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Course: MSANDE 212, Fall 2009
School: Stanford
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Programming Mathematical and Combinatorial Optimization MS&amp;E 212, Spring 2005-2006, Stanford University Instructor: Ashish Goel Handout 14, practice problems for the nal 1. A company is considering n possible new products for next years product line. A decision now needs to be made regarding which products to manufacture and in what quantities. Product i has a xed cost of fi and a marginal cost of mi per...

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Programming Mathematical and Combinatorial Optimization MS&E 212, Spring 2005-2006, Stanford University Instructor: Ashish Goel Handout 14, practice problems for the nal 1. A company is considering n possible new products for next years product line. A decision now needs to be made regarding which products to manufacture and in what quantities. Product i has a xed cost of fi and a marginal cost of mi per unit. Each unit of product i will fetch a revenue of ri . No more than di units of product i can be sold. The total production budget is B. The goal of the company is to maximize its revenue without exceeding the production budget. Give a dynamic programming solution for deciding the amount of each product that should be produced to meet this goal. Assume that B, di , fi , ri , mi are all integers. 2. The following theorem is a classical result due to Knig. Prove it using the max-ow min-cut o theorem. Note: A vertex cover of a graph is a set of vertices S such that each edge in the graph has at least one endpoint in S. Also, a graph is bipartite if the vertices can be partitioned into two sets P and Q such that all edges are between a vertex in P and a vertex in Q. Knigs Theorem: The size of the largest matching in an undirected graph bipartite is the o same as the size of the smallest vertex cover 3. You are given an undirected bipartite graph. Present a polynomial time algorithm to nd the smallest vertex cover of this graph. 4. Present an algorithm to nd the maximum weight spanning tree of a graph. Give its running time and explain why it is correct. 5. You are given a set of eight men and eight women. Man i and woman j are compatible if i is prime and j is composite, or if i is composite and j is prime. Draw the corresponding bipartite graph and nd a maximum matching. Prove that your matching is the largest possible. For this problem, the number 1 is assumed to be prime. 6. Suppose you are given a min-cost ow problem on a graph with n vertices and m ...

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