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MIT15_053S13_lec10

# MIT15_053S13_lec10 - 15.053/8 Introduction to Integer...

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1 15.053/8 March 14, 2013 Introduction to Integer Programming Integer programming models

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2 Quotes of the Day “Somebody who thinks logically is a nice contrast to the real world.” -- The Law of Thumb Take some more tea, the March Hare said to Alice, very earnestly. I’ve had nothing yet, Alice replied in an offended tone, so I can’t take more. You mean you can’t take less, said the Hatter. It’s very easy to take more than nothing. -- Lewis Carroll in Alice in Wonderland
Combinatorial optimization problems INPUT : A description of the data for an instance of the problem FEASIBLE SOLUTIONS : there is a way of determining from the input whether a given solution x’ (assignment of values to decision variables) is feasible. Typically in combinatorial optimization problems there is a finite number of possible solutions. OBJECTIVE FUNCTION : For each feasible solution x’ there is an associated objective f(x’) . Minimization problem. Find a feasible solution x* that minimizes f( ) among all feasible solutions. 3

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Example 1: Traveling Salesman Problem INPUT : a set N of n points in the plane FEASIBLE SOLUTION: a tour that passes through each point exactly once. OBJECTIVE : minimize the length of the tour. 4
Example 2: Balanced Partition INPUT : A set of positive integers a 1 , …, a n FEASIBLE SOLUTION : a partition of {1, 2, … n} into two disjoint sets S and T. S ∩ T = , S T = {1, … , n} OBJECTIVE : minimize | ∑ i S a i - i T a i | 5 Example: 7, 10, 13, 17, 20, 22 These numbers sum to 89 The best split is {10, 13, 22} and {7, 17, 20}.

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Example 3. Exam Scheduling INPUT : a list of subjects with a final exam; class lists for each of these subjects, and a list of times that final exams can be scheduled. Let a ij denote the number of students that are taking subjects i and j. FEASIBLE SOLUTION : An assignment of subjects to exam periods OBJECTIVE : minimize {a ij : i and j are scheduled at the same time} 6
Example 4: Maximum Clique Problem INPUT : a friendship network G = (N, A). If persons i and j are friends, then (i, j) A. FEASIBLE SOLUTION : a set S of people such that every pair of persons in S are friends. OBJECTIVE : maximize |S| 7

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Example 5: Integer programming INPUT : a set of variables x 1 , …, x n and a set of linear inequalities and equalities, and a subset of variables that is required to be integer. FEASIBLE SOLUTION : a solution x’ that satisfies all of the inequalities and equalities as well as the integrality requirements. OBJECTIVE : maximize ∑ i c i x i 8 Example: maximize 3x + 4y subject to 5x + 8y ≤ 24 x, y ≥ 0 and integer
Which of the following is false? 9 1. The Traveling Salesman Problem is a combinatorial optimization problem.

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MIT15_053S13_lec10 - 15.053/8 Introduction to Integer...

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