34,140 research outputs found

    On the Complexity of Anchored Rectangle Packing

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    Dagstuhl Reports : Volume 1, Issue 2, February 2011

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    Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

    Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

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    We present a deterministic distributed algorithm that computes a (2Δ1)(2\Delta-1)-edge-coloring, or even list-edge-coloring, in any nn-node graph with maximum degree Δ\Delta, in O(log7Δlogn)O(\log^7 \Delta \log n) rounds. This answers one of the long-standing open questions of \emph{distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were 2O(logn)2^{O(\sqrt{\log n})} by Panconesi and Srinivasan [STOC'92] and O~(Δ)+O(logn)\tilde{O}(\sqrt{\Delta}) + O(\log^* n) by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2Δ1)(2\Delta-1)-edge-coloring to poly(loglogn)(\log\log n) rounds. The key technical ingredient is a deterministic distributed algorithm for \emph{hypergraph maximal matching}, which we believe will be of interest beyond this result. In any hypergraph of rank rr --- where each hyperedge has at most rr vertices --- with nn nodes and maximum degree Δ\Delta, this algorithm computes a maximal matching in O(r5log6+logrΔlogn)O(r^5 \log^{6+\log r } \Delta \log n) rounds. This hypergraph matching algorithm and its extensions lead to a number of other results. In particular, a polylogarithmic-time deterministic distributed maximal independent set algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a ((logΔ/ε)O(log(1/ε)))((\log \Delta/\varepsilon)^{O(\log (1/\varepsilon))})-round deterministic algorithm for (1+ε)(1+\varepsilon)-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting λ\lambda-arboricity graphs with out-degree at most (1+ε)λ(1+\varepsilon)\lambda, for any constant ε>0\varepsilon>0, hence partially answering Open Problem 10 of Barenboim and Elkin's book

    Approximating max-min linear programs with local algorithms

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    A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise minkvckvxv\min_k \sum_v c_{kv} x_v subject to vaivxv1\sum_v a_{iv} x_v \le 1 for each ii and xv0x_v \ge 0 for each vv. Here ckv0c_{kv} \ge 0, aiv0a_{iv} \ge 0, and the support sets Vi={v:aiv>0}V_i = \{v : a_{iv} > 0 \}, Vk={v:ckv>0}V_k = \{v : c_{kv}>0 \}, Iv={i:aiv>0}I_v = \{i : a_{iv} > 0 \} and Kv={k:ckv>0}K_v = \{k : c_{kv} > 0 \} have bounded size. In the distributed setting, each agent vv is responsible for choosing the value of xvx_v, and the communication network is a hypergraph H\mathcal{H} where the sets VkV_k and ViV_i constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if Vi|V_i| and Vk|V_k| are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in H\mathcal{H}.Comment: 16 pages, 2 figure
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