1 Optimal Online Algorithms for Minimax Resource Scheduling Imen BOURGUIBA CAS 744 McMaster University

2 Outline Introduction Objectives The ORMP problem The HLBP problem Comparison Problem definition Analysis Randomized algorithms Conclusion

3 Introduction The objective is to minimize the maximum level of resource allocated at any time during the planning period, this problem is called Online Resource Minimization Problem (ORMP) The objective is to minimize the maximum amount of work assigned to any machine, the problem is called the Hierarchical Line Balancing Problem (HLBP)

4 Introduction Both ORMP and HLBP are special cases of a more general problem, the Online Min- Max Problem (OMMP) The quality of the algorithm is evaluated by the competitive ratio

5 Objectives A simple parameterized deterministic algorithm, called the  -policy, with parameter  and competitive ratio , provided it produces a feasible solution The  -policy is also optimal among all randomized algorithms

6 The ORMP problem Work with different deadlines arrives over time and has to be performed using a resource. The quantities of work that arrive as well as their deadlines become known only at the times of arrival. At a given set of time points, the decision maker decides how much resource to allocate and what part of the available work to perform at that time

7 The HLBP problem Work with different requirements arrives over time and has to be assigned to a collection of machines with different capabilities. The machines form a linear hierarchy based on their capabilities The amount of work that arrives as well as the required machine capabilities become known only at the time of arrival

8 Comparison ORMP appears to be a new problem, and there is no existing literature discussing it For HLBP, the optimal competitive ratio tends to e when the number of machines goes to infinity. When the machines have same capability, the competitive ratio is 2/1-m for the m identical parallel machines [Graham]

9 Problem definition An instance  of the OMMP is a finite sequence (a(1), ….a(T)) of length T The set of all instances with parameter  is denoted by   For example, the set of all instances of length T is denoted by  T let  t = (a(1 ), ….a(T)) denote the first t elements of instance  ; that is,  t denotes the history of instance  up to time t<=T For any instance of length T, a solution r is a sequence (r(1), …...r(T))  R + T of T nonnegative real numbers

10 Problem definition  (  )(t): the decision at time t for instance  under  r  (t) denotes the decision  (  )(t) when  is fixed For any instance    T and any deterministic algorithm  the value   (  ) = max {r  (1), r  (2), …, r  (t)} and the optimal value is  * (  ) = inf r ( {max {r(1), r(2), …, r(t)}} The  -policy is an online policy with parameter  >= 1, defined by: r i  (  ) =   * (  i ) for all i    is the worst-case competitive ratio of policy  over all instances with time     = Sup  {   (  )/  * (  )} An instance  of the OMMP is a finite sequence (a(1), ….a(T)) of length T The set of all instances with parameter  is denoted by   For example, the set of all instances of length T is denoted by  T let  t = (a(1 ), ….a(T)) denote the first t elements of instance  ; that is,  t denotes the history of instance  up to time t For any instance of length T, a solution r is a sequence (r(1), …...r(T))  R + T of T nonnegative real numbers

11 Analysis A simple parameterized algorithm, called the  -policy, with parameter   and competitive ratio  , provided that it is feasible To be feasible a solution must satisfy 3 constraints: 1) the total amount of work performed at time i cannot exceed the amount of work accomplished with r i resources 2) all work must be performed with the respective deadlines 3) the work cannot be performed before it has arrived

12 Theorem For any algorithm    DO if    <  then the  -policy   with parameter   =    achieves the same competitive ratio,    =    Proof:

13 Theorem (cont)  (  )(t): the decision at time t for instance  under  r  (t) denotes the decision  (  )(t) when  is fixed For any instance    T and any deterministic algorithm  the value   (  ) = max {r  (1), r  (2), …, r  (t)} and the optimal value is  * (  ) = inf r ( max {r(1), r(2), …, r(t)}} The  -policy is an online policy with parameter  >= 1, defined by: r i  (  ) =   * (  i ) for all i

14 Randomized algorithms For some problems, randomized algorithms can have better competitive ratios than deterministic algorithms (Motwani and Raghavan and Hoogeveen and Vestjens) A randomized algorithm can have better competitive ratios for the OMMP than any deterministic algorithm.

15 Conclusion The  -policy theory developed in this paper is a powerful tool for finding worst-case optimal algorithms for online min-max problems With an appropriate choice of parameters, the  -policy has as good a competitive ratio as any other deterministic algorithm Under mild conditions an optimal parameter value exists, so the  -policy is

16 References B. Hunsaker y, A. J. Kleywegt, M. W. P. Savelsbergh, andC. A. Tovey, Optimal Online Algorithms for Minimax Resource Scheduling, SIAM J. Discrete Math. Vol. 16, No. 4, 2003, pp A. J. Kleywegt, V. S. Nori, M. W. P. Savelsbergh, and C. A. Tovey, Online resource minimization, in Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, January 1999, SIAM, Philadelphia, 1999, pp