Approx - CSE 4101/5101 Prof. Andy Mirzaian Approximation...

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CSE 4101/5101 Approximation Algorithms for NP-hard Optimization Problems Prof. Andy Mirzaian
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References: [CLRS] chapter 35 Lecture Notes 9, 10 AAW Sanjoy Dasgupta, Christos Papadimitriou, Umesh Vazirani, "Algorithms,“ McGraw-Hill, 2008. Jon Kleinberg and Eva Tardos, "Algorithm Design," Addison Wesley, 2006. Vijay Vazirani , Approximation Algorithms ,” Springer, 2001. Dorith Hochbaum (editor), "Approximation Algorithms for NP-Hard Problems," PWS Publishing Company, 1997. Michael R. Garey, David S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness," W.H. Freeman and Company, 1979. 2
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COMBINATORIAL OPTIMIZATION find the best solution out of finitely many possibilities. 3
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Mr. Roboto: Find the best path from A to B avoiding obstacles A B There are infinitely many ways to get from A to B. We can’t try them all. But you only need to try finitely many critical paths to find the best. With brute-force there are exponentially many paths to try. There may be a simple and fast (incremental) greedy strategy. 4
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Mr. Roboto: Find the best path from A to B avoiding obstacles The Visibility Graph: 4n + 2 vertices (n = # rectangular obstacles) A B 5
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Combinatorial Optimization Problem (COP) INPUT : Instance I to the COP. Feasible Set: FEAS( Ι29 = τηε σετ οφ αλλ φεασιβλε (ορ ωαλιδ29 σολυτιονσ φορ ινστανχε Ι, υσυαλλψ εξπρεσσεδ βψ α σετ οφ χονστραιντσ. Οβϕεχτιωε Χοστ Φυνχτιον Ινστανχε Ι ινχλυδεσ α δεσχριπτιον οφ τηε οβϕεχτιωε χοστ φυνχτιον, Χοστ[ I] τηατ μαπσ εαχη σολυτιον Σ (φεασιβλε ορ νοτ29 το α ρεαλ νυμβερ ορ ± ∞. Γοαλ Οπτιμιζε (ι.ε., μινιμιζε ορ μαξιμιζε29 τηε οβϕεχτιωε χοστ φυνχτιον. Οπτιμυμ Σετ ΟΠΤ(Ι29 = {Σολ ∈ ΦΕΑΣ(Ι29 | Χοστ[ I] (Σολ29 Χοστ[ I] (Σολ 29, 2200Σολ ∈ΦΕΑΣ(Ι29 } τηε σετ οφ αλλ μινιμυμ χοστ φεασιβλε σολυτιονσ φορ ινστανχε Ι Χομβινατοριαλ Μεανσ τηε προβλεμ στρυχτυρε ιμπλιεσ τηατ ονλψ α φινιτε νυμβερ οφ σολυτιονσ νεεδ το βε εξαμινεδ το φινδ τηε οπτιμυμ. 6
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Polynomial vs Exponential time Time complexity Running time n 1 sec. n log n 20 sec. n2 12 days 2n 40 quadrillion (1015) years Assume: Computer speed 106 IPS and input size n = 106 7
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COP Examples q “Easy” (polynomial-time solvable): Ø Shortest (simple) Path [AAW, Dijkstra, Bellman-Ford, Floyd-Warshall …] Ø Minimum Spanning Tree [AAW, Prim, Kruskal, Cheriton-Tarjan, …] Ø Graph Matching [Jack Edmonds, …] q “NP-Hard” (no known polynomial-time solution): Ø Longest (simple) Path Ø Traveling Salesman Ø Vertex Cover Ø Set Cover Ø K-Cluster Ø 0/1 Knapsack 8
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Example: Primality Testing Given integer N  2, is N a prime number?
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Approx - CSE 4101/5101 Prof. Andy Mirzaian Approximation...

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