1. Integer Programming 2013 1 I.5. Computational Complexity  Nemhauser and Wolsey, p 114 - Ref: Computers and Intractability: A Guide to the Theory of NP- Completeness, M. Garey and D. Johnson, 1979, Freeman  Purpose: classification of problems according to their difficulties ( polynomial time solvability). Many problems look similar, but have quite different complexity.  e.g.) Shortest Path Problem (with nonnegative arc weights, arbitrary arc weights). Chinese Postman Problem ( graph undirected, directed, mixed) and TSP. Matching and Node Packing (Stable Set) in graphs. Spanning Tree, Steiner Tree. Uncapacitated Lot Sizing, Capacitated Lot Sizing. Uncapacitated Facility Location, Capacitated Facility Location.

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3. 3 2.Measuring alg efficiency and prob complexity

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6. 6 3. Some Problems Solvable in Polynomial Time

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11. Integer Programming 2013 11 4. Remarks on 0-1 and Pure-Integer Prog.

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15. Integer Programming 2013 15 5. Nondeterministic Polynomial-Time Algorithms and NP Problems

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17. Integer Programming 2013 17 Equivalence of Optimization and Feasibility Problem

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19. Integer Programming 2013 19 Turing Machine Model  Deterministic Turing Machine : mathematical model of algorithm (refer GJ p.23 - ) Finite State Control -2 -3 3 1 0 2 4 …. Read-write head Tape (Deterministic one-tape Turing machine)

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21. Integer Programming 2013 21 01  This DTM program accepts 0-1 strings with rightmost two symbols are zeroes. ( check with 10100 ), i. e. it solves the problem of integer divisibility by 4.)

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24. Integer Programming 2013 24 Certificate of Feasibility, the Class NP, and Nondeterministic Algorithms  Nondeterministic Turing Machine model Finite State Control -2 -3 3 1 0 2 4 …. Read-write head Tape (Nondeterministic one-tape Turing machine) Guessing Module Guessing head

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29. Integer Programming 2013 29 The Class CoNP

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