Lecture19

Lecture19 - C&O 355 Lecture 19 N. Harvey Topics Solving...

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Lecture 19 N. Harvey
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Topics Solving Integer Programs Basic Combinatorial Optimization Problems Bipartite Matching, Minimum s-t Cut, Shortest Paths, Minimum Spanning Trees 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 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 . 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 The blue edges are a max-weight perfect 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 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
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Minimum s-t Cut in a Graph (from Lecture 12) Let G=(V,E) be a graph. Fix two vertices s,t 2 V. An
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Lecture19 - C&O 355 Lecture 19 N. Harvey Topics Solving...

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