## Existence of a pseudo-polynomial time algorithm for a counting problem.

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Let T={1,...,n} be a set of tasks. Each task i has associated a non negative processing time p_i and a deadline d_i. A feasible schedule of the tasks consists of a permutation of n elements pi, such that \sum_i=1^k p_(pi(i)) <= d_(pi(i)) for all k=1,...n.

Does there exists a pseudo-polynomial time algorithm for computing the total number of feasible schedules?

A pseudo-polynomial time algorithm is an algorithm whose running time is bounded by a polynomial on the size of the input, given that the input is written in unary notation (2=II, 3 =III). (e.g., the size of a number n in unary notation is O(n), and not O(log(n)).

This is an open question from an article published in 2009 at Operations Research Letters.

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Usually asking open problems is not really considered appropriate for MO; see the FAQ. That said, you might ask something like "what is the current state of knowledge about this problem?" – Daniel Litt Jul 29 2010 at 14:57
Should the RHS read $d_{\pi(k)}$ instead of $d_{\pi(i)}$? I also agree with Daniel, it seems inappropriate to ask an open question (especially such a recent one) on MO. – Artem Kaznatcheev Aug 27 2010 at 13:00

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At the time I write this comment, the question has been edited and now makes no sense whatsoever – Yemon Choi Sep 13 2010 at 15:40

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I've rolled the question back to it's previous version, which at least made sense. I don't know anything about this question, but it seems perfectly reasonable to ask whether any progress has been made on it. – David Speyer Sep 13 2010 at 15:51

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Actually, David, could you further roll it back to the version by Gerry, which had the same content but had grammar corrected and used LaTeX? – JBL Sep 13 2010 at 17:51
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## 1 Answer

If you need to compute all solution for a specific instance, you could generate a an IP formulation of the problem and use a lattice point enumeration code such as LattE. This might be a good problem for the operations research QA

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