458 research outputs found

    Graphs with large generalized (edge-)connectivity

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    The generalized kk-connectivity κk(G)\kappa_k(G) of a graph GG, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized kk-edge-connectivity λk(G)\lambda_k(G). In this paper, graphs of order nn such that κk(G)=nk21\kappa_k(G)=n-\frac{k}{2}-1 and λk(G)=nk21\lambda_k(G)=n-\frac{k}{2}-1 for even kk are characterized.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1207.183

    Packing Steiner Trees

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    Let TT be a distinguished subset of vertices in a graph GG. A TT-\emph{Steiner tree} is a subgraph of GG that is a tree and that spans TT. Kriesell conjectured that GG contains kk pairwise edge-disjoint TT-Steiner trees provided that every edge-cut of GG that separates TT has size 2k\ge 2k. When T=V(G)T=V(G) a TT-Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved that Kriesell's conjecture holds when 2k2k is replaced by 24k24k, and recently West and Wu have lowered this value to 6.5k6.5k. Our main result makes a further improvement to 5k+45k+4.Comment: 38 pages, 4 figure

    Streaming Complexity of Spanning Tree Computation

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    The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an n-node input graph to be read sequentially in p passes using Õ(n) space. If the list of edges includes deletions, then the model is called the turnstile model; otherwise it is called the insertion-only model. In both models, some graph problems, such as spanning trees, k-connectivity, densest subgraph, degeneracy, cut-sparsifier, and (Δ+1)-coloring, can be exactly solved or (1+ε)-approximated in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require Ω̃(n) passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees, including BFS, DFS, and maximum-leaf spanning trees. Our results, in both the insertion-only and the turnstile models, are as follows. Maximum-Leaf Spanning Trees: This problem is known to be APX-complete with inapproximability constant ρ ∈ [245/244, 2). By constructing an ε-MLST sparsifier, we show that for every constant ε > 0, MLST can be approximated in a single pass to within a factor of 1+ε w.h.p. (albeit in super-polynomial time for ε ≤ ρ-1 assuming P ≠ NP) and can be approximated in polynomial time in a single pass to within a factor of ρ_n+ε w.h.p., where ρ_n is the supremum constant that MLST cannot be approximated to within using polynomial time and Õ(n) space. In the insertion-only model, these algorithms can be deterministic. BFS Trees: It is known that BFS trees require ω(1) passes to compute, but the naïve approach needs O(n) passes. We devise a new randomized algorithm that reduces the pass complexity to O(√n), and it offers a smooth tradeoff between pass complexity and space usage. This gives a polynomial separation between single-source and all-pairs shortest paths for unweighted graphs. DFS Trees: It is unknown whether DFS trees require more than one pass. The current best algorithm by Khan and Mehta [STACS 2019] takes Õ(h) passes, where h is the height of computed DFS trees. Note that h can be as large as Ω(m/n) for n-node m-edge graphs. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for k-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to O(√n), and it also offers a smooth tradeoff between pass complexity and space usage.ISSN:1868-896

    Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

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    The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E)G=(V,E) and a set DV×V\mathcal{D}\subseteq V\times V of kk demand pairs. The aim is to compute the cheapest network NGN\subseteq G for which there is an sts\to t path for each (s,t)D(s,t)\in\mathcal{D}. It is known that this problem is notoriously hard as there is no k1/4o(1)k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parametrizing the runtime by kk [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter kk. For the bi-DSNPlanar_\text{Planar} problem, the aim is to compute a planar optimum solution NGN\subseteq G in a bidirected graph GG, i.e., for every edge uvuv of GG the reverse edge vuvu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for kk. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSNPlanar_\text{Planar}, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network NGN\subseteq G needs to strongly connect a given set of kk terminals. It has been observed before that for SCSS a parameterized 22-approximation exists when parameterized by kk [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for kk no parameterized (2ε)(2-\varepsilon)-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for kk
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