NDMI084 Introduction to approximation and randomized algorithms

Winter semester 2022/23

Petr Kolman

The weekly lecture takes place every Wednesday in Lecture room S3 (Mala Strana) from 2 - 3:30 pm.

The tutorials take place once in two weeks on Tuesday from 2:00 - 3:30 pm and are led by Matej Lieskovský.

Czech Lectures take place on Monday at 12:20 - 13:50 pm, lecture room S3, and are given by Jiří Sgall.

Lecture notes by J. Sgall (in Czech)

Syllabus

This course covers techniques for design and analysis of algorithms, demonstrated on concrete combinatorial problems. For many optimization problems it is impossible (or NP-hard) to design algorithms that finds an optimal solution fast. In such a case we study approximation algorithms that work faster, at the cost that we find a good solution, not necessarily an optimal one. Often we use randomness in design of (approximation and other) algorithms, which allows to solve problems more efficiently or even to solve problems that are otherwise intractable. Recommended for the 3rd year.

Covered topics

• October 5, 2022
  • Introduction.
  • Randomized protocol for computing the average salary.
  • Optimization and NP-optimization problems.
  • Approximation ratio [WS 1.1, V1].
  • TSP - limits on approximation in the general setting.
• October 12
  • Metric TSP - 2-approximation
  • Christofides' 1.5-approximation [WS 2.4, V 3.2].
  • Discrete probability - review of elementary definitions and properties (see old notes or more detailed Notes on probability by J. Matousek).
• October 19
  • Randomized Quicksort [MU 2.5, MR 1, KT 13.5].
  • Contention resolution in a distributed system - randomized protocol [KT 13.1].
  • Randomized global minimum cut algorithm [KT 13.2].
• October 26
  • Randomized global minimum cut algorithm [KT 13.2].
  • Scheduling on identical machines - local search [WS 2.3].
• November 2
  • Greedy algorithm for scheduling on identical machines (list scheduling, longest processing time first), online [WS 2.3].
  • Greedy algorithms for bin packing (first fit, best fit, any fit) [WS 3.3, V 9].
• November 16
  • Algorithms for MAX SAT [WS 5.1-5.5, V 16].
    
    • Simple 1/2-approximation (RAND SAT).
    • Derandomization of RAND SAT by the method of conditional expectation [WS 5.3].
    • Biased randomized approximation algorithm.
    • Approximation based on linear programming relaxation and randomized rounding.
• November 23
  • Randomized algorithms for MAX SAT [WS 5.1-5.5, V 16].
    
    • Approximation of MAX SAT based on LP relaxation - analysis.
    • Choosing the better of two solutions - 3/4 approximation for MAX SAT [WS 5.5].
  • Greedy algorithm for edge disjoint paths in graphs [KT 11.5].
• November 30
  • Greedy algorithm for paths in graphs with capacities [KT 11.5].
  • Algorithms for vertex and set cover [WS 1.2-1.6, 7.1, V 13-14].
    
    • The greedy algorithm and its analysis using duality of linear programming.
• December 7
  • Algorithms for vertex and set cover [WS 1.2-1.6, 7.1, V 13-14].
    
    • The primal algorithm, the dual algorithm, the primal-dual algorithm.
  • Parallel randomized algorithm for maximum independent set problem - description [MR 12.3, also nice notes by Eric Vigoda at http://www.cc.gatech.edu/~vigoda/7530-Spring10/MIS.pdf].
• December 14
  • Parallel randomized algorithm for maximum independent set problem - analysis [MR 12.3, the notes by E. Vigoda].
  • Parallel randomized algorithm for maximum independent set problem - pairwise independence [MR 12.3, the notes by E. Vigoda].
• December 21
  • Derandomization of the parallel alg. for max IS
  • Hashing
    • The dictionary problem and universal hashing [MR 8.4, MU 13.3].
    • 2-universal hashing and dynamic dictionary with expected constant time per operation.
    • Perfect hashing and static dictionary with constant time per lookup in the worst case.
• January 4, 2023
  • Randomized testing of polynomial identities [MR 7.1, MU 1.3].
  • Testing the existance of perfect matchings in bipartite and general graphs [MR 7.3, 12.4].
  • Parallel randomized algorithm for a perfect matching [MR 7.3, 12.4].

Study Material

Notes and recordings from edition of the course in 2020/21.

Textbooks

There is no required textbook but the material presented in the course is covered by several good books.

[WS] D. P. Williamson, D. B. Shmoys: The Design of Approximation Algorithms, Cambridge University Press, 2011.

[MR] R. Motwani, P. Raghavan: Randomized algorithms, Cambridge University Press, 1995.

[MU] M. Mitzenmacher, E. Upfal: Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press, 2005.

[V] V. V. Vazirani: Approximation Algorithms, Springer, 2001.

[KT] J. Kleinberg, E. Tardos: Algorithm Design, Pearson, 2006.

Other courses on approximation or randomized algorithms

• Approximation Algorithms by Chandra Chekuri
• Approximation Algorithms and Approximability by Mohammad R. Salavatipour
• Approximation Algorithms by Anupam Gupta and R. Ravi