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Unformatted text preview: C&O 355 Mathematical Programming Fall 2010 Lecture 17 N. Harvey Topics Integer Programs Computational Complexity Basics What is Combinatorial Optimization? Bipartite Matching Combinatorial Analysis of Extreme Points Total Unimodularity Mathematical Programs Weve Seen Linear Program (LP) Convex Program Semidefinite Program (SDP) Integer Program (IP) (where f is convex) (where X is symmetric matrix corresponding to x ) Can be efficiently solved e.g., by Ellipsoid Method Cannot be efficiently solved assuming P NP Computational Complexity If you could efficiently (i.e., in polynomial time) decide if every integer program is feasible, then P = NP And all of modern cryptography is broken And you win $1,000,000 P NP coNP Sorting, string matching, breadthfirst search, NP coNP Is LP feasible? Is integer program feasible? Can graph be colored with k colors? Does every coloring of graph use > k colors? Is integer program infeasible? Combinatorial Optimization Study of optimization problems that have discrete solutions and some combinatorial flavor (e.g., involving graphs) Why are we interested in this? Applications: OR (planning, scheduling, supply chain) , Computer networks (shortest paths, lowcost trees) , Compilers (coloring) , Online advertising (matching) ... Rich theory of what can be solved efficiently and what cannot Underlying math can be very interesting: highdimensional geometry, polyhedra, discrete probability, Banach space theory, Fourier analysis, ... Combinatorial IPs are often nice Maximum Bipartite Matching (from Lecture 2) Given bipartite graph G=(V, E) Find a maximum size matching A set M E s.t. every vertex has at most one incident edge in M Combinatorial IPs are often nice Maximum Bipartite Matching (from Lecture 2) Given bipartite graph G=(V, E) Find a maximum size matching A set M E s.t. every vertex has at most one incident edge in M The blue edges are a matching M Combinatorial IPs are often nice The natural integer program This IP can be efficiently solved, in many different ways Maximum Bipartite Matching (from Lecture 2) Given bipartite graph G=(V, E) Find a maximum size matching A set M E s.t. every vertex has at most one incident edge in M Combinatorial IPs are often nice MaxWeight Perfect Matching Given bipartite graph G=(V, E). Every edge e has a weight w e . Find a maximumweight perfect matching A set M E s.t. every vertex has exactly one incident edge in M 5 3 1 2 4 2 2 1 Combinatorial IPs are often nice MaxWeight Perfect Matching Given bipartite graph G=(V, E). Every edge e has a weight w e ....
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This note was uploaded on 06/16/2011 for the course CO 355 taught by Professor Harvey during the Winter '10 term at Waterloo.
 Winter '10
 Harvey

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