On the price of anarchy for high-price links

We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker in 2003. This is a selfish network creation model where players correspond to nodes in a network and each of them can create links to the other n−1 players at a prefixed price α>0. The player’s goal is to minimise the sum of her cost buying edges and her cost for using the resulting network. One of the main conjectures for this model states that the price of anarchy, i.e. the relative cost of the lack of coordination, is constant for all α. This conjecture has been confirmed for α=O(n1−δ) with δ≥1/logn and for α>4n−13. The best known upper bound on the price of anarchy for the remaining range is 2O(logn√) . We give new insights into the structure of the Nash equilibria for α>n and we enlarge the range of the parameter α for which the price of anarchy is constant. Specifically, we prove that for any small ϵ>0, the price of anarchy is constant for α>n(1+ϵ) by showing that any biconnected component of any non-trivial Nash equilibrium, if it exists, has at most a constant number of nodes.This work has been partially supported by funds from the Spanish Ministry for Economy and Competitiveness (MINECO) and the European Union (FEDER funds) under grant GRAMM (TIN2017-86727-C2-1-R) and from the Catalan Agency for Management of University and Research Grants (AGAUR, Generalitat de Catalunya) under project ALBCOM 2017-SGR-786.Peer ReviewedPostprint (author's final draft

On bipartite sum basic equilibria

A connected and undirected graph G of size n≥1 is said to be a sum basic equilibrium iff for every edge uv from G and any node v′ from G, when performing the swap of the edge uv for the edge uv′ the sum of the distances from u to all the other nodes is not strictly reduced. This concept comes from the so called Network Creation Games, a wide subject inside Algorithmic Game Theory that tries to better understand how Internet-like networks behave. It has been shown that the diameter of sum basic equilibria is 2O(logn√) in general and at most 2 for trees. In this paper we see that the upper bound of 2 not only holds for trees but for bipartite graphs, too. Specifically, we show that the only bipartite sum basic equilibrium networks are the complete bipartite graphs Kr,s with r,s≥1 .This work has been partially supported by funds from the Spanish Ministry for Economy and Competitiveness (MINECO) and the European Union (FEDER funds) under grant GRAMM (TIN2017-86727-C2-1-R) and from the Catalan Agency for Management of University and Research Grants (AGAUR, Generalitat de Catalunya) under project ALBCOM 2017-SGR-786.Peer ReviewedPostprint (author's final draft

Network creation games: structure vs anarchy

We study Nash equilibria and the price of anarchy in the classical model of Network Creation Games introduced by Fabrikant et al. In this model every agent (node) buys links at a prefixed price a > 0 in order to get connected to the network formed by all the n agents. In this setting, the reformulated tree conjecture states that for a > n, every Nash equilibrium network is a tree. Since it was shown that the price of anarchy for trees is constant, if the tree conjecture were true, then the price of anarchy would be constant for a > n. Moreover, Demaine et al. conjectured that the price of anarchy for this model is constant. Up to now the last conjecture has been proven in (i) the lower range, for a = O(n1-o¿) with o¿ = 1 and (ii) in the upper range, for a > 65n. In ¿log n contrast, the best upper bound known for the price of anarchy for the remaining range is 2O(vlog n). In this paper we give new insights into the structure of the Nash equilibria for different ranges of a and we enlarge the range for which the price of anarchy is constant. Regarding the upper range, we prove that every Nash equilibrium is a tree for a > 17n and that the price of anarchy is constant even for a > 9n. In the lower range, we show that any Nash equilibrium for a 4, induces an o¿-distance-almost- uniform graph.Postprint (published version

On dlogtime and polylogtime reductions

We investigate properties of the relativized NC and AC hierarchies in their DLOGTIME-. respectively, ALOGTIME-uniform setting and show that these hierarchies can be characterized in terms of adaptive reducibility in logarithmic or polylogarithmic time, i.e. O (log_n)〗⁡2 for i ≥ 0. As a corollary, the relationship between AC^i and NC^i+1 reducibility is clarified by the result stating that if DLOGTIME-uniform AC' and ALOGTIME-u11iform NC•+1 reducibility coincide for i = o when applied to an arbitrary function class F, then they coincide on F for all i 2 O. Our result.substantially generalize various previous results (Wi 90), (ABJ 91), (Ba 91)

