Liam Roditty

On nash equilibria for a network creation game

We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n 0 , there exists a graph built by n ≥ n 0 players which contains cycles and forms a non-transient Nash equilibrium, for any α with 1 < α ≤ √n/2. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for α ≥ 12n[log n], every Nash equilibrium forms a tree.Without relying on the tree conjecture, Fabrikant et al. proved an upper bound on the price of anarchy of O(√α), where α ∈ [2, n2]. We improve this bound. Specifically, we derive a constant upper bound for α ∈ O(√n) and for α ≥ 12n[log n]. For the intermediate values we derive an improved bound of O(1 + (min{α2/n, n2/α})1/3).Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing.

Deterministic constructions of approximate distance oracles and spanners

Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn

A fully dynamic reachability algorithm for directed graphs with an almost linear update time

^{\rm 1/{\it k}}

Improved Dynamic Reachability Algorithms for Directed Graphs

) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn

Dynamic approximate all-pairs shortest paths in undirected graphs

^{\rm 1+1/{\it k}}

Replacement paths and k simple shortest paths in unweighted directed graphs

) capable of returning in O(k) time an approximation

SINR diagrams: towards algorithmically usable SINR models of wireless networks

\hat{\delta}(u,v)

Distributed algorithms for network diameter and girth

of the distance δ(u,v) from u to v in G that satisfies

An Optimal Dynamic Spanner for Doubling Metric Spaces

\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)

Improved dynamic reachability algorithms for directed graphs

, for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn