Algoritmi on-line e risoluzione di problemi complessi Carlo Fantozzi

Summary & Caveats From Professor Ausiello’s course Structure of the talk:  approximation algorithms (core)  on-line algorithms (a few words) 16:1 compression ratio! Heterogeneous audience (& dumb speaker) No details: only the framework of the course

Complexity Theory What is efficiently computable? Efficient = “Polynomial time” Decision problems: yes/no answer The P class: poly-time solution The NP class: poly-time verification Poly-time (Karp) reductions  NP-hard problems  NP-complete problems P  NP; is it also P=NP?

Optimization problems Optimization problem: a quadruple  Set of instances I  Set of feasible solutions S  Cost function m(x,y), x  I, y  S  min/MAX We are interested in optimal solutions Example: min vertex cover

Optimization: 3 Approaches Decision problem: “is there a solution with cost  m?  m?” Evaluation problem: “find optimal value of cost function” Constructive optimization problem: “find the optimal solution” Approaches are not (proved to be) equivalent. They are equivalent whenever the decision problem is NP-complete.

Complexity Revised (1) Solutions have polynomial size Cost function computable in poly time PO class: poly-time algorithm returns optimal solution and optimal cost NPO class: poly-time verification alg. for feasible solutions PO  NPO; is it also PO=NPO?

Complexity Revised (2) Poly-time Turing reductions; P1  T P2 if there exists an algorithm for P1 which queries an oracle for P2 NP-hard optimization problem: every decision problem is T-reducible to it in poly time PO=NPO if and only if P=NP: do not hope for poly-time algs…

Approximation Algs (1) Poly-time algs that give approx solutions Guaranteed performance (1970s): maximum “distance” from optimum over all problem instances Absolute approximation: cost within an additive constant from opt. Relative error: E(x,y)=|opt(x)-m(x,y)|/MAX{opt(x),m(x,y)} Performance ratio: R(x,y)=MAX{m(x,y)/opt(x),opt(x)/m(x,y)}

Approximation Algs (2) r-approximate algorithm: R(x,y)  r  x, i.e. cost is at most r times the optimum Example: min vertex cover (NP-hard)  Pick an edge  Include nodes of the edge into the cover  Remove nodes from the graph; repeat Greedy algorithm is 2-approximate

Approximation classes Sometimes things are not so easy Some problems are “more approximable” than others Many approximation classes exist  The APX class  The PTAS class  The FPTAS class These classes form a hierarchy if P  NP

The APX class Class of all NPO problems that admit a poly-time r-approximate algorithm The general TSP is not in the APX class (unless P=NP) The metric TSP is in the APX class (best algorithm: Cristofides', with a 3/2 performance ratio)

The PTAS class Class of NPO problems that admit a poly-time approximation scheme An approximaxion scheme is a tunable alg: can trade running time for performance An algorithm is a PTAS for P if it gives an r-approx. solution in poly time for any r>1 The metric TSP is not in the PTAS class: no (129/128)-approximate algorithm exists The Euclidean TSP is in the PTAS class

The FPTAS class The approximation scheme must be polynomial in both |x| and r Problems in this class are optimally solvable for most practical purposes The Euclidean TSP is not in FPTAS Maximum Knapsack is in FPTAS Not many problems belong to FPTAS

Class Hierarchy NPO e.g. TSP APX e.g. MTSP PTAS e.g. ETSP FPTAS e.g. Knapsack

Approximation Techniques Greedy approach Linear programming Dynamic programming Randomization Computational geometry A compendium of NPO problems:

On-Line Algorithms An on-line algorithm  does not know entire input in advance  must take a decision each time it receives an input piece; each decision has a cost  total cost must be maximized/minimized The constraint on the input makes these optimization problems much more difficult! Off-line algorithm: knows entire input as usual

Performance Evaluation Competitive analysis: compare on-line solution & optimal off-line sol. On-line algorithm A is c-competitive if there exists a constant a such that Worst-case analysis: A may perform much better on real-world inputs m A (x)  cm OPT (x)+a  x

Examples Ski rental problem: for how long should a novice skier rent skis before buying? Buy when the total cost of rental becomes higher than the cost of skis Greedy strategy is 2-competitive Task scheduling problem Graham’s algorithm (1966): assign new task to the less loaded processor Greedy strategy is (2-1/p)-competitive if the durations of tasks are known in advance