## Algorithmic Methods - 0368.4139

#### Yossi Azar ( azar@tau.ac.il ) 2nd Semester, 2022/23 - Time: Mon 4-7pm School of Computer Science, Tel-Aviv University For the outline of the course given in 2021 see course2021

Previous Exams

Exercises

Ex1:

Ex2:

Ex3:

30% exercises
70% exam (on July 24, 2023 at 9:00am)

Text books (only couple of chapters from each book)

(1) Linear Programming by H. Karloff, Birkhauser, 1991.
(2) Introduction to Algorithms by T. Cormen, C. Leiserson and R. Rivest, MIT Press, 1990
(3) Approximation Algorithms for NP-hard problems edited by S. Hochbaum, PWS Publishing company, 1997.
(4) Approximation Algorithms by Vijay Vazirani, Springer, 2003.
(5) Survey on Local Ratio Survey

### Course syllabus:

Linear Programming - simplex, duality, the ellipsoid algorithm, applications.
Approximation algorithms,

Randomized algorithms, De-randomization,

Distributed and Parallel algorithms,
On-line algorithms.

### Course outline

1.      Mar 13:
Introduction
Examples of linear programming problems
Basic definitions (canonical, standard, general forms, polyhedron, polytope, basic feasible solution)
Theorems A, B, C on polyhedrons and their vertices

2.      Mar  20:
The simplex method
Initialization of the simplex method
The dual

3.      Mar  27:
The dual
Complementary slackness
Economic interpretation
Feasible vs. Optimal solutions
Farkas Lemma
The minimax theorem

4.      Apr 17:
The Ellipsoid algorithm (Yamanitsky-Levin 1982 variant)

5.      Apr 24:
The Ellipsoid algorithm with oracle
Theorem D
Bi-stochastic matrices
2-approximation for weighted vertex cover

6.      May 1:
Approximations for MAX-SAT (randomized and deterministic algorithms)
De-randomization
Approximations for Routing

7.      May 8:
Approximations for Routing
Approximations for Machine Scheduling (identical + related machines)
Online Algorithms

8.      May 15:
Reduction from optimality to feasibility (non-polynomial number of constraints)
Approximations for Machine Scheduling (restricted + unrelated machines)

9.      May 22:
Distributed coloring of a circle (upper bound)
Distributed coloring of a circle (lower bound)

10.  May 29:
Local ratio - vertex cover
Local ratio - Interval scheduling

11.  Jun 5:
Interval scheduling
Steiner tree
Generalized Steiner forest

12.  Jun 12:
Generalized Steiner forest
PTAS for scheduling

13.  Jun 19:
PTAS for scheduling
Exercises

14.  Jun 26: extra