On the proper intervalization of colored caterpillar trees

This paper studies the computational complexity of the Proper interval colored graph problem (picg), when the input graph is a colored caterpillar, parameterized by hair length. In order prove our result we establish a close relationship between the picg and a graph layout problem the Proper colored layout problem (pclp). We show a dichotomy: the picg and the pclp are NP-complete for colored caterpillars of hair length ≥ 2, while both problems are in P for colored caterpillars of hair length < 2. For the hardness results we provide a reduction from the Multiprocessor Scheduling problem, while the polynomial time results follow from a characterization in terms of forbidden subgraphs.Preprin

Maximum congestion games on networks: How can we compute their equilibria?

We study Network Maximum Congestion Games, a class of network games where players choose a path between two given nodes in order to minimize the congestion of the bottleneck (the most congested link) of their path. For single-commodity games, we provide an algorithm which computes a Pure Nash Equilibrium in polynomial time. If all players have the same weight, the obtained equilibrium has optimum social cost. If players are allowed to have different weights, the obtained equilibrium has social cost at most 4/3 times worst than the optimum. For multi-commodity games with a fixed number of commodities and a particular graph topology, we also provide an algorithm which computes a Pure Nash Equilibria in polynomial time. We also study some issues related to the quality of the equilibria in this kind of games.Postprint (published version

Firefighting as a game

The Firefighter Problem was proposed in 1995 [16] as a deterministic discrete-time model for the spread (and containment) of a fire. Its applications reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks. In this work, we study the problem from a game-theoretical perspective. Such a context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination. We model the Firefighter Problem as a strategic game where there is one player for each time step who decides where to place the firefighters. We show that the Price of Anarchy is linear in the general case, but at most 2 for trees. We prove that the quality of the equilibria improves when allowing coalitional cooperation among players. In general, we have that the Price of Anarchy is in T(n/k) where k is the coalition size. Furthermore, we show that there are topologies which have a constant Price of Anarchy even when constant sized coalitions are considered.Peer ReviewedPostprint (author’s final draft

The Complexity of pure Nash equilibria in weighted Max-Congestion Games

We study Network Max-Congestion Games (NMC games, for short), a class of network games where each player tries to minimize the most congested edge along the path he uses as strategy. We focus our study on the complexity of computing a pure Nash equilibria in weighted NMC games. We show that, for single-commodity games with non-decreasing delay functions, this problem is in P when either all the paths from the source to the target node are disjoint or all the delay functions are equal. For the general case, we prove that the computation of a PNE belongs to the complexity class PLS through a new technique based on semi-potential functions and a slightly modified definition of the usual local search neighborhood. We further apply this technique to a different class of games (which we call Pareto-efficient) with restricted cost functions. Finally, we also prove some PLS-hardness results, showing that computing a PNE for Pareto-efficient NMC games is indeed a PLS- complete problem.Postprint (published version

Communication tree problems

In this paper, we consider random communication requirements and several cost measures for a particular model of tree routing on a complete network. First we show that a random tree does not give any approximation. Then give approximation algorithms for the case for two random models of requirements.Postprint (published version

Celebrity games

We introduce Celebrity games, a new model of network creation games. In this model players have weights (W being the sum of all the player's weights) and there is a critical distance ß as well as a link cost a. The cost incurred by a player depends on the cost of establishing links to other players and on the sum of the weights of those players that remain farther than the critical distance. Intuitively, the aim of any player is to be relatively close (at a distance less than ß ) from the rest of players, mainly of those having high weights. The main features of celebrity games are that: computing the best response of a player is NP-hard if ß>1 and polynomial time solvable otherwise; they always have a pure Nash equilibrium; the family of celebrity games having a connected Nash equilibrium is characterized (the so called star celebrity games) and bounds on the diameter of the resulting equilibrium graphs are given; a special case of star celebrity games shares its set of Nash equilibrium profiles with the MaxBD games with uniform bounded distance ß introduced in Bilò et al. [6]. Moreover, we analyze the Price of Anarchy (PoA) and of Stability (PoS) of celebrity games and give several bounds. These are that: for non-star celebrity games PoA=PoS=max{1,W/a}; for star celebrity games PoS=1 and PoA=O(min{n/ß,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when ß=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for ß=2.Peer ReviewedPostprint (author's final draft