One of the key results in Robertson and Seymour’s seminal work on graph minors is the grid-minor theorem (also called the excluded grid theorem ). The theorem states that for every grid H , every graph whose treewidth is large enough relative to | V ( H )| contains H as a minor. This theorem has found many applications in graph theory and algorithms. Let f ( k ) denote the largest value such that every graph of treewidth k contains a grid minor of size ( f ( k ) × f ( k )). The best previous quantitative bound, due to recent work of Kawarabayashi and Kobayashi, and Leaf and Seymour, shows that f ( k )=Ω(√log k /log log k ). In contrast, the best known upper bound implies that f ( k ) = O (√ k /log k ). In this article, we obtain the first polynomial relationship between treewidth and grid minor size by showing that f ( k ) = Ω( k δ ) for some fixed constant δ > 0, and describe a randomized algorithm, whose running time is polynomial in | V ( G )| and k , that with high probability finds a model of such a grid minor in G .
We consider a multicast game with selfish non-cooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NP-hard. We focus on the price of anarchy of a Nash equilibrium resulting from the best-response dynamics of a game course, where the players join the game sequentially. For a game with n players, we establish an upper bound of O(√ n log 2 n) on the price of anarchy, and a lower bound of Ω(log n/ log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium. Index Terms Multicast game, Nash equilibrium, Price of anarchy, Price of stability.. 2 I. INTRODUCTION In many networking scenarios, including the Internet, network users are free to act according to their individual interests, without taking into account overall network performance. Users thus may make selfish decisions (strategy choices) based on the state of the network, which depends (among other factors) on the behavior of other users, resulting in a non-cooperative game. Naturally, these scenarios call for a game-theoretic approach for studying both the behavior of such non-cooperative users, as well as their impact on the network performance. More specifically, we are interested in the properties of Nash equilibrium solutions which are the stable outcomes of a non-cooperative game. We note that there is a considerable amount of research dealing with non-cooperative games in networks [16], [19], [26], [29], [30], [32]. A scenario frequently encountered is the situation where each edge has a load-dependent latency function, and each user aims to minimize the total latency from its source to its destination. In this framework, both simple and general network topologies were studied, as well as various types of latency functions and different constraints on the strategies of the users [16], [26], [29], [30], [32]. While unicast is the traditional form of routing, it results in a waste of re...
Given a set A of m agents and a set I of n items, where agent A ∈ A has utility u A,i for item i ∈ I, our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of the utilities for items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention recently, its approximability has not been well-understood thus far: the best known approximation algorithm achieves anÕ( √ m)-approximation, and in contrast, the best known hardness of approximation stands at 2.Our main result is an approximation algorithm that achieves anÕ(n ǫ ) approximation for any ǫ = Ω(log log n/ log n) in time n O(1/ǫ) . In particular, we obtain poly-logarithmic approximation in quasi-polynomial time, and for every constant ǫ > 0, we obtainÕ(n ǫ )-approximation in polynomial time. An interesting technical aspect of our algorithm is that we use as a building block a linear program whose integrality gap is Ω( √ m). We bypass this obstacle by iteratively using the solutions produced by the LP to construct new instances with significantly smaller integrality gaps, eventually obtaining the desired approximation.We also investigate the special case of the problem, where every item has a non-zero utility for at most two agents. We show that even in this restricted setting the problem is hard to approximate upto any factor better than 2, and show a factor (2 + ǫ)-approximation algorithm running in time poly(n, 1/ǫ) for any ǫ > 0. This special case can be cast as a graph edge orientation problem, and our algorithm can be viewed as a generalization of Eulerian orientations to weighted graphs.
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