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Lecture17 - C&O 355 Mathematical Programming Fall 2010...

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C&O 355 Mathematical Programming Fall 2010 Lecture 17 N. Harvey
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Topics Integer Programs Computational Complexity Basics What is Combinatorial Optimization? Bipartite Matching Combinatorial Analysis of Extreme Points Total Unimodularity
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Mathematical Programs We’ve 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
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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, breadth-first search, … NP Å 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?
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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, low-cost trees) , Compilers (coloring) , Online advertising (matching) ... Rich theory of what can be solved efficiently and what cannot Underlying math can be very interesting: high-dimensional geometry, polyhedra, discrete probability, Banach space theory, Fourier analysis, ...
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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 µ E s.t. every vertex has at most one incident edge in M
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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 µ E s.t. every vertex has at most one incident edge in M The blue edges are a matching M
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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 µ E s.t. every vertex has at most one incident edge in M
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Combinatorial IPs are often nice Max-Weight Perfect Matching Given bipartite graph G=(V, E). Every edge e has a weight w e . Find a maximum-weight perfect matching A set M µ E s.t. every vertex has exactly one incident edge in M 5 3 1 2 4 2 2 1
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Combinatorial IPs are often nice Max-Weight Perfect Matching Given bipartite graph G=(V, E). Every edge e has a weight w e .
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