We present an algorithm for the asymmetric traveling salesman problem on instances which satisfy the triangle inequality. Like several existing algorithms, it achieves approximation ratio O(log n). Unlike previous algorithms, it uses randomized rounding.

In the title compound, C(13)H(17)NO, the cyclo-hexane ring adopts a chair conformation and the amide C(=O)-N moiety is almost coplanar with the phenyl ring [C-N-C-O = 4.1 (2)°]. In the crystal, mol-ecules are linked to form a C(4) infinite [001] chain via N-H⋯O hydrogen bonds, unlike the cyclic motif seen in related structures.

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres Theorem: For every k ≥ 1, every order-n k-dimensional permutation contains a monotone subsequence of length Ωk (√ n ) , and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random kdimensional permutation of order n is...

In this paper, we introduce a conceptually very simple and demonstrative algorithm for finding small solutions (x, y) of ax + y = c mod N , where gcd(a, N) = 1. Our new algorithm is a variant of the Euclidian algorithm. Unlike former methods, it finds a small solution whenever such a solution exists. Further it runs in time O((log N)), which is the same as the best known previous techniques, e....