There exist problems that have a polynomial time promise solution, while being NP-hard if required to be robust. This is to be contrasted with the “promise” version of solving problems on restricted domains, in which there is a guarantee that the input is in the class, and an algorithm to “solve” the problem need not function correctly or even terminate if this guarantee is not met. Under this definition, an algorithm is required to be “robust,” i.e., it must produce correct output regardless of whether the input actually belongs to the restricted domain or not. We introduce a new definition of efficient algorithms for restricted domains. Number of gradient evaluations needed in any possible approximation method. An appendix proves a well-known folklore result, which gives lower bounds on the Of a simple coloring heuristic are given. Some worst-case bounds on the performance A theorem is proved showing that the general distance-k graph coloring problem is NP-Complete for all fixedk≥2, and hence that the optimal non-overlapping direct cover problem is also NP-Complete. Graph coloring problem-the distance-2 graph coloring problem. The problem of finding a non-overlapping direct cover is shown to be equivalent to a generalized Examples are given that show the relationships among We study the class of Direct Methodsįor this problem, and propose two new ways of classifying Direct Methods. We consider the problem of approximating the Hessian matrix of a smooth non-linear function using a minimum number of gradientĮvaluations, particularly in the case that the Hessian has a known, fixed sparsity pattern. We finish the paper by discussing some open problems. We show that the computational complexity of the problem "Given a graph $G$, a spanning tree $T$ of $G$, and an integer $\ell$, is there a backbone coloring for $G$ and $T$ with at most $\ell$ colors?" jumps from polynomial to NP-complete between $\ell = 4$ (easy for all spanning trees) and $\ell = 5$ (difficult even for spanning paths). R: V \rightarrow \mathbb R^+ that maps each point v in V to the disk of radius Consider an n-point metric M = (V,delta), and a transmission range assignment
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