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Algoritmi on-line e risoluzione di problemi complessi

Algoritmi on-line e risoluzione di problemi complessi. Carlo Fantozzi (carlo.fantozzi@dei.unipd.it). Summary & Caveats. From Professor Ausiello’s course Structure of the talk: approximation algorithms (core) on-line algorithms (a few words) 16:1 compression ratio!

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Algoritmi on-line e risoluzione di problemi complessi

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  1. Algoritmi on-line e risoluzione di problemi complessi Carlo Fantozzi(carlo.fantozzi@dei.unipd.it)

  2. 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

  3. 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?

  4. 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

  5. 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.

  6. 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?

  7. 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…

  8. Approximation Algs (1) • Poly-time algs that give approx solutions • Guaranteed performance (1970s):maximum “distance” from optimumover 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)}

  9. 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

  10. 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

  11. 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)

  12. The PTAS class • Class of NPO problems that admita 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 anr-approx. solution in poly time for anyr>1 • The metric TSP is not in the PTAS class:no (129/128)-approximate algorithm exists • The Euclidean TSP is in the PTAS class

  13. The FPTAS class • The approximation scheme must bepolynomial 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

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

  15. Approximation Techniques • Greedy approach • Linear programming • Dynamic programming • Randomization • Computational geometry • A compendium of NPO problems: http://www.nada.kth.se/~viggo/wwwcompendium/

  16. On-Line Algorithms • An on-line algorithm • does not know entire input in advance • must take a decision each time it receivesan 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

  17. Performance Evaluation • Competitive analysis: compareon-line solution & optimal off-line sol. • On-line algorithm A is c-competitiveif there exists a constant a such that • Worst-case analysis: A may performmuch better on real-world inputs mA(x) cmOPT(x)+ax

  18. 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)-competitiveif the durations of tasks are known in advance

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