We introduce and analyze greedy equilibria (GE) for the well-known model of selfish network creation by [PODC'03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategy-changes, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NP-hard. Hence, poly-time agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the SUM version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3-approximate NE on general networks. For the latter we also provide a lower bound of 3 2 on the approximation ratio. For the MAX version, where agents attempt to minimize their maximum distance, we show that any GE-star is in 2-approximate NE and any GE-tree having larger diameter is in 6 5-approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the MAX version on general networks in GE. This result implies a locality gap of @math for the metric min-max facility location problem, where n is the number of clients.
A part of our work focuses on tree networks. Such topologies are common outcomes of NCGs if edges are expensive, which led the authors of @cite_3 to conjecture that all (non-transient) stable networks of NCGs are trees if @math is greater than some constant. The conjecture was later disproved by @cite_4 but it was shown to be true for high edge-cost. In particular, the authors of @cite_2 proved that all stable networks are trees if @math in the version or if @math in the version. Experimental evidence suggests that this transition to tree networks already happens at much lower edge-cost and it is an interesting open problem to improve on these bounds.
@cite_4 @cite_3 @cite_2
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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. 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, 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. This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes---primary and secondary---and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4 3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem. Given a connected graph G and a failure probability p(e) for each edge e in G, the reliability of G is the probability that G remains connected when each edge e is removed independently with probability p(e). In this paper it is shown that every n-vertex graph contains a sparse backbone, i.e., a spanning subgraph with O(nlogn) edges whose reliability is at least (1-n^-^@W^(^1^)) times that of G. Moreover, for any pair of vertices s, t in G, the (s,t)-reliability of the backbone, namely, the probability that s and t remain connected, is also at least (1-n^-^@W^(^1^)) times that of G. Our proof is based on a polynomial time randomized algorithm for constructing the backbone. In addition, it is shown that the constructed backbone has nearly the same Tutte polynomial as the original graph (in the quarter-plane x>=1, y>1), and hence the graph and its backbone share many additional features encoded by the Tutte polynomial. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math .