915 research outputs found

    Network flow problems and congestion games : complexity and approximation results

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    This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2006.Includes bibliographical references (p. 155-164).(cont.) We first address the complexity of finding an optimal minimum cost solution to a congestion game. We consider both network and general congestion games, and we examine several variants of the problem concerning the structure of the game and its associated cost functions. Many of the problem variants are NP-hard, though we do identify several versions of the games that are solvable in polynomial time. We then investigate existence and the price of anarchy of pure Nash equilibria in k-splittable congestion games with linear costs. A k-splittable congestion game is one in which each player may split its flow on at most k different paths. We identify conditions for the existence of equilibria by providing a series of potential functions. For the price of anarchy, we show an asymptotic lower bound of 2.4 for unweighted k-splittable congestion games and 2.401 for weighted k-splittable congestion games, and an upper bound of 2.618 in both cases.In this thesis we examine four network flow problems arising in the study of transportation, communication, and water networks. The first of these problems is the Integer Equal Flow problem, a network flow variant in which some arcs are restricted to carry equal amounts of flow. Our main contribution is that this problem is not approximable within a factor of 2n(1-epsilon]), for any fixed [epsilon] > 0, where n is the number of nodes in the graph. We extend this result to a number of variants on the size and structure of the arc sets. We next study the Pup Matching problem, a truck routing problem where two commodities ('pups') traversing an arc together in the network incur the arc cost only once. We propose a tighter integer programming formulation for this problem, and we address practical problems that arise with implementing such integer programming solutions. Additionally, we provide approximation and exact algorithms for special cases of the problem where the number of pups is fixed or the total cost in the network is bounded. Our final two problems are on the topic of congestion games, which were introduced in the area of communications networks.by Carol Meyers.Ph.D

    A Branch and Price Algorithm for the k-splittable Maximum Flow Problem

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    The Maximum Flow Problem with flow width constraints is a NP-hard problem. Two models are proposed: the first model is a compact node-arc model using two flow conservation blocks per path. For each path, one block de?nes the path while the other one send the right amount of flow on it. The second model is an extended arc-path model. It is obtained from the first model after a Dantzig-Wolfe reformulation. It is an extended model as it relies on the set of all the paths between the source and the sink nodes. Some symmetry breaking constraints are used to improve the model. A branch and price algorithm is proposed to solve the problem. The column generation reduces to the computation of a shortest path whose cost depends on weights on the arcs and on the path capacity. A polynomial time algorithm is proposed to solve this subproblem. Computational results are shown on a set of medium-sized instances to show the effectiveness of our approach

    A polynomial time approximation algorithm for the two-commodity splittable flow problem

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    We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity i{1,2}{i \in \{1, 2\}} can be split into a bounded number k i of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k i and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k 1, k 2)-splittable flow without chunk size restrictions for fixed demand ratio

    WiLiTV: A Low-Cost Wireless Framework for Live TV Services

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    With the evolution of HDTV and Ultra HDTV, the bandwidth requirement for IP-based TV content is rapidly increasing. Consumers demand uninterrupted service with a high Quality of Experience (QoE). Service providers are constantly trying to differentiate themselves by innovating new ways of distributing content more efficiently with lower cost and higher penetration. In this work, we propose a cost-efficient wireless framework (WiLiTV) for delivering live TV services, consisting of a mix of wireless access technologies (e.g. Satellite, WiFi and LTE overlay links). In the proposed architecture, live TV content is injected into the network at a few residential locations using satellite dishes. The content is then further distributed to other homes using a house-to-house WiFi network or via an overlay LTE network. Our problem is to construct an optimal TV distribution network with the minimum number of satellite injection points, while preserving the highest QoE, for different neighborhood densities. We evaluate the framework using realistic time-varying demand patterns and a diverse set of home location data. Our study demonstrates that the architecture requires 75 - 90% fewer satellite injection points, compared to traditional architectures. Furthermore, we show that most cost savings can be obtained using simple and practical relay routing solutions

