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1.
The Price of Selfish Routing   总被引:2,自引:0,他引:2  
We study the problem of routing traffic through a congested network. We focus on the simplest caseof a network consisting of m parallel links. We assume a collection of n network users; each user employs a mixed strategy, which is a probability distribution over links, to control the shipping of its own assigned traffic. Given a capacity for each link specifying the rate at which the link processes traffic, the objective is to route traffic so that the maximum (over all links) latency is minimized. We consider both uniform and arbitrary link capacities. How much decrease in global performace is necessary due to the absence of some central authority to regulate network traffic and implement an optimal assignment of traffic to links? We investigate this fundamental question in the context of Nash equilibria for such a system, where each network user selfishly routes its traffic only on those links available to it that minimize its expected latency cost, given the network congestion caused by the other users. We use the Coordination Ratio, originally defined by Koutsoupias and Papadimitriou, as a measure of the cost of lack of coordination among the users; roughly speaking, the Coordination Ratio is the ratio of the expectation of the maximum (over all links) latency in the worst possible Nash equilibrium, over the least possible maximum latency had global regulation been available. Our chief instrument is a set of combinatorial Minimum Expected Latency Cost Equations, one per user, that characterize the Nash equilibria of this system. These are linear equations in the minimum expected latency costs, involving the user traffics, the link capacities, and the routing pattern determined by the mixed strategies. In turn, we solve these equations in the case of fully mixed strategies, where each user assigns its traffic with a strictly positive probability to every link, to derive the firstexistence and uniqueness results for fully mixed Nash equilibria in this setting. Through a thorough analysisand characterization of fully mixed Nash equilibria, we obtain tight upper bounds of no worse than O(ln n/ln ln n) on the Coordination Ratio for (i) the case of uniform capacities and arbitrary traffics and (ii) the case of arbitrary capacities and identical traffics.  相似文献   

2.
We investigate the effectiveness of Stackelberg strategies for atomic congestion games with unsplittable demands. In our setting, only a fraction of the players are selfish, while the rest are willing to follow a predetermined strategy. A Stackelberg strategy assigns the coordinated players to appropriately selected strategies trying to minimize the performance degradation due to the selfish players. We consider two orthogonal cases, namely congestion games with affine latency functions and arbitrary strategies, and congestion games on parallel links with arbitrary non-decreasing latency functions. We restrict our attention to pure Nash equilibria and derive strong upper and lower bounds on the pure Price of Anarchy (PoA) under different Stackelberg strategies.  相似文献   

3.
We consider the well-known problem of randomly allocating mm balls into nn bins. We investigate various properties of single-choice games as well as multiple-choice games in the context of weighted balls. We are particularly interested in questions that are concerned with the distribution of ball weights, and the order in which balls are allocated. Do any of these parameters influence the maximum expected load of any bin, and if yes, then how?  相似文献   

4.
We study here the effect of concurrent greedy moves of players in atomic congestion games where n selfish agents (players) wish to select a resource each (out of m resources) so that her selfish delay there is not much. The problem of “maintaining” global progress while allowing concurrent play is exactly what is examined and answered here. We examine two orthogonal settings: (i) A game where the players decide their moves without global information, each acting “freely” by sampling resources randomly and locally deciding to migrate (if the new resource is better) via a random experiment. Here, the resources can have quite arbitrary latency that is load dependent. (ii) An “organised” setting where the players are pre-partitioned into selfish groups (coalitions) and where each coalition does an improving coalitional move. Our work considers concurrent selfish play for arbitrary latencies for the first time. Also, this is the first time where fast coalitional convergence to an approximate equilibrium is shown.  相似文献   

5.
We study the network routing problem with restricted and related links.There are parallel links with possibly different speeds,between a source and a sink.Also there are users,and each user has a traffic of some weight to assign to one of the links from a subset of all the links,named his/her allowable set.The users choosing the same link suffer the same delay,which is equal to the total weight assigned to that link over its speed.A state of the system is called a Nash equilibrium if no user can decrease his/her delay by unilaterally changing his/her link.To measure the performance degradation of the system due to the selfish behavior of all the users,Koutsoupias and Papadimitriou proposed the notion Price of Anarchy (denoted by PoA),which is the ratio of the maximum delay in the worst-case Nash equilibrium and in an optimal solution.The PoA for this restricted related model has been studied,and a linear lower bound was obtained.However in their bad instance,some users can only use extremely slow links.This is a little artificial and unlikely to appear in a real world.So in order to better understand this model,we introduce a parameter for the system,and prove a better Price of Anarchy in terms of the parameter.We also show an important application of our result in coordination mechanism design for task scheduling game.We propose a new coordination mechanism,Group-Makespan,for unrelated selfish task scheduling game with improved price of anarchy.  相似文献   

