We consider the problem of approximating the Hessian matrix of a smooth non-linear function using a minimum number of gradient evaluations, particularly in the case that the Hessian has a known, fixed sparsity pattern. We study the class of Direct Methods for this problem, and propose two new ways of classifying Direct Methods. Examples are given that show the relationships among optimal method...

Let G = (V, E) be a strongly connected and aperiodic directed graph of uniform out-degree k. A deterministic finite automaton is obtained if the edges are colored with k colors in such a way that each vertex has one edge of each color leaving it. The automaton is called synchronized if there exists an input word that maps all vertices into the same fixed vertex. The road coloring conjecture ask...

We study the problem of coloring the n-dimensional boolean cube with c = 2 s colors such that in every k-dimensional subcube each color appears 2 k =c times. We are interested in the smallest k for which such a coloring exists. This problem is equivalent to the t-resilient function problem. A coloring is locally symmetric if each vertex in the n-cube has the same number of neighbors from each c...

A new variation of the coloring problem, μ-coloring, is defined in this paper. Given a graph G and a function μ, a μ-coloring is a coloring where each vertex v of G must receive a color at most μ(v). It is proved that μ-coloring lies between coloring and list-coloring, in the sense of generalization of problems and computational complexity. The notion of perfection is extended for μ-coloring, l...

Dvo?ák and Postle introduced the concept of DP-coloring to overcome some difficulties in list coloring. Sittitrai Nakprasit combined defective coloring define a new coloring—relaxed DP-coloring. For relaxed DP-coloring, Sribunhung et al. proved that planar graphs without 4- 7-cycles are DP-(0, 2, 2)-colorable. Li 4, 8-cycles or 9-cycles DP-(1, 1, 1)-colorable. Lu Zhu 5-cycles, 6-cycles, In this...

Matrix partitioning problems that arise in the efficient estimation of sparse Jacobians and Hessians can be modeled using variants of graph coloring problems. In a previous work [6], we argue that distance-2 and distance3 2 graph coloring are robust and flexible formulations of the respective matrix estimation problems. The problem size in large-scale optimization contexts makes the matrix esti...

We consider the problem of efficient coloring of the edges of a so-called binomial tree T, i.e. acyclic graph containing two kinds of edges: those which must have a single color and those which are to be colored with L consecutive colors, where L is an arbitrary integer greater than 1. We give an O(n) time algorithm for optimal coloring of such a tree, where n is the number of vertices of T. Al...

We devise a new formulation for the vertex coloring problem. Different from other formulations, decision variables are associated with the pairs of vertices. Consequently, colors will be distinguishable. Although the objective function is fractional, it can be replaced by a piece-wise linear convex function. Numerical experiments show that our formulation has significantly good performance for ...