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I am working on a problem that appears like a variant of the assignment problem. There are tasks that need to be assigned to servers. The sum of costs over servers needs to be minimized. The following conditions hold:

  1. Each task has a unit size.

  2. A task may not be divided among more than one servers. A task must be handled by exactly one server.

  3. A server has a limit on the maximum number of tasks that may be assigned to it.

  4. The cost function for task assignment is a staircase function. A server incurs a minimum cost 'a'. For each task handled by the server, the cost increases by 1. If the number of tasks assigned to a particular server exceeds half of it's capacity, there is a jump in that server's cost equal to a positive number 'd'.

    1. Tasks have preferences, i.e., a given task may be assigned to one of a few of the servers.

I have a feeling that this is an NP-Hard problem, but I can't seem to find an NP-Complete problem to map to it. I've tried Bin Packing, Assignment problem, Multiple Knapsacks, bipartite graph matching but none of these problems have all the key characteristics of my problem. Can you please suggest some problem that maps to it?

Thanks and best regards

Saqib

Muhammad Saqib Ilyas
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1 Answers1

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Have you tried reducing the set partitioning problem to yours?

The SET-PART (stands for "set partitioning") decision problem asks whether there exists a partition of a given set S of numbers into two sets S1 and S2, so that the sum of the elements in S1 equals the sum of elements in S2. This problem is known to be NP-complete.

Your problem seems related to the m-PROCESSOR decision problem. Given a nonempty set A of n>0 tasks {a1,a2,...,an} with processing times t1,t2,...,tn, the m-PROCESSOR problem asks if you can schedule the tasks among m equal processors so that all tasks finish in at most k>0 time steps. (Processing times are (positive) natural numbers.)

The reduction of SET-PART to m-PROCESSOR is very easy: first show that the special case, with m=2, is NP-complete; then use this to show that m-PROCESSOR is NP-complete for all m>=2. (A reduction in Slovene.)

Hope this helps.

EDIT 1: Oops, this m-PROCESSOR thingy seems very similar to the assignment problem.

blazs
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