Tight Approximation Algorithms for Scheduling with Fixed Jobs and Nonavailability

We study two closely related problems in nonpreemptive scheduling of jobs on identical parallel machines. In these two settings there are either fixed jobs or nonavailability intervals during which the machines are not available; in both cases, the objective is to minimize the makespan. Both formulations have different applications, for example, in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of 3/2. For scheduling with fixed jobs, a lower bound of 3/2 on the approximation ratio has been obtained by Scharbrodt et al. [1999]; for scheduling with nonavailability we provide the same lower bound. We use dual approximation, creation of a gap structure, and a PTAS for the multiple subset sum problem, combined with a postprocessing step to assign large jobs.

Strong LP formulations for scheduling splittable jobs on unrelated machines

Ola Nils Anders Svensson

A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to , where is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than , which is larger than the inapproximability bound of 1.5 for the original problem.

Springer Heidelberg2015

Strengths and Limitations of Linear Programming Relaxations

Abbas Bazzi

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than

2

and

\omega(1)

, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than

1.5

for the Vertex Cover Problem, and better than

2

for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of

2-\varepsilon

for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

EPFL2017

Santa Claus Schedules Jobs On Unrelated Machines

Ola Nils Anders Svensson

One of the classic results in scheduling theory is the 2-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines; i.e., job j requires time p(ij) if processed on machine i. More than two decades after its introduction it is still the algorithm of choice even in the restricted model where processing times are of the form p(ij) is an element of {p(j), infinity}. This problem, also known as the restricted assignment problem, is NP-hard to approximate within a factor less than 1.5, which is also the best known lower bound for the general version. Our main result is a polynomial time algorithm that estimates the optimal makespan of the restricted assignment problem within a factor 33/17 + epsilon approximate to 1.9412 + epsilon, where epsilon > 0 is an arbitrarily small constant. The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as the configuration LP, that was previously successfully used for the related Santa Claus problem. Similar to the strongest analysis for that problem our proof is based on a local search algorithm that will eventually find a schedule of the mentioned approximation guarantee but is not known to converge in polynomial time.

Siam Publications2012

Strengths and Limitations of Linear Programming Relaxations

Abbas Bazzi

2

and

\omega(1)

1.5

for the Vertex Cover Problem, and better than

2

2-\varepsilon

EPFL2017