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Publication# Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation

Résumé

The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, Asadpour et al. [2012] proved that a certain configuration LP can be used to estimate the optimal value within a factor of 1/(4 + epsilon), for any epsilon > 0, but at the same time it is not known how to efficiently find a solution with a comparable performance guarantee. A natural question that arises from their work is if the difference between these guarantees is inherent or results from a lack of suitable techniques. We address this problem by giving a quasi-polynomial approximation algorithm with the mentioned performance guarantee. More specifically, we modify the local search of Asadpour et al. [2012] and provide a novel analysis that lets us significantly improve the bound on its running time: from 2(O(n)) to n(O(log n)). Our techniques also have the interesting property that although we use the rather complex configuration LP in the analysis, we never actually solve it and therefore the resulting algorithm is purely combinatorial.

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Publications associées (28)

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Concepts associés (11)

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

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

Combinatoire

En mathématiques, la combinatoire, appelée aussi analyse combinatoire, étudie les configurations de collections finies d'objets ou les combinaisons d'ensembles finis, et les dénombrements.
Géné

Lukas Polacek, Ola Nils Anders Svensson

The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, Asadpour et al. [1] proved that a certain configuration LP can be used to estimate the optimal value within a factor 1/(4 + ε), for any ε > 0, but at the same time it is not known how to efficiently find a solution with a comparable performance guarantee. A natural question that arises from their work is if the difference between these guarantees is inherent or because of a lack of suitable techniques. We address this problem by giving a quasi-polynomial approximation algorithm with the mentioned performance guarantee. More specifically, we modify the local search of [1] and provide a novel analysis that lets us significantly improve the bound on its running time: from 2O(n) to nO(logn). Our techniques also have the interesting property that although we use the rather complex configuration LP in the analysis, we never actually solve it and therefore the resulting algorithm is purely combinatorial. © 2012 Springer-Verlag.

Chidambaram Annamalai, Christos Kalaitzis, Ola Nils Anders Svensson

We study the basic allocation problem of assigning resources to players to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain Configuration-LP can be used to estimate the value of the optimal allocation to within a factor of 4 + epsilon. In contrast, however, the best-known approximation algorithm for the problem has an unspecified large constant guarantee. In this article, we significantly narrow this gap by giving a 13-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [13] for hypergraph matchings and later used in this context by Asadpour, Feige, and Saberi [2]. For our local search procedure to terminate in polynomial time, we introduce several new ideas, such as lazy updates and greedy players. Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the Configuration-LP only in the analysis.

Chidambaram Annamalai, Christos Kalaitzis, Ola Nils Anders Svensson

We study the basic allocation problem of assigning resources to players so as to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain configuration-LP can be used to estimate the value of the optimal allocation to within a factor of 4 + ε. In contrast, however, the best known approximation algorithm for the problem has an unspecified large constant guarantee. In this paper we significantly narrow this gap by giving a 13-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [Hax95] for hypergraph matchings, and later used in this context by Asadpour, Feige, and Saberi [AFS12]. For our local search procedure to terminate in polynomial time, we introduce several new ideas such as lazy updates and greedy players. Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the configuration-LP only in the analysis.

2014