# IEOR 6614, Spring 2009 : Homework 3

Assigned: Thursday, February 5, 2009
Due: Thursday, February 12, 2009

## General Instructions

1. Please review the course information.
2. You must write down with whom you worked on the assignment. If this changes from problem to problem, then you should write down this information separately with each problem.
3. Numbered problems are all from the textbook Network Flows .

## Problems

1. Problem 4.10. Multiple Knapsack problem.
2. Problem 4.34, Problem 4.36. K-shortest paths.
3. Given a directed acyclic graph G, with source c let ct(v) be the number of distinct paths from s to v. Give an efficient algorithm to compute ct(v) for all vertices v.
4. Problem 5.51, 5.52. Bit-scaling for shortest paths
5. Suppose that you are given a graph G with real-valued weights c, which is guaranteed to have exactly one negative cost cycle. For any set of node prices p, we define the reduced cost of edge (i,j) by c'(i,j) = c(i,j+ p(i) - p(j). Given a set of reduced costs, we define v(c') to be the min(i,j) {c'(i,j)} Give a polynomial-time algorithm that computes a set of prices p, such that v(c') is maximized. Prove your algorithm is correct.

### Switch to:

cliff@ieor.columbia.edu