    Optimizing Emergency Transportation through Multicommodity Quickest Paths

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    In transportation networks with limited capacities and travel times on the arcs, a class of problems attracting a growing scientific interest is represented by the optimal routing and scheduling of given amounts of flow to be transshipped from the origin points to the specific destinations in minimum time. Such problems are of particular concern to emergency transportation where evacuation plans seek to minimize the time evacuees need to clear the affected area and reach the safe zones. Flows over time approaches are among the most suitable mathematical tools to provide a modelling representation of these problems from a macroscopic point of view. Among them, the Quickest Path Problem (QPP), requires an origin-destination flow to be routed on a single path while taking into account inflow limits on the arcs and minimizing the makespan, namely, the time instant when the last unit of flow reaches its destination. In the context of emergency transport, the QPP represents a relevant modelling tool, since its solutions are based on unsplittable dynamic flows that can support the development of evacuation plans which are very easy to be correctly implemented, assigning one single evacuation path to a whole population. This way it is possible to prevent interferences, turbulence, and congestions that may affect the transportation process, worsening the overall clearing time. Nevertheless, the current state-of-the-art presents a lack of studies on multicommodity generalizations of the QPP, where network flows refer to various populations, possibly with different origins and destinations. In this paper we provide a contribution to fill this gap, by considering the Multicommodity Quickest Path Problem (MCQPP), where multiple commodities, each with its own origin, destination and demand, must be routed on a capacitated network with travel times on the arcs, while minimizing the overall makespan and allowing the flow associated to each commodity to be routed on a single path. For this optimization problem, we provide the first mathematical formulation in the scientific literature, based on mixed integer programming and encompassing specific features aimed at empowering the suitability of the arising solutions in real emergency transportation plans. A computational experience performed on a set of benchmark instances is then presented to provide a proof-of-concept for our original model and to evaluate the quality and suitability of the provided solutions together with the required computational effort. Most of the instances are solved at the optimum by a commercial MIP solver, fed with a lower bound deriving from the optimal makespan of a splittable-flow relaxation of the MCQPP

    On the Existence of Pure Strategy Nash Equilibria in Integer-Splittable Weighted Congestion Games

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    We study the existence of pure strategy Nash equilibria (PSNE) in integer–splittable weighted congestion games (ISWCGs), where agents can strategically assign different amounts of demand to different resources, but must distribute this demand in fixed-size parts. Such scenarios arise in a wide range of application domains, including job scheduling and network routing, where agents have to allocate multiple tasks and can assign a number of tasks to a particular selected resource. Specifically, in an ISWCG, an agent has a certain total demand (aka weight) that it needs to satisfy, and can do so by requesting one or more integer units of each resource from an element of a given collection of feasible subsets. Each resource is associated with a unit–cost function of its level of congestion; as such, the cost to an agent for using a particular resource is the product of the resource unit–cost and the number of units the agent requests.While general ISWCGs do not admit PSNE [(Rosenthal, 1973b)], the restricted subclass of these games with linear unit–cost functions has been shown to possess a potential function [(Meyers, 2006)], and hence, PSNE. However, the linearity of costs may not be necessary for the existence of equilibria in pure strategies. Thus, in this paper we prove that PSNE always exist for a larger class of convex and monotonically increasing unit–costs. On the other hand, our result is accompanied by a limiting assumption on the structure of agents’ strategy sets: specifically, each agent is associated with its set of accessible resources, and can distribute its demand across any subset of these resources.Importantly, we show that neither monotonicity nor convexity on its own guarantees this result. Moreover, we give a counterexample with monotone and semi–convex cost functions, thus distinguishing ISWCGs from the class of infinitely–splittable congestion games for which the conditions of monotonicity and semi–convexity have been shown to be sufficient for PSNE existence [(Rosen, 1965)]. Furthermore, we demonstrate that the finite improvement path property (FIP) does not hold for convex increasing ISWCGs. Thus, in contrast to the case with linear costs, a potential function argument cannot be used to prove our result. Instead, we provide a procedure that converges to an equilibrium from an arbitrary initial strategy profile, and in doing so show that ISWCGs with convex increasing unit–cost functions are weakly acyclic
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