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08 - Five Representative Problems

08 - Five Representative Problems - ∑ v i i ∈ F...

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1/28/08 - Five Representative Problems Lecture: Five representative problems Reading: Chapter 1.2 Earliest Start Time with Pre-emption (1) Preprocess the input to remove jobs whose interval contains another job’s interval (2) Sort the remaining jobs by increasing start (finish) time (3) t = 0 (4) while t < T find earliest start time s i t. Schedule job i. (among jobs starting at s i t, find the time with earliest t i ) t <- t i + 1 endwhile Lemma. After pre-processing, there is a way to renumber the remaining jobs (s i , t i ), ..., (s k , t k ) such that 1 i < j k, s i s j , and t i t j Proof. Let (s 1 , t 1 ), ... , (s k , t k ) be the lexicographic (dictionary) ordering of jobs (sort by s i , break ties using t i ) If i < j then s i s j by construction. Suppose t i > t j Job i is thrown away in step 1. <=> 6-1 6
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Five Representative Problems (1) Interval Scheduling (greedy alg’s) (2) Weighted interval scheduling. Input: Jobs (s i , t i , v i ) v i 0 Output: Feasible schedule F maximizing
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Unformatted text preview: ∑ v i i ∈ F Earliest fnish time will not work (Greedy algorithms won’t work) Solved by dynamic programming “Greedy stays ahead without the greediness.” ±or t = 0, 1, 2, . .., T fnd the best schedule ±or the subinterval end±or (3) Maximum Matching Input. A bipartite graph Output. A matching (a set o± edges with distinct vertices) o± maximum cardinality 6-2 Solution uses max-f ow (4) Maximum independent set Input: An undirected graph G = (GE) Output: An independent set oF max cardinality (set oF vertices with no edge between them) Brute ±orce: O(k 2 n 2 ) Best known O(n 0.79. ..k + O(1) ) Poly-time algo? Equiv’t P = NP (5) Competitive Facility Location: Input: A graph with vertex weights w i ≥ A: 15 + 10 B: 10 Does player B have a strategy that guarantees ≥ p points? PSPACE-hard 6-3...
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