Algorithms - COMPLESSITÀ (2012/2013)

Course code
Name of lecturer
Roberto Posenato
Number of ECTS credits allocated
Academic sector
Language of instruction
I semestre dal Oct 1, 2012 al Jan 31, 2013.
Web page

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

Lesson timetable

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.

Recommended Prerequisite
To attend the course lessons in a productive way, a student should be confident with the following concepts:
1. Basic data structures as list, stack, queue, tree, heap.
2. Graph representation and fundamental graph algorithms:
2.1 Graph visit: BFS, DFS.
2.2 Topological ordering. Connected component.
2.3 Minimal spanning tree. Kruskal and Prim algorithm.
2.4 Single-source shortest path: Dijkstra algorithm and Bellman-Ford one.
2.5 All-pairs shortest path: Floyd-Warshall algorithm and Johnson one.
2.6 Max flow: Ford-Fulkerson algorithm.
A recommended book to revise the above concepts is ``Introduction to Algorithms" di T. H. Cormen, C. E. Leiserson, R. L. Rivest e C. Stein (3 ed.).


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.

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.

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

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.