On the Configuration-LP of the Restricted Assignment Problem

Abstract

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.

Approximation algorithms for minimum norm and ordered optimization problems

“…Depending on the application, oftentimes one minimizes either the sum of the entries of the cost vector, or the largest entry of the cost vector. For example, in the load balancing setting, the largest entry of the load vector is the makespan of the assignment, and minimizing makespan has been extensively studied [34,40,20,41,15,28]. Similarly, in the clustering setting, the problem of minimizing the sum of assignment costs is the k-median problem, and the problem of minimizing the largest assignment cost is the k-center problem.…”

Scheduling to Minimize Total Weighted Completion Time via Time-Indexed Linear Programming Relaxations

“…Some efforts have been put on a special case of the problem, where each job j has a size p j and p i,j ∈ {p j , ∞} for every i ∈ M (the model is called restricted assignment model.) [13,40,7,21].…”

“…Constraint (19) says that if j ≺ j ′ , then C j ′ is at least C j plus the processing time of j ′ on the machine it is assigned to. Constraint (20) says that the makespan D is at least the total processing time of all jobs assigned to i, for every machine i. Constraint (21) says that the makespan D is at least the completion time of any job j. Constraint (22) requires the x and C variables to be non-negative.…”

“…where H is over all precedence chains of jobs. The first inequality is by Claim 4.2 and the second inequality is by Constraints (19) and (21) in the LP.…”

A Quasi-Polynomial Approximation for the Restricted Assignment Problem

“…In a recent breakthrough, Svensson has proved that the configuration-LP, a natural linear programming relaxation, has an integrality gap of at most 33/17 [10]. We have later improved this bound to 11/6 [7]. By approximating the configuration-LP this yields an (11/6 + ǫ)-estimation algorithm for every ǫ > 0.…”