Algorithms - COMPLESSITÀ (2011/2012)

Course code
4S02709
Name of lecturer
Roberto Posenato
Number of ECTS credits allocated
6
Academic sector
ING-INF/05 - INFORMATION PROCESSING SYSTEMS
Language of instruction
Italian
Location
VERONA
Period
I semestre dal Oct 3, 2011 al Jan 31, 2012.
Web page
https://elearning.univr.it/j/index.php?option=com_wrapper&Itemid=60&url=/sso/accedi_e.php?go=/e/course/view.php?id=811

To show the organization of the course that includes this module, follow this link * Course organization

Lesson timetable

I semestre
Day Time Type Place Note
Monday 2:30 PM - 4:30 PM lesson Lecture Hall I  
Tuesday 2:30 PM - 4:30 PM lesson Lecture Hall I  
Thursday 9:30 AM - 10:30 AM lesson Lecture Hall I  

Learning outcomes

The goal of this module is to introduce students to computational complexity theory in general, to the NP-completeness theory in detail and to computational analysis of problems with respect to their approximability.

Syllabus

Introduction.
Computational model concept, computational resource, efficient algorithm and tractable problem.

Computational models
Turing Machine (MdT): definition, behavior, configuration, production and computation concepts. MdT examples. MdT and languages: the difference between accepting and deciding a language. MdT extension: multi-tape MdT (k-MdT)

Time Complexity
Time computational resource. Computational class TIME(). Theorem about polynomial relation between k-MdT computations and MdT ones (sketch of proof).
Introduction to Random Access Machine (RAM) computational model: configuration, program and computation concepts. RAM: computation time by uniform cost criterion and by logarithmic cost one. Example of a RAM program that determines the product of two integers.
Theorem about simulation cost of a MdT by a RAM.
Theorem about simulation cost of a RAM program by a MdT.
Sequential Computation Thesis and its consequences.
Linear Speed-up Theorem and its consequences.

P Computational Class.
Problems in P: PATH, MAX FLOW, PERFECT MATCHING.

Extension of MdT: non-deterministic MdT (NMdT).
Time resource for k-NMdT. NTIME() computational class.
Example of non-deterministic algorithm computable by a NMdT: algorithm for Satisfiability (SAT).

Relation between MdT and NMdT.

NP Computational Class.
Relation between P and NP. Example of a problem into NP: Travel-salesman Problem (TSP).
An alternative characterization of NP: polynomial verifiers.

EXP Computation Class.

Space Complexity.
Space complexity concept. MdT with I/O. Computational Classes: SPACE() and NSPACE().
Compression Theorem.
Computational Classes: L and NL.
Example of problems: PALINDROME ∈ L and PATH ∈ NL.

Theorems about relations between space and time for a MdT with I/O.
Relations betwee complexity classes.
Proper function concept and example of proper functions.
Borodin Gap Theorem.

Reachability method. Theorem about space-time classes: NTIME(f(n)) ⊆ SPACE(f(n)), NSPACE(f(n)) ⊆ TIME(k(log n+f(n))).

Universal MdT.
The Hf set.
Lemma 1 and 2 for time hierarchy theorem.
Time Hierarchy Theorem: strict and no-strict versions.
P ⊂ EXP Corollary.

Space Hierarchy Theorem. L ⊂ PSPACE Corollary.
Savitch Theorem. SPACE(f(n))=SPACE(f(n)^2) corollary. PSPACE=NPSPACE Corollary.

Reductions and completeness.
Reduction concept and logarithmic space reduction. HAMILTON PATH ≤log SAT, PATH ≤log CIRCUIT VALUE, CIRCUIT SAT ≤log SAT.
Language completeness concept.
Closure concept with respect to reduction.
Class reduction of L, NL, P, NP, PSPACE and EXP.
Computation Table concept.
Theorem about P-completeness of CIRCUIT VALUE problem.
Cook Theorem: an alternative proof.
Gadget concept and completeness proof of: INDEPENDENT SET, CLIQUE, VERTEX COVER and others.


Randomized computational complexity.
Probabilistic Turing machines.
Computational classes: BPP, RP, coRP e ZPP. Relationship between BPP and other classes.
Randomized reductions.

Approximation algorithms and approximate complexity classes.

Reference books
Author Title Publisher Year ISBN Note
Christos H. Papadimitriou Computational complexity Addison Wesley 1994 0201530821
S. Arora, B. Barak Computational Complexity. A modern approach (Edizione 1) Cambridge University Press 2009 9780521424264 principale

Assessment methods and criteria

The examination consists of a written test. The grade in this module is worth 1/3 of the grade in the Algorithms examination.