6.
In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models selfish routing over a network consisting of mm parallel links. We assume a collection of nn users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of her own traffic. In a Nash equilibrium, each user selfishly routes her traffic on those links that minimize her expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.  相似文献   

7.
The classic balls-into-bins game considers the experiment of placing m balls independently and uniformly at random (i.u.r.) in n bins. For m=n , it is well known that the maximum load, i.e., the number of balls in the fullest bin is Θ(log n/log log n) , with high probability. It is also known (see [S2]) that a maximum load of O( m / n ) can be obtained for all m≥ n if each ball is allocated in one (suitably chosen) of two (i.u.r.) bins. Stemann presents a distributed algorithm to find the ``suitable' bin for each ball. The algorithm uses r communication rounds to achieve a maximum load of , with high probability. Adler et al. [ACMR] show that Stemann's protocol is optimal up to a constant factor for constant r . In this paper we extend the above results in two directions: we generalize the lower bound to arbitrary r≤log log n . This implies that Stemann's protocol is optimal for all r . Our key result is a generalization of Stemann's upper bound to weighted balls: Let W A (resp. W M ) denote the average (resp. maximum) weight of the balls. Furthermore, let Δ=W A /W M . Then the optimal maximum load is Ω(m/n⋅ W A +W M ) . We present a protocol that achieves a maximum load of γ⋅( m / n ⋅ W A +W M ) using O( log log n / log (γ⋅(m/n⋅Δ+1)) ) communication rounds. For uniform weights this matches the results of Stemann. In particular, we achieve a load of O( m / n ⋅ W A +W M ) using log log n communication rounds, which is optimal up to a constant factor. An extension of our lower bound shows that our algorithm also reaches a load which is within a constant factor of the optimal load in the case of weighted balls. All the balls-into-bins games model load balancing problems: the balls are jobs, the bins are resources, the task is to allocate the jobs to the resources in such a way that the maximum load is minimized. Our extension to weighted balls allows us to extend previous bounds to models where resource requirements may vary. For example, if the jobs are computing tasks, their running times may vary. Applications of such load balancing problems occur, e.g., for client-server networks and for multimedia-servers using disk arrays. Received December 23, 1997, and in final form September 9, 1998.  相似文献   

8.
9.
We study the following distributed access problem which arises naturally in many settings: given a set of n data items shared among n nodes in a distributed network, all nodes want to access all (or a subset of) the items residing on different nodes in a conflict-free manner. In addition, items may move from one node to the other during access. Our goal is to design distributed protocols so that all nodes access all the desired items as quickly as possible, while at the same time not overloading the storage space of any one node. Using centralized coordination among the nodes it is easy to design an optimal scheme in which all nodes can access all the items in n−1 steps storing only one item at any time. We show that a simple randomized distributed protocol performs almost as well as the optimal (centralized) scheme but with no coordination overhead. Our protocol takes O(n) time with high probability to access all n items which is asymptotically as good as the optimal centralized scheme. The protocol guarantees that the maximum load (the maximum number of items stored in any node) at any time is at most O(log n/log log n) with high probability which is only slightly larger compared to the Ω(1) load of the optimal scheme. Our analysis involves a stochastic analysis of a “balls into bins” problem in a dynamic setting where balls (data items) move into bins (nodes) on request and we study the time and load requirements to move all the balls to the requested bins. A short version of this paper appeared in the Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing (PODC), 2005.  相似文献   

