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Publication# Santa Claus Schedules Jobs On Unrelated Machines

Résumé

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.

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Concepts associés (15)

Algorithme d'approximation

En informatique théorique, un algorithme d'approximation est une méthode permettant de calculer une solution approchée à un problème algorithmique d'optimisation. Plus précisément, c'est une heuristiq

Algorithme

thumb|Algorithme de découpe d'un polygone quelconque en triangles (triangulation).
Un algorithme est une suite finie et non ambiguë d'instructions et d’opérations permettant de résoudre une classe de

Résolution de problème

vignette|Résolution d'un problème mathématique.
La résolution de problème est le processus d'identification puis de mise en œuvre d'une solution à un problème.
Méthodologie
Dans l'ind

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Approximation algorithms are a commonly used tool for designing efficient algorithmic solutions for intractable problems, at the expense of the quality of the output solution. A prominent technique for designing such algorithms is the use of Linear Programming (LP) relaxations. An optimal solution to such a relaxation provides a bound on the objective value of the optimal integral solution, to which we compare the integral solution we return. In this context, when studying a specific problem, two natural questions often arise: What is a strong LP relaxation for this problem, and how can we exploit it? Over the course of the past few decades, a significant amount of effort has been expended by the research community in order to answer these questions for a variety of interesting intractable problems. Although there exist multiple problems for which we have designed LP relaxations that achieve best-possible guarantees, there still exist numerous problems for which we either have no strong LP relaxations, or do not know how to use them. The main focus of this thesis is extending our understanding of such strong relaxations. We focus on designing good approximation algorithms for certain allocation problems, by employing a class of strong LP relaxations, called configuration-LPs. For many such allocation problems, the best-known results are derived by using simple and natural LP relaxations, whereas configuration-LPs have been used successfully on several occasions in order to break pre-existing barriers set by weaker relaxations. However, our understanding of configuration-LPs is far from complete for many problems. Therefore, understanding and using these relaxations to the farthest extent possible is a quite intriguing question. Answering this question could result in improved approximation algorithms for a wide variety of allocation problems. The first problem we address in this thesis is the restricted max-min fair allocation problem. Prior to our work, the best known result provided an $\Omega(1)$-approximation that ran in polynomial time. Also, it was known how to estimate the value of an optimal solution to the problem within a factor of $1/(4+c)$, for any $c>0$, by solving the corresponding configuration-LP. Our first contribution in this thesis is the design of a $1/13$-approximation algorithm for the problem, using the configuration-LP. Specifically, although our algorithm is fully combinatorial, it consists of a local-search procedure that is guaranteed to succeed only when the configuration-LP is feasible. In order to establish the correctness and running time of the algorithm, it is crucial to use the configuration-LP in our analysis. The second problem we study is the scheduling of jobs on unrelated machines in order to minimize the sum of weighted completion times. For this problem, the best known approximation algorithm achieves a ratio of $3/2-r$, for some small $r>0$. Our second contribution in this thesis is the improvement of this ratio to $(1+\sqrt{2})/2+c$, for any $c>0$, for the special case of the problem where the jobs have uniform Smith ratios. To achieve this ratio, we design a randomized rounding algorithm that rounds solutions to the corresponding configuration-LP. Through a careful examination of the distribution this randomized algorithm outputs, we identify the one that maximizes the approximation ratio, and we then upper bound the ratio this worst-case distribution exhibits by $(1+\sqrt{2})/2+c$.

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 pij ∈ pj, ∞. 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 + ∈ ∼ 1.9412 + ∈, where ∈ > 0 is an arbitrarily small constant. The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as 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. © 2011 ACM.

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.