Gianpiero Monaco

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.

Improved Lower Bounds on the Price of Stability of Undirected Network Design Games

Bounding the price of stability of undirected network design games with fair cost allocation is a challenging open problem in the Algorithmic Game Theory research agenda. Even though the generalization of such games in directed networks is well understood in terms of the price of stability (it is exactly Hn, the n-th harmonic number, for games with n players), far less is known for network design games in undirected networks. The upper bound carries over to this case as well while the best known lower bound is 42/23≈1.826. For more restricted but interesting variants of such games such as broadcast and multicast games, sublogarithmic upper bounds are known while the best known lower bound is 12/7≈1.714. In the current paper, we improve the lower bounds as follows. We break the psychological barrier of 2 by showing that the price of stability of undirected network design games is at least 348/155≈2.245. Our proof uses a recursive construction of a network design game with a simple gadget as the main building block. For broadcast and multicast games, we present new lower bounds of 20/11≈1.818 and 1.862, respectively.

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.

Improved lower bounds on the price of stability of undirected network design games

Bounding the price of stability of undirected network design games with fair cost allocation is a challenging open problem in the Algorithmic Game Theory research agenda. Even though the generalization of such games in directed networks is well understood in terms of the price of stability (it is exactly Hn, the n-th harmonic number, for games with n players), far less is known for network design games in undirected networks. The upper bound carries over to this case as well while the best known lower bound is 42/23 ≅ 1.826. For more restricted but interesting variants of such games such as broadcast and multicast games, sublogarithmic upper bounds are known while the best known lower bound is 12/7 ≅ 1.714. In the current paper, we improve the lower bounds as follows. We break the psychological barrier of 2 by showing that the price of stability of undirected network design games is at least 348/155 ≅ 2.245. Our proof uses a recursive construction of a network design game with a simple gadget as the main building block. For broadcast and multicast games, we present new lower bounds of 20/11 ≅ 1.818 and 1.862, respectively.

Approximating the Traffic Grooming Problem with Respect to ADMs and OADMs

We consider the problem of switching cost in optical networks, where messages are sent along lightpaths. Given lightpaths, we have to assign them colors, so that at most glightpaths of the same color can share any edge (gis the grooming factor). The switching of the lightpaths is performed by electronic ADMs (Add-Drop-Multiplexers) at their endpoints and optical ADMs (OADMs) at their intermediate nodes. The saving in the switching components becomes possible when lightpaths of the same color can use the same switches. Whereas previous studies concentrated on the number of ADMs, we consider the cost function - incurred also by the number of OADMs - of f(i¾?) = i¾?|OADMs| + (1 i¾? i¾?)|ADMs|, where 0 ≤ i¾?≤ 1. We concentrate on chain networks, but our technique can be directly extended to ring networks. We show that finding a coloring which will minimize this cost function is NP-complete, even when the network is a chain and the grooming factor is g= 2, for any value of i¾?. We then present a general technique that, given an r-approximation algorithm working on particular instances of our problem, i.e. instances in which all requests share a common edge of the chain, builds a new algorithm for general instances having approximation ratio ri¾?logni¾?. This technique is used in order to obtain two polynomial time approximation algorithms for our problem: the first one minimizes the number of OADMs (the case of i¾?= 1), and its approximation ratio is 2 i¾?logni¾?; the second one minimizes the combined cost f(i¾?) for 0 ≤ i¾?< 1, and its approximation ratio is

Nash Stability in Fractional Hedonic Games

2 \sqrt{g}~\lceil \log n \rceil

The ring design game with fair cost allocation

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Some Anomalies of Farsighted Strategic Behavior

Cluster formation games are games in which self-organized groups (or clusters) are created as a result of the strategic interactions of independent and selfish players. We consider fractional hedonic games, that is, cluster formation games in which the happiness of each player in a group is the average value she ascribes to its members. We adopt Nash stable outcomes, where no player can improve her utility by unilaterally changing her own group, as the target solution concept and study their existence, complexity and performance for games played on general and specific graph topologies.

Optimizing regenerator cost in traffic grooming

In this paper we study the network design game when the underlying network is a ring. In a network design game we have a set of players, each of them aims at connecting nodes in a network by installing links and equally sharing the cost of the installation with other users. The ring design game is the special case in which the potential links of the network form a ring. It is well known that in a ring design game the price of anarchy may be as large as the number of players. Our aim is to show that, despite the worst case, the ring design game always possesses good equilibria. In particular, we prove that the price of stability of the ring design game is at most 3/2, and such bound is tight. Moreover, we observe that the worst Nash equilibrium cannot cost more than 2 times the optimum if the price of stability is strictly larger than 1. We believe that our results might be useful for the analysis of more involved topologies of graphs, e.g., planar graphs.

Approximating the revenue maximization problem with sharp demands

We investigate the loss in optimality due to the presence of selfish players in sequential games, a relevant subclass of extensive form games with perfect information recently introduced in Paes Leme et al. (Proceedings of innovations in theoretical computer science (ITCS), ACM, New York, pp. 60–67, 2012). In such a setting, the notion of subgame perfect equilibrium is preferred to that of Nash equilibrium since it better captures the farsighted rationality of the players who can anticipate future strategic opportunities. We prove that the sequential price of anarchy, that is the worst-case ratio between the social performance at a subgame perfect equilibrium and that of the best possible solution, is exactly 3 in cut and consensus games. Moreover, we improve the known Ω(n) lower bound for unrelated scheduling to