## Distributed selfish load balancing (2006)

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Venue: | In Proc. 17th Ann. ACM–SIAM Symp. on Discrete Algorithms (SODA |

Citations: | 29 - 1 self |

### BibTeX

@INPROCEEDINGS{Berenbrink06distributedselfish,

author = {Petra Berenbrink and Tom Friedetzky and Leslie Ann Goldberg and Paul W. Goldberg and Zengjian Hu and Russell Martin},

title = {Distributed selfish load balancing},

booktitle = {In Proc. 17th Ann. ACM–SIAM Symp. on Discrete Algorithms (SODA},

year = {2006},

pages = {354--363},

publisher = {ACM Press}

}

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### Abstract

Abstract. Suppose that a set of m tasks are to be shared as equally as possible amongst a set of n resources. A gametheoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent”, and require each agent to select a resource, with the cost of a resource being the number of agents to select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For m ≫ n, the system becomes approximately balanced (an ǫ-Nash equilibrium) in expected time O(log log m). We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time O(log log m + n 4). We also give a lower bound of Ω(max{log log m, n}) for the convergence time. 1. Introduction. Suppose

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Citation Context ...ult from a sufficiently long betterresponse sequence, many algorithms for finding them are based on such sequences. An important sub-class is the Elementary step system (ESS), proposed in Orda et al. =-=[26]-=-, which consists of best-response moves (where a migrating agent switches not to any improved choice, but to one that is optimal at the time of migration). For matroid games (a class of congestion gam... |

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Citation Context ...inition we gave above, but many ideas carry over. A widely-studied version we introduced by Koutsoupias and Papadimitriou [19], and subsequently studied extensively in different contexts (for example =-=[21, 7, 14, 5, 29]-=-). In this generalisation, each task may have a numerical weight (sometimes called traffic, or demand) and each resource has a speed (or capacity). The cost of using a resource is the total weight of ... |

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Citation Context ...hm that purports to be a realistic model of agents’ behavior. Regarding the first of these questions, the answer is no in the general setting (the problem is PLS-complete for general congestion games =-=[10]-=-, see also [1, 3]). PLS-completeness (introduced in [18]) is a generally-accepted criterion for intractability of computational problems in which we seek a local optimum of a given objective function.... |

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Citation Context ...inition we gave above, but many ideas carry over. A widely-studied version we introduced by Koutsoupias and Papadimitriou [19], and subsequently studied extensively in different contexts (for example =-=[21, 7, 14, 5, 29]-=-). In this generalisation, each task may have a numerical weight (sometimes called traffic, or demand) and each resource has a speed (or capacity). The cost of using a resource is the total weight of ... |

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Citation Context ...inition we gave above, but many ideas carry over. A widely-studied version we introduced by Koutsoupias and Papadimitriou [19], and subsequently studied extensively in different contexts (for example =-=[21, 7, 14, 5, 29]-=-). In this generalisation, each task may have a numerical weight (sometimes called traffic, or demand) and each resource has a speed (or capacity). The cost of using a resource is the total weight of ... |

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Citation Context ...ns. It is well-known that a fairly even distribution can be achieved if every ball is allowed to randomly choose d bins and then the ball is allocated to the least loaded amongst the chosen bins (see =-=[24]-=- for an overview). Czumaj et al. [6] consider such an allocation where each ball initially chooses two bins. They show that, in a polynomial number of steps, the reallocation process ends up in a stat... |

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Citation Context ...′ = τ + ⌈10B2 /V ⌉. Lemma 4.17. Given any starting state X(0) = x, the probability that X(t ′ ) is a Nash equilibrium is at least 3/4. Proof. The proof is based on a standard martingale argument, see =-=[20]-=-. Suppose that Φ(X(τ)) ≤ 720n. Let Wt = Φ(X(t+τ))−r(1−r/n) and let Dt = min(Wt, B). Note that D0 ≤ 720n. Together, Observation 4.5 and Corollary 4.14 tell us that Wt is a supermartingale. This implies... |

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Citation Context ...llocate their tasks at the same time. Thus, we do not (and cannot) use the traditional Elementary Step System, where the assumption is that at most one user may reallocate her task at any given stage =-=[15, 8]-=-. Throughout we let m denote the number of tasks (in the above discussion, customers) and n the number of resources (power suppliers). As hinted in the above discussion, we assume that typically m ≫ n... |

76 | Selfish Traffic Allocation for Server Farms
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Citation Context |

