In this paper we give improved approximation algorithms for some network design problems. In the Bounded-Diameter or Shallow-Light k-Steiner tree problem (SLkST), we are given an undirected graph G = (V,E) with terminals T ⊆ V containing a root r ∈ T , a cost function c : E → R, a length function l : E → R, a bound L > 0 and an integer k ≥ 1. The goal is to find a minimum c-cost r-rooted Steint...

Given a set of n circular arcs, we provide an optimal parallel algorithm (on the CREW PRAM model of computation) for finding a minimum number of circular arcs whose union covers the circle. The algorithm runs in O(log n) time with O(n) processors and uses O(n) space. This is a significant improvement over the recent algorithm by Bertossi that runs in O(log n) time with 0( n2) processors and use...

We present the first fully dynamic algorithm for maintaining a minimum spanning tree in time o( √ n) per operation. To be precise, the algorithm uses O(n1/3 log n) amortized time per update operation. The algorithm is fairly simple and deterministic. An immediate consequence is the first fully dynamic deterministic algorithm for maintaining connectivity and, bipartiteness in amortized time O(n1...

We study the problem of computing the minimum cut in a weighted distributed messagepassing networks (the CONGEST model). Let λ be the minimum cut, n be the number of nodes (processors) in the network, and D be the network diameter. Our algorithm can compute λ exactly in O(( √ n log∗ n+D)λ log n) time. To the best of our knowledge, this is the first paper that explicitly studies computing the ex...

A. Gyarfas COMPUTER AND AUTOMATION INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES BUDAPEST_ HUNGARY J. Koml6s and E. Szemeredi MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES BUDAPEST_ HUNGARY The set of different cycle lengths of a graph G is denoted by C(G). We study how the distribution of C(G) depends on the minimum degree of G. We prove two results indicating that C(G) is dense...

Given an input consisting of an n-vertex convex polygon with k hole vertices or an n-vertex planar straight line graph (PSLG) with k holes and/or reflex vertices inside the convex hull, the parameterized minimum number convex partition (MNCP) problem asks for a partition into a minimum number of convex pieces. We give a fixedparameter tractable algorithm for this problem that runs in the follow...

Assume that n tasks must be processed by one machine in a fixed sequence. The processing time, the preferred starting time and the earliness and tardiness costs per time unit are known for each task. The problem is to allocate each task a starting time such that the total cost incurred by the early and tardy tasks is minimum. Garey et al. have proposed a nice O(n log n) algorithm for the specia...

In an undirected graph, the feedback vertex set problem is to find a set of vertices of minimum cardinality whose removal makes the graph acyclic (connected or disconnected). This problem is known to be NP-hard for general graph. In this paper, we proposed two algorithms for finding the minimum feedback vertex set on interval graphs. The first algorithm is for unweighted case and takes O(n + m)...

In this paper we show that for discounted MDPs with discount factor, > 1/2 the asymptotic rate of convergence of Q-Iearning is O(1/tR(1-1') if R(1 ,) < 1/2 and O( Jlog log tit) otherwise provided that the state-action pairs are sampled from a fixed probability distribution. Here R = Pmin/Pmax is the ratio of the minimum and maximum state-action occupation frequencies. The results extend to conv...

Let G = (V (G), E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆ V (G) is called a global offensive k-alliance if |N(v) ∩ S| ≥ |N(v)−S|+k for every v ∈ V (G)−S, where N(v) is the neighborhood of v. The global offensive k-alliance number γ o (G) is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on γ o (G) in terms of order, maximum degree, inde...