We consider the classical problem of Scheduling on Unrelated Machines. In this problem a set of jobs is to be distributed among a set of machines and the maximum load (makespan) is to be minimized. The processing time pij of a job j depends on the machine i it is assigned to. Lenstra, Shmoys and Tardos gave a polynomial time 2-approximation for this problem [8]. In this paper we focus on a prominent special case, the Restricted Assignment problem, in which pij ∈ {pj, ∞}. The configuration-LP is a linear programming relaxation for the Restricted Assignment problem. It was shown by Svensson that the multiplicative gap between integral and fractional solution, the integrality gap, is at most 2 − 1/17 ≈ 1.9412 [11]. In this paper we significantly simplify his proof and achieve a bound of 2 − 1/6 ≈ 1.8333. As a direct consequence this provides a polynomial (2 − 1/6 + )-estimation algorithm for the Restricted Assignment problem by approximating the configuration-LP. The best lower bound known for the integrality gap is 1.5 and no estimation algorithm with a guarantee better than 1.5 exists unless P = NP.
We study an important case of ILPs max{c T x | Ax = b, l ≤ x ≤ u, x ∈ Z nt } with n • t variables and lower and upper bounds , u ∈ Z nt. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s × t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and ∆, where ∆ is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Ω(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs∆) O(r 2 s+s 2) L 2 • nt log O(1) (nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L.
Scheduling jobs on unrelated machines and minimizing the makespan is a classical problem in combinatorial optimization. In this problem a job j has a processing time pij for every machine i. The best polynomial algorithm known goes back to Lenstra et al. and has an approximation ratio of 2. In this paper we study the Restricted Assignment problem, which is the special case where pij ∈ {pj , ∞}. We present an algorithm for this problem with an approximation ratio of 11/6 + ǫ and quasi-polynomial running time n O(1/ǫ log(n)) for every ǫ > 0. This closes the gap to the best estimation algorithm known for the problem with regard to quasi-polynomial running time.
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