Luca Moscardelli

Tight bounds for selfish and greedy load balancing

We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it selects to run its job to the server among its permissible servers having the smallest latency given the assignments of the jobs of other clients to servers. In online load balancing, clients appear online and, when a client appears, it has to make an irrevocable decision and assign its job to one of its permissible servers. Here, we assume that the clients aim to optimize some global criterion but in an online fashion. A natural local optimization criterion that can be used by each client when making its decision is to assign its job to that server that gives the minimum increase of the global objective. This gives rise to greedy online solutions. The aim of this paper is to determine how much the quality of load balancing is affected by selfishness and greediness. We characterize almost completely the impact of selfishness and greediness in load balancing by presenting new and improved, tight or almost tight bounds on the price of anarchy and price of stability of selfish load balancing as well as on the competitiveness of the greedy algorithm for online load balancing when the objective is to minimize the total latency of all clients on servers with linear latency functions.

On the complexity of the regenerator placement problem in optical networks

Placement of regenerators in optical networks has attracted the attention of recent research works in optical networks. In this problem, we are given a network with an underlying topology of a graph G and with a set of requests that correspond to paths in G. There is a need to put a regenerator every certain distance, because of a decrease in the power of the signal. In this paper, we investigate the problem of minimizing the number of locations to place the regenerators. We present analytical results regarding the complexity of this problem, in four cases, depending on whether or not there is a bound on the number of regenerators at each node, and depending on whether or not the routing is given or only the requests are given (and part of the solution is also to determine the actual routing). These results include polynomial time algorithms, NP-completeness results, approximation algorithms, and inapproximability results.

Tight Bounds for Selfish and Greedy Load Balancing

We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it chooses, among its permissible servers, to run its job on the server having the smallest latency given the assignments of the jobs of other clients to servers. In online load balancing, clients appear online and, when a client appears, it has to make an irrevocable decision and assign its job to one of its permissible servers. Here, we assume that the clients aim to optimize some global criterion but in an online fashion. A natural local optimization criterion that can be used by each client when making its decision is to assign its job to that server that gives the minimum increase of the global objective. This gives rise to greedy online solutions. The aim of this paper is to determine how much the quality of load balancing is affected by selfishness and greediness.We characterize almost completely the impact of selfishness and greediness in load balancing by presenting new and improved, tight or almost tight bounds on the price of anarchy of selfish load balancing as well as on the competitiveness of the greedy algorithm for online load balancing when the objective is to minimize the total latency of all clients on servers with linear latency functions. In addition, we prove a tight upper bound on the price of stability of linear congestion games.

Minimizing total busy time in parallel scheduling with application to optical networks

We consider a scheduling problem in which a bounded number of jobs can be processed simultaneously by a single machine. The input is a set of n jobs J = {J1, … , Jn}. Each job, Jj, is associated with an interval [sj, cj] along which it should be processed. Also given is the parallelism parameter g ≥ 1, which is the maximal number of jobs that can be processed simultaneously by a single machine. Each machine operates along a contiguous time interval, called its busy interval, which contains all the intervals corresponding to the jobs it processes. The goal is to assign the jobs to machines such that the total busy time of the machines is minimized. The problem is known to be NP-hard already for g = 2. We present a 4-approximation algorithm for general instances, and approximation algorithms with improved ratios for instances with bounded lengths, for instances where any two intervals intersect, and for instances where no interval is properly contained in another. Our study has important application in optimizing the switching costs of optical networks.

On nash equilibria in non-cooperative all-optical networks

In this paper we investigate the problem in which an all-optical network provider must determine suitable payment functions for non-cooperative agents wishing to communicate so as to induce routings in Nash equilibrium using a low number of wavelengths. We assume three different information levels specifying the local knowledge that agents may exploit to compute their payments. While under complete information of all the agents and their routing strategies we show that functions can be determined that perform how centralized algorithms preserving their time complexity, knowing only the used wavelengths along connecting paths (minimal level) or along the edges (intermediate level) the most reasonable functions either do not admit equilibria or equilibria with a different color assigned to each agent, that is with the worst possible ratio between the Nash versus optimum performance, also called price of anarchy. However, by suitably restricting the network topology, a price of anarchy 25.72 has been obtained for chains and 51.44 for rings under the minimal level, and further reduced respectively to 3 and 6 under the intermediate level, up to additive factors converging to 0 as the load increases. Finally, again under the minimal level, a price of anarchy logarithmic in the number of agents has been determined also for trees.

Graphical Congestion Games

We consider congestion games with linear latency functions in which each player is aware only of a subset of all the other players. This is modeled by means of a social knowledge graph G in which nodes represent players and there is an edge from i to j if i knows j . Under the assumption that the payoff of each player is affected only by the strategies of the adjacent ones, we first give a complete characterization of the games possessing pure Nash equilibria. We then investigate the impact of the limited knowledge of the players on the performance of the game. More precisely, given a bound on the maximum degree of G , for the convergent cases we provide tight lower and upper bounds on the price of stability and asymptotically tight bounds on the price of anarchy. All the results are then extended to load balancing games.

The Price of Anarchy in All-Optical Networks

In this paper we consider all-optical networks in which a service provider has to satisfy a given set of communication requests. Each request is charged a cost depending on its wavelength and on the wavelengths of the other requests met along its path in the network. Under the assumption that each request is issued by a selfish agent, we seek for payment strategies which can guarantee the existence of a pure Nash equilibrium, that is an assignment of paths to the requests so that no request can lower its cost by choosing a different path in the network. For such strategies, we bound the loss of performance of the network (price of anarchy) by comparing the number of wavelengths used by the worst pure Nash equilibrium with that of a centralized optimal solution.

When ignorance helps: Graphical multicast cost sharing games

In non-cooperative games played on highly decentralized networks the assumption that each player knows the strategy adopted by any other player may be too optimistic or even infeasible. In such situations, the set of players of which each player knows the chosen strategy can be modeled by means of a social knowledge graph in which nodes represent players and there is an edge from i to j if i knows j. Following the framework introduced in [7], we study the impact of social knowledge graphs on the fundamental multicast cost sharing game in which all the players want to receive the same communication from a given source in an undirected network. In the classical complete information case, such a game is known to be highly inefficient, since its price of anarchy can be as high as the total number of players @r. We first show that, under our incomplete information setting, pure Nash equilibria always exist only if the social knowledge graph is directed acyclic (DAG). We then prove that the price of stability of any DAG is at least 12log@r and provide a DAG lowering the classical price of anarchy to a value between 12log@r and log^2@r. If specific instances of the game are concerned, that is if the social knowledge graph can be selected as a function of the instance, we show that the price of stability is at least 4@r@r+3, and that the same bound holds also for the price of anarchy of any social knowledge graph (not only DAGs). Moreover, we provide a nearly matching upper bound by proving that, for any fixed instance, there always exists a DAG yielding a price of anarchy less than 4. Our results open a new window on how the performances of non-cooperative systems may benefit from the lack of total knowledge among players.

The Speed of Convergence in Congestion Games under Best-Response Dynamics

We investigate the speed of convergence of congestion games with linear latency functions under best response dynamics. Namely, we estimate the social performance achieved after a limited number of rounds, during each of which every player performs one best response move. In particular, we show that the price of anarchy achieved after krounds, defined as the highest possible ratio among the total latency cost, that is the sum of all players latencies, and the minimum possible cost, is

The Price of Stability for Undirected Broadcast Network Design with Fair Cost Allocation Is Constant

O(\sqrt[2^{k-1}] {n})