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A layered graph is a connected graph whose vertices are partitioned into sets L 0 =s, L 1 , L 2 ,..., and whose edges, which have nonnegative integral weights, run between consecutive layers. Its width is {|L i |}. In the on-line layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i-1 and i are only revealed when the searcher reaches layer i-1.We give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. We give a deterministic on-line algorithm which is O(9 w )-competitive on width-w graphs and prove that for no w can a deterministic on-line algorithm have a competitive ratio better than 2 w-2 on width-w graphs. We prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized on-line layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, we give a randomized on-line algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor. Abstract. A layered graph is a connected graph whose vertices are partitioned into sets L 0 = {s}, L 1 , L 2 , ..., and whose edges, which have nonnegative integral weights, run between consecutive layers. Its width is max{|L i |}. In the on-line layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i−1 and i are only revealed when the searcher reaches layer i−1.We give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. We give a deterministic on-line algorithm which is O(9 w )-competitive on width-w graphs and prove that for no w can a deterministic on-line algorithm have a competitive ratio better than 2 w−2 on width-w graphs. We prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized on-line layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, we give a randomized on-line algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor.

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The input to the asymmetric p-center problem consists of an integer p and an n = n distance matrix D defined on a vertex set V of size n, where d gives the i j distance from i to j. The distances are assumed to obey the triangle inequality. For a subset S : V the radius of S is the minimum distance R such that every point in V is at a distance at most R from some point in S. The p-center problem consists of picking a set S : V of size p to minimize the radius. This problem is known to be NP-complete.For the symmetric case, when d s d , approximation algorithms that deliver a i j ji w x solution to within 2 of the optimal are known. David Shmoys, in his article 11 , mentions that nothing was known about the asymmetric case. We present an Ž U . algorithm that achieves a ratio of O log n . ᮊ