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midterm - T j = ∑ i ∈ A j t i will be the load of...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) Midterm – 02/16/10 Problem 1. Show that a graph has a unique minimum spanning tree if, for every cut of the graph, the edge with the smallest cost across that cut is unique. Show that the converse is not true by giving a counterexample. Problem 2. The edge connectivity λ ( G ) of an undirected graph G ( V,E ) is defined as the cardinality of a minimum set of edges S E , whose removal disconnects G . Give a polynomial-time algorithm for computing λ ( G ). Problem 3. Recall the job scheduling problem. We have m machines and n jobs such that any machine j takes time t i to process job i . Let A j be the set of jobs assigned to machine j . In that case,
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Unformatted text preview: T j = ∑ i ∈ A j t i will be the load of machine j . Our goal is to find an assignment of jobs to machines that would minimize max j T j . Denote this minimum value OPT . Consider a slightly modified greedy approach to the one showed in class. First, we sort the jobs so that t 1 ≥ t 2 ≥ ··· ≥ t n . Then, we assign them iteratively to the machines, every time to the machine with the smallest load. Show that the approximation factor of this algorithm is at most 3 / 2. Hint: Note that if n > m then t m +1 ≤ OPT/ 2. Why is this true? Extra Credit: Prove that the approximation factor of the algorithm is actually equal to 4 / 3....
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