In this note, we point out two major errors in the paper "Minimizing total tardiness on parallel machines with preemptions" by Kravchenko and Werner [2010]. More precisely, they proved that both problems P|pmtn|sum(Tj) and P|rj, pj = p, pmtn|sum(Tj) are NP-Hard. We give a counter-example to their proofs, letting the complexity of these two problems open.
In Kravchenko and Werner [2010], the authors propose a reduction of P |pmtn| T j (in Section 3) and P |r j , p j = p, pmtn| T j (in Section 4) from Partition: given a set of positive integers a 1 , . . . , a k , b with k i=1 a i = 2b, does there exist a subset I ⊂ {1, . . . , k} such that i∈I a i = b? Given an instance of Partition, we only detail the reduction of P |pmtn| T j since it is used in the proof of the N P-Hardness of P |r j , p j = p, pmtn| T j . The instance of P |pmtn| T j is composed of 2k 2 + k + 1 jobs and k machines, and a constant L = 4kb 3 +2b k is used. The authors define three classes of jobs:
the a-jobs: it is composed of k jobs a i with p i = a i and d i = L, i ∈ {1, . . . , k}.
the ba-jobs: it is composed of 2k 2 jobs, 2k equivalent jobs ba i with p i = b 2 a i and
one long job b 3 , with processing time b 3 and due date b 3 .
The authors claim that Partition has a solution if and only if there exists a schedule with T j ≤ b 3 +b. The necessary part of this result is quite obvious. The sufficient part is more complex, and the authors claim that if there exists a schedule with T j ≤ b 3 + b, then the set of ba-jobs completed after time point L defines the solution for Partition. We show in the next section that this sufficient part does not hold, i.e. starting with a solution of P |pmtn| T j such that T j ≤ b 3 + b may not lead to a solution of Partition.
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The counter-example to the reduction is an instance of Partition with k = 3: a 1 = 1, a 2 = 2, and a 3 = 3. The corresponding instance of P |pmtn| T j is hence composed of 22 jobs and 3 machines, and the constant L is equal to 110. The jobs have the following characteristics:
the a-jobs: a 1 ,a 2 and a 3 have a processing time of 1, 2, and 3 and a common due date L = 110.
the ba-jobs: there are six jobs ba 1 of processing 9 and due date 109, six jobs ba 2 of processing 18 and due date 108, six jobs ba 3 of processing 27 and due date 107.
one long job b 3 with processing 27 and due date 27.
A schedule such that T j ≤ b 3 + b = 30 is proposed in Figure 1. There are only three late jobs, of type ba 1 , each of them finishing 10 time units after its due date. According to Kravchenko and Werner [2010], the corresponding solution of Partition is I = {a 1 }, which is obviously wrong.
As a consequence, the proposed reduction of P |pmtn|sumT j from Partition does not hold; the reduction of P |r j , p j = p, pmtn| T j from Partition being based on the same construction, it is also wrong. Hence, it is still an open question to know whether P |pmtn|sumT j and P |r j , p j = p, pmtn| T j are N P-Hard problems or not.
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