Probability Theory and Combinatorial Optimization

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

“…Inspired in part by Karp's result, Dyer, Frieze and McDiarmid [19] developed a general bound for the objective function of linear programming problems with random costs, and they were able to recapture Karp's bound without recourse to special bases. A probabilist's interpretation of the Dyer-Frieze-McDiarmid inequality forms the basis of Chapter 4 of Steele [60] where one can find further information on the early history of the assignment problem with random costs.…”

Concentration inequalities for functions of independent variables

“…The next step is to integrate inequalities (15) and (16) to obtain a bound on H ( ) H (0) in terms of an integral of the thermal expectation of + . Uniform bounds on the integrand lead to Theorem 1, substitution of the self-bounding condition (5) gives Theorem 2.…”

“…Talagrand's convex distance inequality (see [16], [17], [12], [15] or [3]) asserts that (questions of measurability aside), for A and t > 0 we have…”

“…Using space-…lling curves it can be shown (see [15]), that there is a constant c 2 such that for all n and x 2 there is a tour c (x) with…”

“…The now classical applications (see McDiarmid [12] and Steele [15]) include concentration inequalities for con…guration functions, such as the length of the longest increasing subsequence in a sample, or for geometrical constructions, such as the length of an optimal travelling salesman tour or an optimal Steiner tree.…”