CMPSCI 611: Advanced Algorithms Micah Adler Summary: CMPSCI 611: Advanced Algorithms Micah Adler Problem Set 5 Out: December 3, 2002 Due: December 10, 2002 1. Recall the Bin-Packing problem: INPUT: A set of positive integers A = fa 1 : : : an g, a positive bin size B, and a positive integer k. QUESTION: Can we partition A into k disjoint sets such that the sum of the integers in any set is at most B. (a) Use the Subset-Sum problem to prove that Bin-Packing is NP-Complete. (b) Does your proof from part (a) show that Bin-Packing is strongly NP-Complete? Explain why or why not. (c) De ne an appropriate optimization version of the Bin-Packing problem, and prove that if there is a ( 3 2 )-approximation to this problem, for any  > 0, then P=NP. 2. [CLRS] 35-5: Parallel machine scheduling. 3. In class, we saw how to express the Max-Flow problem as a Linear Programming problem. In this question, you are asked to extend this construction to a more general ow problem, called the Multi- commodity Flow problem. This is de ned as follows. We are given a directed graph G with positive capacities C(e) on its edges, Collections: Computer Technologies and Information Sciences