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Lecture17

# 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
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?
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
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
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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