DMP87 - Scheduling, Timetabling and Routing
Weekly Notes
Lecture 3, Fall 2008

Lecture February 4

We treated the following topics.

• Mathematical programming formulations:
  • Formulation for $Qm\vert p_j=1\vert\sum h_j(C_j)$ and relation with transportation problems (pages 384 and 386 of text book)

  • Totally unimodular matrices and sufficient conditions for total unimodularity i) two ones per column and ii) consecutive 1's property (pages 37-50 of [3]).

  • Formulation of $1\vert prec\vert \sum w_j C_j$ and $Rm\vert\vert \sum C_j$ as weighted bipartite matching and assignment problems.

  • Formulation of set covering, set partitioning and set packing. (Slides)

  • Formulation of Traveling Salesman Problem (Slides, pages 81-89 [1]).

  • Formulation of $1\vert prec\vert \sum w_j C_j$ and how to deal with disjunctive constraints (text book, pages 391, 392)

• Dynamic programming, general underlying ideas (pages 396, 397 of the text book).
• Branch and bound, general underlying ideas (pages 400, 403 of the text book).

• Constraint Programming (Slides, chp 5 of [2]).
  • Constraint Satisfaction Problem

  • Examples as Graph Coloring, $n$-queens and cryptoarithmetic and the First order logic as a language to express constraints

  • Types of Variables, Constraints and backtracking search.

The next lecture we will finish Constraint Programming and treat Heuristics. Reading material Appendix C and D of the text book.

I have been asked to move again the Monday lecture to another timeslot. I will check the feasibility of the slot 14-16 in the days days and give notice.

Bibliography

1

David L. Applegate, Robert E. Bixby, Vasek Chvátal, and William J. Cook.
The Traveling Salesman Problem: A Computational Study.
Princeton University Press, 2006.

2

Stuart Russell and Peter Norvig.
Artificial Intelligence: A Modern Approach.
Prentice Hall, Englewood Cliffs, New Jersey, USA, second edition, 2003.

3

Laurence A. Wolsey.
Integer programming.
Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, USA, 1998.

Assignments

Exercise 1

Give at least two integer programming formulations for the problem of coloring, legally, a graph with the minimal number of colours.