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  1. Moritz Buchem: Welcome everybody to my talk on additive approximation schemes for load balancing problems. My name is Moritz Buchem and this is joint work together with Lars Rohwedder, Tjark Vredeveld and Andreas Wiese.
  2. Moritz Buchem: Before I go into the details of our results, I would first like to shortly define the problems that we're considering.
  3. Moritz Buchem: A solution or a schedule for these problems is given as an assignment of jobs to machines.
  4. Moritz Buchem: We define the individual load of machine I as the sum of the processing times of the jobs assigned to machine I which is simply the time that it takes machine I to finish all jobs assigned to it.
  5. Moritz Buchem: In particular, we consider three different load balancing objectives. The first objective we consider is the makespan where we want to minimize the maximum of the machine loads.
  6. Moritz Buchem: These load balancing problems have been studied quite extensively in the literature and they're all known to be strongly NP hard in the setting that we consider with a number of machines as part of the input.
  7. Moritz Buchem: There has also been some
  8. Moritz Buchem: Research on multiplicative approximation schemes. So for makespan minimization and Santa Claus, we know that polynomial time approximation schemes exist, and these have also been improved to efficient polynomial time
  9. Moritz Buchem: approximations schemes. For the envy minimizing Santa Claus problem, however, the status quo is a bit different.
  10. Moritz Buchem: As we know that no constant factor approximation is possible unless P is NP.
  11. Moritz Buchem: Now, what do we do? We take an alternative point of view on approximations, namely additive approximations.
  12. Moritz Buchem: So the question that we want to answer is whether it's possible to given the maximum processing time Pmax
  13. Moritz Buchem: to find a scheduled such that the absolute deviation of the value of our obtained scheduled from the value of the optimal schedule is at most Pmax.
  14. Moritz Buchem: So, this would imply that for makespan minimization we want to find a schedule of value at most OPT plus Pmax. Whereas for the Santa Claus problem, we want to find a schedule with value, at least OPT minus Pmax.
  15. Moritz Buchem: Similar to multiplicative approximation, we also define additive approximation schemes and an additive approximation scheme is a family of epsilon Pmax additive approximations for every epsilon larger zero.
  16. Moritz Buchem: Now, why do we consider this point of view or this concept of additive oxygenation schemes. First of all, compared to multiplicative approximation schemes
  17. Moritz Buchem: We can obtain stronger guarantees when Pmax is much smaller than the optimal solution value.
  18. Moritz Buchem: And secondly, we have an alternative to obtain an approximation guarantee whenever multiplicative approximation is not possible, such as for the envy minimizing Santa Claus problem.
  19. Moritz Buchem: Now our result is that for the three load balancing problems we consider we can indeed find an additive polynomial time approximation scheme.
  20. Moritz Buchem: And this runs polynomial in M and N and it is double exponential in one over epsilon. And in this talk, I would like to present you the ideas and the main steps that we need to obtain this result.