6,065 research outputs found

    Integral Symmetric 2-Commodity Flows

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    Let G=(V,A)G=(V,A) be a symmetric digraph, and κ:AN\kappa:A\rightarrow\mathbb{N} be a symmetric capacity. Let s1,s2,t1,t2Vs_1,s_2,t_1,t_2\in V and v1,v2Nv_1,v_2\in\mathbb{N}. An integral symmetric 2-commodity flow in GG from (s1,s2)(s_1,s_2) to (t1,t2)(t_1,t_2) of value (v1,v2)(v_1,v_2) is an integral 4-commodity flow from (s1,t1,s2,t2)(s_1,t_1,s_2,t_2) to (t1,s1,t2,s2)(t_1,s_1,t_2,s_2) of value (v1,v1,v2,v2)(v_1,v_1,v_2,v_2). The Integral Symmetric 2-Commodity Flow Problem consists in finding a symmetric 2-commodity flow (f1,f1,(f_1,f_{-1}, f2,f2)f_2,f_{-2}) from (s1,t1)(s_1,t_1) to (s2,t2)(s_2,t_2) of value (v1,v2)(v_1,v_2) such that Σfiκ\Sigma f_i\leq\kappa. It is known that the Integral 2-Commodity Flow Problem is NP-complete for both directed and undirected graphs (\cite{FHW80} and \cite{EIS76}). We prove that the cut criterion is a necessary and sufficient condition for the existence of a solution to the Integral Symmetric 2-Commodity Flow Problem, and give a polynomial-time algorithm in that provides a solution to this problem. The time complexity of our algorithm is \textbf{6Cflow+O(A)6C_{flow}+O(|A|)}, where CflowC_{flow} is the time complexity of your favorite flow algorithm (usually in O(V×A)O(|V|\times|A|))

    New and simple algorithms for stable flow problems

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    Stable flows generalize the well-known concept of stable matchings to markets in which transactions may involve several agents, forwarding flow from one to another. An instance of the problem consists of a capacitated directed network, in which vertices express their preferences over their incident edges. A network flow is stable if there is no group of vertices that all could benefit from rerouting the flow along a walk. Fleiner established that a stable flow always exists by reducing it to the stable allocation problem. We present an augmenting-path algorithm for computing a stable flow, the first algorithm that achieves polynomial running time for this problem without using stable allocation as a black-box subroutine. We further consider the problem of finding a stable flow such that the flow value on every edge is within a given interval. For this problem, we present an elegant graph transformation and based on this, we devise a simple and fast algorithm, which also can be used to find a solution to the stable marriage problem with forced and forbidden edges. Finally, we study the stable multicommodity flow model introduced by Kir\'{a}ly and Pap. The original model is highly involved and allows for commodity-dependent preference lists at the vertices and commodity-specific edge capacities. We present several graph-based reductions that show equivalence to a significantly simpler model. We further show that it is NP-complete to decide whether an integral solution exists

    Cut-Matching Games on Directed Graphs

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    We give O(log^2 n)-approximation algorithm based on the cut-matching framework of [10, 13, 14] for computing the sparsest cut on directed graphs. Our algorithm uses only O(log^2 n) single commodity max-flow computations and thus breaks the multicommodity-flow barrier for computing the sparsest cut on directed graph

    Hardness Results for Structured Linear Systems

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    We show that if the nearly-linear time solvers for Laplacian matrices and their generalizations can be extended to solve just slightly larger families of linear systems, then they can be used to quickly solve all systems of linear equations over the reals. This result can be viewed either positively or negatively: either we will develop nearly-linear time algorithms for solving all systems of linear equations over the reals, or progress on the families we can solve in nearly-linear time will soon halt

    Complexity and Approximation of the Continuous Network Design Problem

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    We revisit a classical problem in transportation, known as the continuous (bilevel) network design problem, CNDP for short. We are given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed. The goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing cost of the induced Wardrop equilibrium and the investment cost. While this problem is considered as challenging in the literature, its complexity status was still unknown. We close this gap showing that CNDP is strongly NP-complete and APX-hard, both on directed and undirected networks and even for instances with affine latencies. As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions (Mathematical Programming, Vol.~34, 1986). Specifically, we derive a closed form expression of its approximation guarantee for arbitrary sets S of allowed latency functions. Second, we propose a different approximation algorithm and show that it has the same approximation guarantee. As our final -- and arguably most interesting -- result regarding approximation, we show that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we give a closed form expression. For affine latencies, e.g., this algorithm achieves a 1.195-approximation which improves on the 5/4 that has been shown before by Marcotte. We finally discuss the case of hard budget constraints on the capacity investment.Comment: 27 page

    Routing Games with Progressive Filling

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    Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived

    Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows

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    The maximum multicommodity flow problem is a natural generalization of the maximum flow problem to route multiple distinct flows. Obtaining a 1ϵ1-\epsilon approximation to the multicommodity flow problem on graphs is a well-studied problem. In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem. As the underlying linear systems in the electrical problems of multicommodity flow problems are no longer Laplacians, our approach is tailored to generate specialized systems which can be preconditioned and solved efficiently using Laplacians. Given an undirected graph with m edges and k commodities, we give algorithms that find 1ϵ1-\epsilon approximate solutions to the maximum concurrent flow problem and the maximum weighted multicommodity flow problem in time \tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
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