10.
We consider one-dimensional and multidimensional vector covering with variable sized bins. In the one-dimensional case, we consider variable sized bin covering with bounded item sizes. For every finite set of bins B, and upper bound 1/m on the size of items for some integer m, we define a ratio r(B, m). We prove this is the best possible competitive ratio for the set of bins B and the parameter m by giving both an algorithm with competitive ratio r(B, m) and an upper bound of r(B, m) on the competitive ratio of any online deterministic or randomized algorithm. The ratio satisfies r(B, m)≥m/(m+1) and equals this number if all bins are of size 1. For multidimensional vector covering we consider the case where each bin is a binary d-dimensional vector. It was shown by N. Alon, Y. Azar, J. Csirik, L. Epstein, S. V. Sevastianov, A. P. A. Vestjens, and G. J. Woeginger (1998, Algorithmica 21, 104–118) that if B contains a single bin which is all 1, then the best competitive ratio is Θ(1/d). We show an upper bound of 1/2d(1−o(1)) for the general problem, and consider four special case variants. We show an algorithm with optimal competitive ratio 1/2 for the model where each bin in B is a standard basis vector. We consider the model where B consists of all unit prefix vectors. A unit prefix vector has i leftmost components of 1, and all other components are 0. We show that this model is harder than the case of standard basis vector bins by giving an upper bound of O(1/log d) on the competitive ratio of any deterministic or randomized algorithm. Next, we discuss the model where B contains all binary vectors. We show this model is easier than the model of one bin type which is all 1 by giving an algorithm with competitive ratio Ω(1/log d). The most interesting multidimensional case is d=2. The results of N. Alon et al. give a 0.25-competitive algorithm for B={(1, 1)} and an upper bound of 0.4 on the competitive ratio of any algorithm. In this paper we consider all other models for d=2. For standard basis vectors, we give an algorithm with optimal competitive ratio 1/2. For unit prefix vectors we give an upper bound of 4/9 on the competitive ratio of any deterministic or randomized algorithm. For the model where B consists of all binary vectors, we design an algorithm with ratio larger than 0.4. These results show that all above relations between models hold for d=2 as well.  相似文献   

11.
If an element is inserted into or removed from a set, then the set covering problem can be reoptimized with some ratio ( 2 - \frac1lnm + 1 ) \left( {2 - \frac{1}{{\ln m + 1}}} \right) , where m is the number of elements of the set. A similar result holds if an arbitrary number 1 < p < m of elements of the set is inserted or removed.  相似文献   

12.
We study a multicast game in non-cooperative directed networks in which a source sends the same message or service to a set of r receiving users and the cost of the used links is divided among the receivers according to a given cost sharing method. By following the approach recently proposed by Chen et al. (Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 854–863, 2008), we analyze the performances of a family of methods satisfying certain desiderata, namely, weak and strong budget-balance, fairness and separability. We show that any fair method may require an arbitrary number of selfish moves in order to converge to a pure Nash equilibrium, hence we focus on the solutions obtained after one round of selfish moves. We evaluate their quality according to two global social functions: the overall cost of the solution and the maximum shared cost of users. The only method satisfying all the properties is the well-known Shapley value for which we show an approximation ratio of the solutions reached after a one round walk equal to Θ(r 2). We then prove that relaxing the strong budget balance and separability properties (we call feasible any method satisfying weak budget balance and fairness) leads to improved performances since we determine a feasible method achieving an approximation ratio of the solutions reached after a one round walk equal to O(r). This bound is asymptotically optimal since we also show that any method satisfying weak budget balance cannot achieve an approximation ratio of the solutions reached after a one round walk smaller than r. Finally, we prove the NP-hardness of computing the sequence of moves leading to the best possible global performance and extend most of the results to undirected networks.  相似文献   

13.
We consider the following load balancing problem. Jobs arrive on-line and must be assigned to one of m machines thereby increasing the load on that machine by a certain weight. Jobs also depart on-line. The goal is to minimize the maximum load on any machine, the load being defined as the sum of the weights of the jobs assigned to the machine divided by the machine capacity. The scheduler also has the option of preempting a job and reassigning it to another machine. Whenever a job is assigned or reassigned to a machine, the on-line algorithm incurs a reassignment cost depending on the job. For arbitrary reassignment costs and identical machines, we present an on-line algorithm with a competitive ratio of 3.5981 against current load , i.e., the maximum load at any time is less than 3.5981 times the lowest achievable load at that time. Our algorithm also incurs a reassignment cost less than 6.8285 times the cost of assigning all the jobs. For arbitrary reassignment costs and related machines we present an algorithm with a competitive ratio of 32 and a reassignment factor of 79.4. We also describe algorithms with better performance guarantees for some special cases of the problem. Received August 24, 1996; revised August 6, 1997.  相似文献   

14.
15.
Online bin stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for online bin stretching with a stretching factor of \(11/8 = 1.375\) for three bins. Additionally, we present a lower bound of \(45/33 = 1.\overline{36}\) for online bin stretching on three bins and a lower bound of 19/14 for four and five bins that were discovered using a computer search.  相似文献   

