Online algorithms with advice for bin packing and scheduling problems

We consider the setting of online computation with advice and study the bin packing problem and a number of scheduling problems. We show that it is possible, for any of these problems, to arbitrarily approach a competitive ratio of 1 with only a constant number of bits of advice per request. For the bin packing problem, we give an online algorithm with advice that is ( 1 + e ) -competitive and uses O ( 1 e log ? 1 e ) bits of advice per request. For scheduling on m identical machines, with the objective function of any of makespan, machine covering and the minimization of the ? p norm, p 1 , we give similar results. We give online algorithms with advice which are ( 1 + e ) -competitive ( ( 1 / ( 1 - e ) ) -competitive for machine covering) and also use O ( 1 e log ? 1 e ) bits of advice per request. We complement our results by giving a lower bound that shows that for any online algorithm with advice to be optimal, for any of the above scheduling problems, a non-constant number (namely, at least ( 1 - 2 m n ) log ? m , where n is the number of jobs and m is the number of machines) of bits of advice per request is needed.

[1]  Adi Rosén,et al.  On Online Algorithms with Advice for the k-Server Problem , 2011, WAOA.

[2]  Yossi Azar,et al.  On‐line machine covering , 1998 .

[3]  Dennis Komm,et al.  On the Advice Complexity of the k-Server Problem , 2011, ICALP.

[4]  Alejandro López-Ortiz,et al.  A Survey of Performance Measures for On-line Algorithms , 2005, SIGACT News.

[5]  Gerhard J. Woeginger,et al.  A polynomial-time approximation scheme for maximizing the minimum machine completion time , 1997, Oper. Res. Lett..

[6]  Marek Chrobak,et al.  SIGACT news online algorithms column 8 , 2005, SIGA.

[7]  P. Strevens.  Iii , 1985 .

[8]  József Békési,et al.  New lower bounds for certain classes of bin packing algorithms , 2012, Theor. Comput. Sci..

[9]  Yossi Azar,et al.  Ancient and New Algorithms for Load Balancing in the lp Norm , 1998, SODA '98.

[10]  Susanne Albers,et al.  On randomized online scheduling , 2002, STOC '02.

[11]  Reza Dorrigiv,et al.  On the Advice Complexity of Buffer Management , 2012, ISAAC.

[12]  Rob van Stee,et al.  Reordering buffer management with advice , 2013, WAOA.

[13]  Harald Sack,et al.  SOFSEM 2013: Theory and Practice of Computer Science , 2013, Lecture Notes in Computer Science.

[14]  Ramaswamy Chandrasekaran,et al.  Improved Bounds for the Online Scheduling Problem , 2003, SIAM J. Comput..

[15]  Christoph Dürr,et al.  Online Bin Packing with Advice of Small Size , 2015, WADS.

[16]  Jérôme Dohrau,et al.  Online Makespan Scheduling with Sublinear Advice , 2015, SOFSEM.

[17]  Barun Chandra.  Does Randomization Help in On-Line Bin Packing? , 1992, Inf. Process. Lett..

[18]  Rudolf Fleischer,et al.  Online Scheduling Revisited , 2000, ESA.

[19]  Gerhard J. Woeginger,et al.  A Lower Bound for Randomized On-Line Scheduling Algorithms , 1994, Information Processing Letters.

[20]  Adi Rosén,et al.  On Online Algorithms with Advice for the k-Server Problem , 2011, Theory of Computing Systems.

[21]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[22]  Jirí Sgall.  A Lower Bound for Randomized On-Line Multiprocessor Scheduling , 1997, Inf. Process. Lett..

[23]  Alejandro López-Ortiz,et al.  Online Bin Packing with Advice , 2012, Algorithmica.

[24]  Alejandro López-Ortiz,et al.  On the Advice Complexity of the k-server Problem Under Sparse Metrics , 2013, Theory of Computing Systems.

[25]  Noga Alon,et al.  Approximation schemes for scheduling , 1997, SODA '97.

[26]  David S. Johnson,et al.  Near-optimal bin packing algorithms , 1973 .

[27]  Dennis Komm,et al.  On the Advice Complexity of the Knapsack Problem , 2011 .

[28]  Dennis Komm,et al.  Advice Complexity and Barely Random Algorithms , 2011, RAIRO Theor. Informatics Appl..

[29]  David B. Shmoys,et al.  Using dual approximation algorithms for scheduling problems: Theoretical and practical results , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[30]  G. S. Lueker,et al.  Bin packing can be solved within 1 + ε in linear time , 1981 .

[31]  Dennis Komm,et al.  On the Advice Complexity of Online Problems , 2009, ISAAC.

[32]  Dennis Komm,et al.  The online knapsack problem: Advice and randomization , 2014, Theor. Comput. Sci..

[33]  Pierre Fraigniaud,et al.  Online computation with advice , 2009, Theor. Comput. Sci..

[34]  Steven S. Seiden,et al.  On the online bin packing problem , 2001, JACM.