Lec4 - COT 6936 Topics in Algorithms Giri Narasimhan ECS...

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1 1/12/10 COT 6936 1 COT 6936: Topics in Algorithms Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 [email protected] http://www.cs.fiu.edu/~giri/teach/COT6936_S10.html https://online.cis.fiu.edu/portal/course/view.php?id=427 Optimization Problems • Problem: – A problem is a function (relation) from a set I of instances of the problem to a set S of solutions. p: I ± S • Decision Problem: – Problem with S = {TRUE, FALSE} • Optimization Problem: – Problem with a mapping from set S of solutions to a positive rational number called the solution value I ± S ± m(I,S) 1/12/10 COT 6936 2 Optimization Versions of NP-Complete Problems • TSP • CLIQUE • Vertex Cover & Set Cover • Hamiltonian Cycle • Hamiltonian Path • SAT & 3SAT • 3-D matching 1/12/10 COT 6936 3
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2 Optimization Versions of NP-Complete Problems • Computing a minimum TSP tour is NP-hard (every problem in NP can be reduced to it in polynomial time) • BUT, it is not known to be in NP • If P is NP-Complete, then its optimization version is NP-hard (i.e., it is at least as hard as any problem in NP, but may not be in NP) – Proof by contradiction! 1/12/10 COT 6936 4 Performance Ratio • Approximation Algorithm A – A( I ) • Optimal Solution – OPT( I ) • Performance Ratio on input I for minimization problems – R A ( I ) = max {A( I )/OPT( I ), OPT( I )/A( I )} • Performance Ratio of approximation algorithm A – R A = inf {r ± 1| R A ( I ) ² r, for all instances} 1/12/10 COT 6936 5 Metric Space • It generalizes concept of Euclidean space • Set with a distance function (metric) defined on its elements – D : M X M R (assigns a real number to distance between every pair of elements from the metric space M ) • D( x , y ) = 0 iff x = y • D( x , y ) ± 0 • D( x , y ) = D( y , x ) • D( x , y ) + D( y , z ) ± D( x , z ) 1/12/10 COT 6936 6
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Examples of metric spaces • Euclidean distance • L p metrics • Graph distances – Distance between elements is the length of the shortest path in the graph 1/12/10 COT 6936 7 TSP • TSP in general graphs cannot be approximated to within a constant ( Why ?) – What is the approach? • Prove that it is hard to approximate! • TSP in general metric spaces holds promise! – NN heuristic [Rosenkrantz, et al. 77] • NN(I) ± ² (ceil(log 2 n) + 1) OPT(I) – 2-OPT, 3-OPT, k-OPT, Lin-Kernighan Heuristic • Can TSP in general metric spaces be approximated to within a constant? 1/12/10 COT 6936 8 TSP in Euclidean Space • TSP in Euclidean space can be approximated. – MST Doubling (DMST) Algorithm • Compute a MST, M • Double the MST to create a tour, T 1 • Modify the tour to get a TSP tour, T – Theorem : DMST is a 2-approximation algorithm for Euclidean metrics, i.e., DMST( I ) < 2 OPT( I ) – Analysis: • L(T) ± L(T 1 ) = 2L(M) ± 2L(T OPT ) – Is the analysis tight? 1/12/10
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Lec4 - COT 6936 Topics in Algorithms Giri Narasimhan ECS...

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