16.
Summary We consider a variant of the classical one-dimensional bin-packing problem: The number of bins is fixed and the object is to maximize the number of pieces packed from some given set. Both problems have applications in processor and storage allocation in computer systems in addition to a broad application in operations research.It can easily be shown that both problems are NP-complete; our approach will be to propose and analyze very fast heuristics. We consider a class of algorithms and bound the performance of an arbitrary algorithm in that class. Finally we propose an algorithm, the first-fit-increasing algorithm, and analyze its running time and relative performance.This research was supported in part by NSF Grant No. 28290  相似文献   

17.
We investigate the convergence of the price of anarchy after a limited number of moves in the classical multicast communication game when the underlying communication network is directed. Namely, a subset of nodes of the network are interested in receiving the transmission from a given source node and can share the cost of the used links according to fixed cost sharing methods. At each step, a single receiver is allowed to modify its communication strategy, that is to select a communication path from the source, and assuming a selfish or rational behavior, it will make a best response move, that is it will select a solution yielding the minimum possible payment or shared cost. We determine lower and upper bounds on the price of anarchy, that is the highest possible ratio among the overall cost of the links used by the receivers and the minimum possible cost realizing the required communications, after a limited number of moves under the fundamental Shapley cost sharing method. In particular, assuming that the initial set of connecting paths can be arbitrary, we show an $O(r\sqrt{r})We investigate the convergence of the price of anarchy after a limited number of moves in the classical multicast communication game when the underlying communication network is directed. Namely, a subset of nodes of the network are interested in receiving the transmission from a given source node and can share the cost of the used links according to fixed cost sharing methods. At each step, a single receiver is allowed to modify its communication strategy, that is to select a communication path from the source, and assuming a selfish or rational behavior, it will make a best response move, that is it will select a solution yielding the minimum possible payment or shared cost. We determine lower and upper bounds on the price of anarchy, that is the highest possible ratio among the overall cost of the links used by the receivers and the minimum possible cost realizing the required communications, after a limited number of moves under the fundamental Shapley cost sharing method. In particular, assuming that the initial set of connecting paths can be arbitrary, we show an O(r?r)O(r\sqrt{r}) upper bound on the price of anarchy after 2 rounds, during each of which all the receivers move exactly once, and a matching lower bound, that we also extend to W(rk?{r})\Omega(r\sqrt[k]{r}) for any number k≥2 rounds, where r is the number of receivers. Similarly, exactly matching upper and lower bounds equal to r are determined for any number of rounds when starting from the empty state in which no path has been selected. Analogous results are obtained also with respect to other three natural cost sharing methods considered in the literature, that is the egalitarian, path-proportional and egalitarian-path proportional ones. Most results are also extended to the undirected case in which the communication links are bidirectional.  相似文献   

18.
We consider a version of the on-line bounded-space bin-packing problem where repacking the items within the active bins is allowed. For this problem, the 1.69103 lower bound of Lee and Lee [7] for the worst case ratios of bounded-space approximation algorithms still applies. We present a polynomial time approximation algorithm that reaches the best possible worst case ratio matching the Lee and Lee lower bound while using onlythree active bins.  相似文献   

19.
We consider the problem of dynamically reallocating (or re-routing) m weighted tasks among a set of n uniform resources (one may think of the tasks as selfish players). We assume an arbitrary initial placement of tasks, and we study the performance of distributed, natural reallocation algorithms. We are interested in the time it takes the system to converge to an equilibrium (or get close to an equilibrium). Our main contributions are (i) a modification of the protocol in 2006 that yields faster convergence to equilibrium, together with a matching lower bound, and (ii) a non-trivial extension to weighted tasks.  相似文献   

20.
List scheduling for jobs with arbitrary release times and similar lengths   总被引:1,自引:0,他引:1  
This paper considers the problem of on-line scheduling a list of independent jobs in which each job has an arbitrary release time and length in [1,r] with r≥1 on m parallel identical machines. For the list scheduling algorithm, we give an upper bound of the competitive ratio for any m≥1 and show that the upper bound is tight when m=1. When m=2, we present a tight bound for r≥4. For r<4, we give a lower bound and show that 2 provides an upper bound. Li’s work was supported in part by the National Natural Science Foundation of China (No. 10771060).  相似文献   

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