By Schrijver A.

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It can be shown that obtaining a ε-approximate Pareto curve for the bicriteria Min T SP (1, 2) on this modified instance yields a 2ε/(3ε + 1)-approximate Pareto curve for the bicriteria Max T SP (1, 2) problem on the original instance. – Any ε-approximate Pareto curve for the bicriteria Min T SP (1, 2) 2ε -approximate Pareto curve for problem can be polynomially transformed into a 3ε+1 the bicriteria Max T SP (1, 2) problem. Proof. , any ε2ε approximation for Min T SP (1, 2) can be polynomially transformed into a 3ε+1 approximation for Max T SP (1, 2).

N, is occupied by an a[i] − job admits at least n − 1 on-time jobs. Proof. We can prove straightforwardly that in such a sequence the first n − 1 jobs are on-time. – If two jobs 2i and 2i−1 are both on-time, then there exists j, 1 such that jobs 2j and 2j − 1 are both late. j < i, Proof. Suppose by contradiction that there exists a sequence in which for all j, 0 j < i, at least one job 2j or 2j − 1, denoted a[j] -job, is on time and that jobs 2i and 2i − 1 are both on-time. The Complexity of Single Machine Scheduling Problems 31 It is clear that resequencing these on-time jobs in a non-decreasing order of their due dates does not increase the number of late jobs.

1. Here, ε is the error while 1 + ε (resp. , maximization) problem. In the chapter, we equivalently use the error and its associated approximation ratio. Approximation of Multi-criteria Min and Max T SP (1, 2) 41 We now prove that the approximability of the Pareto curve for bicriteria Min T SP (1, 2) and the approximability of the Pareto curve for bicriteria Max T SP (1, 2) are linked by an approximation preserving reduction. The idea is to modify the instance by replacing each edge (2,2) by an edge (1,1), each edge (1,1) by an edge (2,2), and each edge (1,2) by an edge (2,1) and vice versa.

### A Course in Combinatorial Optimization by Schrijver A.

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