A variety V has tolerances permutable with factor congruences if for any A1, A2 of V and every tolerance T on A1 × A2 we have T ◦ Π1 = Π1 ◦ T and T ◦ Π2 = Π2 ◦ T , where Π1,Π2 are factor congruences. If B is a subalgebra of A1×A2, the congruences Θi = Πi∩B are called decomposing congruences. V has tolerances permutable with decomposing congruences if T ◦ Θi = Θi ◦ T (i = 1, 2) for each A1, A2 ∈...

We investigate the hardness of approximating a transshipment problem with a permutable transit vector. We have already known that the transshipment problem of minimizing the total shipping cost is NP-hard. In this paper, we prove that even if the unit shipping cost associated with every arc is given as a positive integer, there is no polynomial time constant factor approximation algorithm for t...