Probability and Problems in Euclidean Combinatorial Optimization

This article summarizes the current status of several streams of research that deal with the probability theory of problems of combinatorial optimization. There is a particular emphasis on functionals of finite point sets. The most famous example of such functionals is the length associated with the Euclidean traveling salesman problem (TSP), but closely related problems include the minimal spanning tree problem, minimal matching problems and others. Progress is also surveyed on (1) the approximation and determination of constants whose existence is known by subadditive methods, (2) the central limit problems for several functionals closely related to Euclidean functionals, and (3) analogies in the asymptotic behavior between worst-case and expected-case behavior of Euclidean problems. No attempt has been made in this survey to cover the many important applications of probability to linear programming, arrangement searching or other problems that focus on lines or planes

Survivable Networks, Linear Programming Relaxations and the Parsimonious Property

We consider the survivable network design problem - the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rateɛ-balanced. Sharp bounds onɛ in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallestɛ in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,ɛ-balanced forɛ≈0.06

Worst-Case Growth Rates of Some Classical Problems of Combinatorial Optimization

A method is presented for determining the asymptotic worst-case behavior of quantities like the length of the minimal spanning tree or the length of an optimal traveling salesman tour of $n$ points in the unit $d$-cube. In each of these classical problems, the worst-case lengths are proved to have the exact asymptotic growth rate of $\beta _n^{{{(d - 1)} / d}}$, where $\beta$ is a positive constant depending on the problem and the dimension. These results complement known results on the growth rates for the analogous quantities under probabilistic assumptions on the points, but the results given here are free of any probabilistic hypotheses

Asymptotic constant-factor approximation algorithm for the Traveling Salesperson Problem for Dubins' vehicle

This article proposes the first known algorithm that achieves a constant-factor approximation of the minimum length tour for a Dubins' vehicle through $n$ points on the plane. By Dubins' vehicle, we mean a vehicle constrained to move at constant speed along paths with bounded curvature without reversing direction. For this version of the classic Traveling Salesperson Problem, our algorithm closes the gap between previously established lower and upper bounds; the achievable performance is of order $n^{2/3}$

Analysis of Linear Programming Relaxations for a Class of Connectivity Problems

We consider the analysis of linear programming (LP) relaxations for a class of connectivity problems. The central problem in the class is the survivable network design problem - the problem of designing a minimum cost undirected network satisfying prespecified connectivity requirements between every pair of vertices. This class includes a number of classical combinatorial optimization problems as special cases such as the Steiner tree problem, the traveling salesman problem, the k-person traveling salesman problem and the k-edge-connected network problem. We analyze a classical linear programming relaxation for this class of problems under three perspectives: structural, worst-case and probabilistic. Our analysis rests mainly upon a deep structural property, the parsimonious property, of this LP relaxation. Roughly stated, the parsimonious property says that, if the cost function satisfies the triangle inequality, there exists an optimal solution to the LP relaxation for which the degree of each vertex is the smallest it can possibly be. The numerous consequences of the parsimonious property make it particularly important. First, several special cases of the parsimonious property are interesting properties by themselves. For example, we derive the monotonicity of the Held-Karp lower bound for the traveling salesman problem and the fact that this bound is a relaxation on the 2-connected network problem. Another consequence is the fact that vertices with no connectivity requirement, such as Steiner vertices in the undirected Steiner tree problem, are unnecessary for the LP relaxation under consideration. From the parsimonious property, it also follows that the LP relaxation bounds corresponding to the Steiner tree problem, the kedge-connected network problem or even the Steiner k-edge-connected network problem can be computed a la Held and Karp.Secondly, we use the parsimonious property to perform worst-case analyses of the duality gap corresponding to these LP relaxations. For this purpose, we introduce two heuristics for the survivable network design problem and present bounds dependent on the actual connectivity requirements. Among other results, we show that the value of the LP relaxation of the Steiner tree problem is within twice the value of the minimum spanning tree heuristic and that several generalizations of the Steiner tree problem, including the k-edge-connected network problem, can also be approximated within a factor of 2 (in some cases, even smaller than 2). We also introduce a new relaxation a la Held and Karp for the k-person traveling salesman problem and show that a variation of an existing heuristic is within times the value of this relaxation. We show that most of our bounds are tight and we investigate whether the bound of 3 for the Held-Karp lower bound is tight.We also perform a probabilistic analysis of the duality gap of these LP relaxations. The model we consider is the Euclidean model. We generalize Steele's theorem on the asymptotic behavior of Euclidean functionals in a way that is particularly convenient for the analysis of LP relaxations. We show that, under the Euclidean model, the duality gap is almost surely a constant and we provide theoretical and empirical bounds on these constants for different problems. From this analysis, we conclude that the undirected LP relaxation for the Steiner tree problem is fairly loose. Finally, we consider the use of directed relaxations for undirected problems. We establish in which settings a related parsimonious property holds and show that, for the Steiner tree problem, the directed relaxation strictly improves upon the undirected relaxation in the worst-case. This latter result uses an elementary but powerful property of linear programs