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Lecture6 - Integer Programming IE418 Lecture 6 Dr Ted...

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Integer Programming IE418 Lecture 6 Dr. Ted Ralphs
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IE418 Lecture 6 1 Reading for This Lecture Wolsey Chapter 6 N&W Sections I.5.3-I.5.6
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IE418 Lecture 6 2 Certificates of Optimality Suppose you had the optimal solution LP and wanted to prove to someone else it was optimal. You could simply produce the primal and dual solutions. Can optimality be verified in polynomial time? In O ( mn ) operations, one could verify optimality. However, what is the magnitude of the numbers? They are the ratio of two integers, each of which can be represented in a size that is polynomially bounded. Information that can be used to check optimality in polynomial time is called a certificate of optimality . If a binary string has a size polynomial in the length of the input, then it is said to be short . Obviously, a certificate of optimality must be short.
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IE418 Lecture 6 3 Importance of Certificates Every polynomially solvable problem has a certificate. It is not known whether every problem with a certificate is polynomially solvable. Until 1979, linear programming was one problem with a certificate that was not known to be polynomially solvable. The Perfect Matching Problem Recall we derived a complete description of the perfect matching polytope. Although the formulation has an exponential number of constraints, this yields a polynomial certificate. This problem can in fact be solved in polynomial time.
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IE418 Lecture 6 4 Problem Reduction Recall that mixed-integer linear programming is a special case of mathematical programming. If we had a fast algorithm for solving general mathematical programs, we would be able to solve integer programs as well. Furthermore, the Traveling Salesman Problem is a special case of pure integer linear programming. Hence, general integer programming is, in some sense, at least as difficult as the TSP . In this way, we can develop a hierarchy of problems. In some cases, we will show that two problem are equally difficult. Our goal is to divide the space of all problems into complexity classes according to relative difficulty.
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IE418 Lecture 6 5 Polynomial Reduction Suppose we are given two problems X 1 and X 2 . We want to show that if we solve one, we can also solve the other.
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  • Spring '08
  • Ralphs
  • Computational complexity theory, optimality, NP-complete, Boolean satisfiability problem, polynomial time, N P-complete

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