the earliest and nicest contributions is contained