54 | Convergence to Approximate Nash Equilibria in Congestion Games
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Citation Context ...s to be a realistic model of agents’ behavior. Regarding the first of these questions, the answer is no in the general setting (the problem is PLS-complete for general congestion games [10], see also =-=[1, 3]-=-). PLS-completeness (introduced in [18]) is a generally-accepted criterion for intractability of computational problems in which we seek a local optimum of a given objective function. However, due to ... |

50 |
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Citation Context ...ntial function of [27] that applies to these games, and which was subsequently used in [15]. For these games however, it seems harder to find polynomial-length best-response sequences. Feldman et al. =-=[11]-=- show how a sequence of steps may lead to NE, under the weaker condition that the maximal cost experienced by agents must not increase, but individual steps need not necessarily be “selfish”. They als... |

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Citation Context ...n allocation where each ball initially chooses two bins. They show that, in a polynomial number of steps, the reallocation process ends up in a state with maximum load at most ⌈m/n⌉+1. Sanders et al. =-=[30]-=- show that a maximum load of ⌈m/n⌉ + 1 is optimal if every ball is restricted to two random choices. In conclusion, this paper sits at one end of a spectrum in which we study a very simple load-balanc... |

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Citation Context ...s to be a realistic model of agents’ behavior. Regarding the first of these questions, the answer is no in the general setting (the problem is PLS-complete for general congestion games [10], see also =-=[1, 3]-=-). PLS-completeness (introduced in [18]) is a generally-accepted criterion for intractability of computational problems in which we seek a local optimum of a given objective function. However, due to ... |

45 | Routing without regret: On convergence to Nash equilibria of regret-minimizing algorithms in routing games
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Citation Context ...task has an incentive to deviate unilaterally from this policy. This implies that in the terminology of [9] it is a Nash rerouting policy. It is also a simple regret-minimizing policy in the sense of =-=[2]-=- since the average cost of resources used by an agent is no higher than the best choice of a single resource to be used repeatedly. Although the above rule is very natural and has the nice properties ... |

41 | Fast Convergence to Wardrop Equilibria by Adaptive Sampling Methods
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Citation Context ...the resources: the overload would oscillate between the two resources, with a load ratio tending towards 2:1. (This observation about the risk of oscillation has also been made in similar contexts in =-=[13, 12]-=-, and we will not elaborate on it further.) It can easily be seen that, if all tasks use the above policy, then the expected load of every resource at ∗ Preliminary version in proceedings of ACM-SIAM ... |

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Citation Context ...s. In this paper we consider the special case of singleton congestion games (where players’ strategies are always single resources, thus the ranks of the matroids is 1). For these games, Ieong et al. =-=[17]-=- give polynomial bounds for best-response and better-response sequences. Chien and Sinclair [3] study a version of the ESS in the context of approximate Nash equilibria, and show that in some cases th... |

26 | Adaptive routing with stale information
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Citation Context ...the resources: the overload would oscillate between the two resources, with a load ratio tending towards 2:1. (This observation about the risk of oscillation has also been made in similar contexts in =-=[13, 12]-=-, and we will not elaborate on it further.) It can easily be seen that, if all tasks use the above policy, then the expected load of every resource at ∗ Preliminary version in proceedings of ACM-SIAM ... |

25 |
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Citation Context ...(t) = m n . This provides a compelling motivation for the policy, which is that as a result, no task has an incentive to deviate unilaterally from this policy. This implies that in the terminology of =-=[9]-=- it is a Nash rerouting policy. It is also a simple regret-minimizing policy in the sense of [2] since the average cost of resources used by an agent is no higher than the best choice of a single reso... |

20 |
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Citation Context ...llocate their tasks at the same time. Thus, we do not (and cannot) use the traditional Elementary Step System, where the assumption is that at most one user may reallocate her task at any given stage =-=[15, 8]-=-. Throughout we let m denote the number of tasks (in the above discussion, customers) and n the number of resources (power suppliers). As hinted in the above discussion, we assume that typically m ≫ n... |

17 | Perfectly Balanced Allocation
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Citation Context ...en distribution can be achieved if every ball is allowed to randomly choose d bins and then the ball is allocated to the least loaded amongst the chosen bins (see [24] for an overview). Czumaj et al. =-=[6]-=- consider such an allocation where each ball initially chooses two bins. They show that, in a polynomial number of steps, the reallocation process ends up in a state with maximum load at most ⌈m/n⌉+1.... |

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