Vittorio Bilò

Geometric clustering to minimize the sum of cluster sizes

We study geometric versions of the min-size k-clustering problem, a clustering problem which generalizes clustering to minimize the sum of cluster radii and has important applications. We prove that the problem can be solved in polynomial time when the points to be clustered are located on a line. For Euclidean spaces of higher dimensions, we show that the problem is NP-hard and present polynomial time approximation schemes. The latter result yields an improved approximation algorithm for the related problem of k-clustering to minimize the sum of cluster diameters.

On the packing of selfish items

In the non cooperative version of the classical minimum bin packing problem, an item is charged a cost according to the percentage of the used bin space it requires. We study the game induced by the selfish behavior of the items which are interested in being packed in one of the bins so as to minimize their cost. We prove that such a game always converges to a pure Nash equilibrium starting from any initial packing of the items, estimate the number of steps needed to reach one such equilibrium, prove the hardness of computing good equilibria and give an upper and a lower bound for the price of anarchy of the game. Then, we consider a multidimensional extension of the problem in which each item can require to be packed in more than just one bin. Unfortunately, we show that in such a case the induced game may not admit a pure Nash equilibrium even under particular restrictions. The study of these games finds applications in the analysis of the bandwidth cost sharing problem in non cooperative 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.

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.

On the Crossing Spanning Tree Problem

Given an undirected n-node graph and a set \({\cal C}\) of m cuts, the minimum crossing spanning tree is a spanning tree which minimizes the maximum crossing of any cut in \({\cal C}\), where the crossing of a cut is the number of edges in the intersection of this cut and the tree. This problem finds applications in fields as diverse as Computational Biology and IP Routing Table Minimization.

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.

BOUNDS ON THE PRICE OF STABILITY OF UNDIRECTED NETWORK DESIGN GAMES WITH THREE PLAYERS

After almost seven years from its definition,2 the price of stability of undirected network design games with fair cost allocation remains to be elusive. Its exact characterization has been achieved only for the basic case of two players2,7 and, as soon as the number of players increases, the gap between the known upper and lower bounds becomes super-constant, even in the special variants of multicast and broadcast games. Motivated by the intrinsic difficulties that seem to characterize this problem, we analyze the already challenging case of three players and provide either new or improved bounds. For broadcast games, we prove an upper bound of 1.485 which exactly matches a lower bound given in Ref. 4; for multicast games, we show new upper and lower bounds which confine the price of stability in the interval [1.524; 1.532]; while, for the general case, we give an improved upper bound of 1.634. The techniques exploited in this paper are a refinement of those used in Ref. 7 and can be easily adapted to deal with all the cases involving a small number of players.

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.

A Unifying Tool for Bounding the Quality of Non-Cooperative Solutions in Weighted Congestion Games ⋆

We present a general technique, based on a primal-dual formulation, for analyzing the quality of self-emerging solutions in weighted congestion games. With respect to traditional combinatorial approaches, the primal-dual schema has at least three advantages: first, it provides an analytic tool which can always be used to prove tight upper bounds for all the cases in which we are able to characterize exactly the polyhedron of the solutions under analysis; secondly, in each such a case, the complementary slackness conditions give us a hint on how to construct matching lower bounding instances; thirdly, proofs become simpler and easy to check. For the sake of exposition, we first apply our technique to the problems of bounding the price of anarchy and stability of exact and approximate pure Nash equilibria, as well as the approximation ratio of the strategy profiles achieved after a one-round walk starting from the empty state, in the case of affine latency functions and we show how all the known upper bounds for these measures (and some of their generalizations) can be easily reobtained under a unified approach. Then, we use the technique to attack the more challenging setting of polynomial latency functions. In particular, we obtain the first known upper bounds on the price of stability of pure Nash equilibria and on the approximation ratio of the strategy profiles achieved after a one-round walk starting from the empty state for unweighted players in the cases of quadratic and cubic latency functions.