89,703 research outputs found

    Spanning trees short or small

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    We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number kk of nodes are required to be connected in the solution. A prototypical example is the kkMST problem in which we require a tree of minimum weight spanning at least kk nodes in an edge-weighted graph. We show that the kkMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2k2\sqrt{k} for the general edge-weighted case and O(k1/4)O(k^{1/4}) for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding kk-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

    A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and Beyond

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    Short spanning trees subject to additional constraints are important building blocks in various approximation algorithms. Especially in the context of the Traveling Salesman Problem (TSP), new techniques for finding spanning trees with well-defined properties have been crucial in recent progress. We consider the problem of finding a spanning tree subject to constraints on the edges in cuts forming a laminar family of small width. Our main contribution is a new dynamic programming approach where the value of a table entry does not only depend on the values of previous table entries, as it is usually the case, but also on a specific representative solution saved together with each table entry. This allows for handling a broad range of constraint types. In combination with other techniques -- including negatively correlated rounding and a polyhedral approach that, in the problems we consider, allows for avoiding potential losses in the objective through the randomized rounding -- we obtain several new results. We first present a quasi-polynomial time algorithm for the Minimum Chain-Constrained Spanning Tree Problem with an essentially optimal guarantee. More precisely, each chain constraint is violated by a factor of at most 1+ε1+\varepsilon, and the cost is no larger than that of an optimal solution not violating any chain constraint. The best previous procedure is a bicriteria approximation violating each chain constraint by up to a constant factor and losing another factor in the objective. Moreover, our approach can naturally handle lower bounds on the chain constraints, and it can be extended to constraints on cuts forming a laminar family of constant width. Furthermore, we show how our approach can also handle parity constraints (or, more precisely, a proxy thereof) as used in the context of (Path) TSP and one of its generalizations, and discuss implications in this context.Comment: A short version of this work appeared in the proceedings of the 30th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019

    Edge-Orders

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    Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a graph decomposition into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1,1)-edge-orders of 2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge- to vertex-connectivity are bound to fail. As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2- and 3-edge independent spanning trees of 2- and 3-edge-connected graphs, the latter of which improves the best known running time from O(n^2) to linear time

    Bounds on the maximum multiplicity of some common geometric graphs

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    We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

    Distributed Connectivity Decomposition

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    We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity kk into (fractionally) vertex-disjoint weighted dominating trees with total weight Ω(klogn)\Omega(\frac{k}{\log n}), in O~(D+n)\widetilde{O}(D+\sqrt{n}) rounds. (II) A decomposition of each undirected graph with edge-connectivity λ\lambda into (fractionally) edge-disjoint weighted spanning trees with total weight λ12(1ε)\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon), in O~(D+nλ)\widetilde{O}(D+\sqrt{n\lambda}) rounds. We also show round complexity lower bounds of Ω~(D+nk)\tilde{\Omega}(D+\sqrt{\frac{n}{k}}) and Ω~(D+nλ)\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}}) for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from O(n3)O(n^3) to near-optimal O~(m)\tilde{O}(m). As corollaries, we also get distributed oblivious routing broadcast with O(1)O(1)-competitive edge-congestion and O(logn)O(\log n)-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal O(logn)O(\log n)-approximation of vertex connectivity: centralized O~(m)\widetilde{O}(m) and distributed O~(D+n)\tilde{O}(D+\sqrt{n}). The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an O(m)O(m) centralized exact algorithm while the latter is the first distributed vertex connectivity approximation
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