Estimating the Makespan of the Two-Valued Restricted Assignment Problem

Abstract

We consider a special case of the scheduling problem on unrelated machines, namely the Restricted Assignment Problem with two different processing times. We show that the configuration LP has an integrality gap of at most 5 3 ≈ 1.667 for this problem. This allows us to estimate the optimal makespan within a factor of 5 3 , improving upon the previously best known estimation algorithm with ratio 11 6 ≈ 1.833 due to Chakrabarty, Khanna, and Li [2].

On the Configuration-LP of the Restricted Assignment Problem

“…On better bounds. The largest gap we have witnessed in an instance is 9/6 (given in [7]). The bound on the integrality gap that was proven in this paper, 11/6, is not necessarily tight.…”

“…In the restricted variant, however, no instance is known to have an integrality gap higher than 1.5. An instance with integrality gap 1.5 is given in [7].…”

“…Further progress has been made on the special case of two different processing times, where an upper bound of 2 − 1/3 was shown for the integrality gap by Land, Maack, and the first author [7] and a (2 − δ)approximation for a very small δ by Chakrabarty, Khanna, and Li [3].…”

A Quasi-Polynomial Approximation for the Restricted Assignment Problem

“…Later Annamalai surpassed this with a (17/9 + ǫ)-approximation for every ǫ > 0 [1]. For this special case it was also shown that the integrality gap is at most 5/3 [6].…”

“…In [10], [7], and [6] the critical idea is to design a local search algorithm, which is then shown to produce good solutions. However, the algorithm has a potentially high running time; so it was only used to prove the existence of such a solution.…”