On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.Comment: 18 pages, 2 figure

Probabilistic analysis of euclidean multi depot vehicle routing and related problems

We consider a generalization of the classical traveling salesman problem: the multi depot vehicle routing problem (MDVRP). Let $D$ be a set of $k$ depots and $P$ be sets $n$ customers in $[0,1]^d$ with the usual Euclidean metric. A multi depot vehicle routing tour is a set of disjoint cycles such that all customers are covered and each cycle contains exactly one depot. The goal is to find a tour of minimum length. $L(D,P)$ denotes the length of an optimal MDVRP tour for depot set $D$ and customer set $P$. It is evident that the asymptotic behavior of \L(D,P) for $n$ tending to infinity depends on the customer-depot ratio $n/k$. We study three cases: $k=o(n)$, $k=\lambda n +o(n)$ for a constant $\lambda >0$, and k=\Omega(n^{1+\ee}) for \ee>0. In the first two cases we show that $L(D,P)$ divided by $n^{(d-1)/d}$ converges completely to a constant if the customers and depots are given by iid random variables. In the last case we prove that the expected tour length divided by $n^{(d-1)/d}$ and multiplied by $k^{1/d}$ converges to a constant if the customers and depots are given by iid random variables with uniform distribution

Comb inequalities for typical Euclidean TSP instances

We prove that even in average case, the Euclidean Traveling Salesman Problem exhibits an integrality gap of $(1+\epsilon)$ for $\epsilon>0$ when the Held-Karp Linear Programming relaxation is augmented by all comb inequalities of bounded size. This implies that large classes of branch-and-cut algorithms take exponential time for the Euclidean TSP, even on random inputs.Comment: 19 pages, 4 figure

Mean field and corrections for the Euclidean Minimum Matching problem

Consider the length $L_{MM}^E$ of the minimum matching of N points in d-dimensional Euclidean space. Using numerical simulations and the finite size scaling law $= \beta_{MM}^E(d) N^{1-1/d}(1+A/N+... )$, we obtain precise estimates of $\beta_{MM}^E(d)$ for $2 \le d \le 10$. We then consider the approximation where distance correlations are neglected. This model is solvable and gives at $d \ge 2$ an excellent ``random link'' approximation to $\beta_{MM}^E(d)$. Incorporation of three-link correlations further improves the accuracy, leading to a relative error of 0.4% at d=2 and 3. Finally, the large d behavior of this expansion in link correlations is discussed.Comment: source and one figure. Submitted to PR

Max-stable random sup-measures with comonotonic tail dependence

Several objects in the Extremes literature are special instances of max-stable random sup-measures. This perspective opens connections to the theory of random sets and the theory of risk measures and makes it possible to extend corresponding notions and results from the literature with streamlined proofs. In particular, it clarifies the role of Choquet random sup-measures and their stochastic dominance property. Key tools are the LePage representation of a max-stable random sup-measure and the dual representation of its tail dependence functional. Properties such as complete randomness, continuity, separability, coupling, continuous choice, invariance and transformations are also analysed.Comment: 28 pages, 1 figur