Luciano Gualà

Bounded-Distance Network Creation Games

A <i>network creation game</i> simulates a decentralized and noncooperative construction of a communication network. Informally, there are <i>n</i> players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, thereby incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about her centrality in the network, we introduce a new setting in which a player who maintains her (maximum or average) distance to the other nodes within a given bound incurs a cost equal to the number of activated edges; otherwise her cost is unbounded. We study the problem of understanding the structure of pure Nash equilibria of the resulting games, which we call M<scp>ax</scp>BD and S<scp>um</scp>BD, respectively. For both games, we show that when distance bounds associated with players are nonuniform, then equilibria can be arbitrarily bad. On the other hand, for M<scp>ax</scp>BD, we show that when nodes have a uniform bound <i>D</i> ≥ 3 on the maximum distance, then the <i>price of anarchy</i> (PoA) is lower and upper bounded by 2 and <i>O</i>(<i>n</i>^{1/⌊log₃ <i>D</i> ⌋+1}), respectively (i.e., PoA is constant as soon as <i>D</i> is Ω(<i>n</i>^ε), for any ε > 0), while for the interesting case <i>D</i>=2, we are able to prove that the PoA is Ω(&sqrt;<i>n</i>) and <i>O</i>(&sqrt;<i>n</i> log <i>n</i>). For the uniform S<scp>um</scp>BD, we obtain similar (asymptotically) results and moreover show that PoA becomes constant as soon as the bound on the average distance is 2^{<i>ω</i>(&sqrt;log <i>n</i>)}.

Improved approximability and non-approximability results for graph diameter decreasing problems

In this paper, we study two variants of the problem of adding edges to a graph so as to reduce the resulting diameter. More precisely, given a graph G=(V,E), and two positive integers D and B, the Minimum-Cardinality Bounded-Diameter Edge Addition (MCBD) problem is to find a minimum-cardinality set F of edges to be added to G in such a way that the diameter of G+F is less than or equal to D, while the Bounded-Cardinality Minimum-Diameter Edge Addition (BCMD) problem is to find a set F of B edges to be added to G in such a way that the diameter of G+F is minimized. Both problems are well known to be NP-hard, as well as approximable within O(lognlogD) and 4 (up to an additive term of 2), respectively. In this paper, we improve these long-standing approximation ratios to O(logn) and to 2 (up to an additive term of 2), respectively. As a consequence, we close, in an asymptotic sense, the gap on the approximability of MCBD, which was known to be not approximable within clogn, for some constant c>0, unless P=NP. Remarkably, as we further show in the paper, our approximation ratio remains asymptotically tight even if we allow for a solution whose diameter is optimal up to a multiplicative factor approaching 53. On the other hand, on the positive side, we show that at most twice of the minimal number of additional edges suffices to get at most twice of the required diameter. Some of our results extend to the edge-weighted version of the problems.

Bounded-Distance network creation games

A network creation game simulates a decentralized and non-cooperative building of a communication network. Informally, there are n players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about its centrality in the network, in this paper we introduce a new setting in which if a player maintains its (either maximum or average) distance to the other nodes within a given bound, then its cost is simply equal to the number of activated edges, otherwise its cost is unbounded. We study the problem of understanding the structure of pure Nash equilibria of the resulting games, that we call MaxBD and SumBD, respectively. For both games, we show that when distance bounds associated with players are non-uniform, then equilibria can be arbitrarily bad. On the other hand, for MaxBD, we show that when nodes have a uniform bound R on the maximum distance, then the Price of Anarchy (PoA) is lower and upper bounded by 2 and

Computational Aspects of a 2-Player Stackelberg Shortest Paths Tree Game

O\left(n^{\frac{1}{\lfloor\log_3 R\rfloor+1}}\right)

Improved Purely Additive Fault-Tolerant Spanners

for R≥3 (i.e., the PoA is constant as soon as R is Ω(ne), for some e>0), while for the interesting case R=2, we are able to prove that the PoA is

On Stackelberg Pricing with Computationally Bounded Consumers

\Omega(\sqrt{n})

Fault-Tolerant Approximate Shortest-Path Trees

and

Locality-based network creation games

O(\sqrt{n \log n} )

On stackelberg pricing with computationally bounded customers

. For the uniform SumBD we obtain similar (asymptotically) results, and moreover we show that the PoA becomes constant as soon as the bound on the average distance is

A truthful (2 - 2/ k )-approximation mechanism for the steiner tree problem with k terminals*

2^{\omega\big({\sqrt{\log n}}\big)}