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lecture 10(1) - Lecture 10 Classically Hard Decision...

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Lecture 10: Classically Hard Decision Problems Daniel Frances c 2012 Contents 1 Two Classical Difficult Problems 2 1.1 The Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 0-1 Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Bounded Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Unbounded Knapsack Problem . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 The Subset Sum Problem . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Traveling Salesman Problem . . . . . . . . . . . . . . . . . . . . . . . . 3 1
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1 Two Classical Difficult Problems Most of the problems covered in traditional under-graduate OR courses deal with problems for which there either exist polynomial time algorithms, or there is significant computational success. For example we have shown that the simplex algorithm is not a polynomial algo- rithm, vast computational experience renders this algorithm of great practical value. Given experience to date it far outweighs the value of the polynomial interior-point method which is generally inferior, except in some isolated large LP problem instances. For Integer Programming problems while there is no existing polynomial time algorithm, many problems of reasonable size, and some very large ones can be solved with existing exponential time algorithms. Yet there are some classical problems for which large versions cannot generally be solved in practice. We want to pay some attention to two of these problems. 1.1 The Knapsack Problem Consider a knapsack, i.e. a backpack, that can hold a maximum weight of W Kg. Suppose we have n items of weights w 1 , . . . , w n whose total weight exceeds the capacity. Suppose the
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