1 Algorithms and Networks Algorithms and Networks 2015/2016 Hans L. Bodlaender Johan M. M. van Rooij

2 Teacher 1: Johan van Rooij Room: BBG 507 Office hours: Tuesday 9:00-17:00 For any questions or whatsoever, please contact me directly after the lecture. Rest of the week, working at CQM.

3 Teacher 2: Hans Bodlander Room: BBG 503 Office hours: Thursday 15:00-17:00 - Or see if I’ve time - Or make appointment by

4 What are we going to do today?  Part 1:  Algorithms and Networks: What and Why?  Course organization.  Euler tour example.  Part 2:  Lecture: Stable marriage problem.

5 WHAT AND WHY? Algorithms and Networks

6 Graphs Started in 1736 with paper by Euler: bridges of Königsberg. Can we make a walk where we cross each bridge once (Euler- tour)? A A B B C C D D E E F F

7 Networks  Graphs are a (mathematical) model for many things in the real world:  Road networks, electrical networks, computer networks, organizational diagrams, social networks, structure of software, data base diagrams, …  Often the real world problem has additional information: the network is graph with some extra info  Weights, lengths, distances, costs, time, …  Labels, sizes, names, …

8 Graph/network problems  Often, we want to compute something on the network, motivated from the application (or theoretical curiosity).  Examples:  Shortest path from A to B.  Maximum flow from A to B.  Does the graph have an Euler tour (all edges exactly once).  Does the graph have a Hamiltonian cycle (all nodes exactly once).  How good and fast can we let the computer do this computation?  Theoretical questions of algorithm design.  If fast algorithms exists, of high practical value.

9 Model and algorithm 1. Real-world problem 2. Mathematical model of real-world problem 3. Algorithm for mathematical model 4. Solution produced by the algorithm 5. Translation of solution to solution for real-world problem 6. Check that the translated solution makes sense for the real- world problem.  If you do this for real problems for real customers, step 6 is often followed by designing a better model and returning to step 2.  This course is on steps 2 and 3: modelling and algorithms.

10 Algorithmic techniques  Combinatorial algorithms  Mathematical programming  (Integer) linear programming.  Quadratic programming.  Non-linear programming.  Pruning an exhaustive search tree  Branch and bound.  Constraint programming.  Heuristics.  Greedy heuristics  Local search (iterative improvement, simulated annealing, tabu search, evolutionary algorithms, …)  …

11 Algorithms and complexity  Algorithms  Exact, heuristic, approximation, special cases, …  Polynomial time, exponential time, …  Polynomial space, exponential space, …  Algorithms that parallelize well, or not, …  …  Complexity (asymptotic – O-notation)  Intrinsic difficulty of a problem.  NP-completeness, other complexity classes.  Fixed parameter tractability.  …

12 COURSE ORGANISATION Algorithms and Networks

13 Algorithms and networks  2 lectures per week  Tuesday 9:00 – 10:45 by Johan van Rooij  Thursday 13:15 – 15:00 by Hans Bodlaender  Approximately 8 sets of exercises  Two weeks time for handing them in  2 partial exams  See the course website ( forwww.cs.uu.nl/docs/vakken/an  Important updates on the course  Powerpoint files  Exercises  Schedules  Dates, etc

14 Purpose of course  Knowing and being able to use and apply…  Important algorithmic techniques.  Important algorithms and their applications.  Modelling.  Analysis of algorithms.  In particular for combinatorial problems involving graphs and networks.

15 Topics of course (1)  Paths, flows, matchings, …  Stable marriage  Shortest paths  Travelling salesman problem  Maximum flow  Minimum cost flow  Matching (bipartite, general graphs)  Hard problems, …

16 Topics of course (2)  Paths, flows, matchings, …  Hard problems, …  NP-completeness and complexity  Parameterized complexity  Exact expontial-time algorithms  Kernelization  Treewidth  Approximation algorithms  To be decided

17 Final grade Final Grade= (Exercise set average) * (Exam 1) * (Exam 2) * 0.35  Assuming:  Exercise sets at least 6  Average exams at least 5  Details see webpage.

18 Studying this course  Visit the lectures!  Materials are scattered through literature: often hard to study at home if you were not at the course.  Make notes!  If you are not at the course: borrow/copy notes of other students.  Next to the lectures, you learn the most from the exercises.  Finish them before the corresponding exam, even when the deadline is later.  Topics in exercises that are not in the lectures are exam content!  Some books are recommended, but not necessary.  Use the powerpoints and pdf’s on the website.

19 Graded exercise sets  8 sets (maybe 7 or 9)  You have two weeks for each set.  Every week a new exercise set will be handed out.  Hand in:  By include [AN1], [AN2] etc in the header of your .  Or, on paper during the lectures in the appropriate folder.  Or, on paper in Hans’ mailbox on the 5th floor of the BBG.  Deadline: Monday 23:59 hours.  After the deadline: grade = 0 (exception: joker rule)  Dutch or English  Write clear, legible, etc.  Unreadable, messy: grade = 0

20 On the exercise sets  Lots of work..., but you learn a lot...  Working in pairs:  Both should have worked on and have understood all parts of what is handed in.  Allowed to mark parts as been done by one student.  Not necessary to always have the same pair.  Working alone is also allowed.  Joker rule (write joker on your work or in your ):  Twice during the course you can extend your deadline with three days (Monday 23:59 becomes Thursday 23:59).  Miss a third deadline: grade = 0.  Unless there are very special circumstances for all missed deadlines.

21 Are there...  Any questions on the organisation of the course?

22 EULER TOUR EXAMPLE Algorithms and Networks

23 Let’s get back to our example  Can we make a walk where we cross each bridge once?  Yes!  Can we make a round tour where we cross each bridge once?  No!  Why? (next slide) A A B B C C D D E E F F

24 Why?  Can we make a walk where we cross each bridge once?  Yes: D-E-A-D-C-A-C-F-A-F-B-F-E-B-A-E.  Can we make a round tour where we cross each bridge once? (Eurler tour).  No. Why?  Theorem: A connected graph has an Euler tour, if and only if, every vertex in the graph has even degree.  If: round trip through an odd degree vertex not possible.  Only if... A A B B C C D D E E F F

25 If... proof continued  Lemma: If every vertex in a connected graph has even degree, then it has an Euler tour that can be found using the following algorithm: 1. From any starting vertex v build up any tour using unvisited edges until returning to v.  It is not possible to get stuck because all vertices have even degree before and after the tour visits the vertex. Hence the tour can only enter vertices of degree at least While there exists unvisited edges, select a vertex u on the tour with unvisited edges. 3. Start another trail from u, following unused edges until returning to u, merge the two tours and return to step 2.  Again we cannot get stuck because of the same reason.

26 Euler paths and tours  Theorem: A connected graph has an Euler tour, if and only if, every vertex in the graph has even degree.  Corrolary: A connected graph has an Euler path, if and only if, the graph has 0 or 2 vertices of odd degree.  So we can see directly that our example graph has an Euler path. Even without the given solution.  D-E-A-D-C-A-C-F-A-F-B-F-E-B-A-E A A B B C C D D E E F F

27 TIME FOR A BREAK... Algorithms and Networks