Lec4_RandomAlgs

Lec4_RandomAlgs - COT 6936 Topics in Algorithms Giri Narasimhan ECS 254A EC 2443 Phone x3748 [email protected]

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COT 6936: Topics in Algorithms Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 [email protected] http://www.cs.fiu.edu/~giri/teach/COT6936_S12.html https://moodle.cis.fiu.edu/v2.1/course/view.php?id= 174
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Polynomial-time algorithms for LP • Simplex is not poly-time in the worst-case • Khachiyan’s ellipsoid algorithm: LP is in P • Karmarkar’s interior-point algorithm • Good implementations for LP exist – Works very well in practice – More competitive than the poly-time methods for LP 2/8/12 COT 6936 2
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COT 6936: Topics in Algorithms Integer Linear Programming
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Integer Linear Programming • LP with integral solutions • NP-hard • If A is a totally unimodular matrix , then the LP solution is always integral. – A TUM is a matrix for which every nonsingular submatrix has determinant 0, +1 or -1. – A TUM is a matrix for which every nonsingular submatrix has integral inverse. 2/8/12 COT 6936 4
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Vertex Cover as an LP? • For vertex v, create variable x v • For edge (u,v), create constraint x u + x v 1 • Objective function : Σ x v • Additional constraints : x v 1 • Doesn’t work because x v needs to be from {0,1} 2/8/12 COT 6936 5
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Set Cover • Given a universe of items U = {e 1 , …, e n } and a collection of subsets S = {S 1 , …, S m } such that each S i is contained in U • Find the minimum set of subsets from S that will cover all items in U (i.e., the union of these subsets must equal U) • Weighted Set Cover : Given universe U and collection S, and a cost c(S i ) for each subset S i in S, find the minimum cost set cover 2/8/12 COT 6936 6
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COT 6936 7
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This note was uploaded on 02/18/2012 for the course CIS 6936 taught by Professor Giri during the Spring '12 term at FIU.

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Lec4_RandomAlgs - COT 6936 Topics in Algorithms Giri Narasimhan ECS 254A EC 2443 Phone x3748 [email protected]

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