42 research outputs found

    Light Spanners

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    A tt-spanner of a weighted undirected graph G=(V,E)G=(V,E), is a subgraph HH such that dH(u,v)tdG(u,v)d_H(u,v)\le t\cdot d_G(u,v) for all u,vVu,v\in V. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters k1k\ge 1 and ϵ>0\epsilon>0, any weighted graph GG on nn vertices admits a (2k1)(1+ϵ)(2k-1)\cdot(1+\epsilon)-stretch spanner of weight at most w(MST(G))Oϵ(kn1/k/logk)w(MST(G))\cdot O_\epsilon(kn^{1/k}/\log k), where w(MST(G))w(MST(G)) is the weight of a minimum spanning tree of GG. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of O(logk)O(\log k).Comment: 10 pages, 1 figure, to appear in ICALP 201

    Exploration of Graphs with Excluded Minors

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    We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g ? 1 and recovers the known tight bound for the planar case (g = 0)

    Exploration of graphs with excluded minors

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    We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g>0 and recovers the known tight bound for the planar case (g=0).Comment: to appear at ESA 202

    Scattering and Sparse Partitions, and Their Applications

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    Using Light Spanning Graphs for Passenger Assignment in Public Transport

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    In a public transport network a passenger’s preferred route from a point x to another point y is usually the shortest path from x to y. However, it is simply impossible to provide all the shortest paths of a network via public transport. Hence, it is a natural question how a lighter sub-network should be designed in order to satisfy both the operator as well as the passengers.We provide a detailed analysis of the interplay of the following three quality measures of lighter public transport networks: - building cost: the sum of the costs of all edges remaining in the lighter network, - routing costs: the sum of all shortest paths costs weighted by the demands, - fairness: compared to the original network, for each two points the shortest path in the new network should cost at most a given multiple of the shortest path in the original network. We study the problem by generalizing the concepts of optimum communication spanning trees (Hu, 1974) and optimum requirement graphs (Wu, Chao, and Tang, 2002) to generalized optimum requirement graphs (GORGs), which are graphs achieving the social optimum amongst all subgraphs satisfying a given upper bound on the building cost. We prove that the corresponding decision problem is NP-complete, even on orb-webs, a variant of grids which serves as an important model of cities with a center. For the case that the given network is a parametric city (cf. Fielbaum et. al., 2017) with a heavy vertex we provide a polynomial-time algorithm solving the GORG-problem. Concerning the fairness-aspect, we prove that light spanners are a strong concept for public transport optimization. We underpin our theoretical considerations with integer programming-based experiments that allow us to compare the fairness-approach with the routing cost-approach as well as passenger assignment approaches from the literature.Peer reviewe
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