On-line and Approximation Algorithms Fall Semester, 2011/12 Exercise 3: Jan 4, 2012 Summary: On-line and Approximation Algorithms Fall Semester, 2011/12 Exercise 3: Jan 4, 2012 Lecturer: Prof. Yossi Azar Write short but full and accurate answers. Each solution should appear on a separate page and each of its parts should not exceed a page. 1. We are given a connected graph G = (V, E). All edges have unit capacity. At step i we receive a request to allocate a bandwidth pi on a specific path Qi. If we allocate the bandwidth on the path we receive benefit of bi otherwise, we receive no benefit. Also for some known value F and for all i, 1 bi/pi F and pi 1 log(2|V |F+2). Our goal is to maximize the total benefit while maintaining the capacity constraints. Design an O(log(|V |F)) competitive algorithm. 2. Consider admission control for the edge disjoint paths problem on a star (a center with k paths each of n edges). The goal is to maximize the number of accepted paths. (a) Design a deterministic preemptive algorithm which is O(log n) competitive. (b) Design a randomized non-preemptive algorithm which is O(log n) competitive. 3. Suppose we are given one machine and a set of jobs that arrive over time. The machine can process one job at a time and may preempt jobs. The duration of a job is known at its release time and the benefit of a job is equal to its duration. In order to get the benefit of a job it must be processed immediately at its release time and should not be preempted until its completion. (a) Design a 4 competitive algorithm for the problem. Collections: Computer Technologies and Information Sciences