18,590 research outputs found

    Improved Approximation Algorithms for Steiner Connectivity Augmentation Problems

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    The Weighted Connectivity Augmentation Problem is the problem of augmenting the edge-connectivity of a given graph by adding links of minimum total cost. This work focuses on connectivity augmentation problems in the Steiner setting, where we are not interested in the connectivity between all nodes of the graph, but only the connectivity between a specified subset of terminals. We consider two related settings. In the Steiner Augmentation of a Graph problem (kk-SAG), we are given a kk-edge-connected subgraph HH of a graph GG. The goal is to augment HH by including links and nodes from GG of minimum cost so that the edge-connectivity between nodes of HH increases by 1. In the Steiner Connectivity Augmentation Problem (kk-SCAP), we are given a Steiner kk-edge-connected graph connecting terminals RR, and we seek to add links of minimum cost to create a Steiner (k+1)(k+1)-edge-connected graph for RR. Note that kk-SAG is a special case of kk-SCAP. All of the above problems can be approximated to within a factor of 2 using e.g. Jain's iterative rounding algorithm for Survivable Network Design. In this work, we leverage the framework of Traub and Zenklusen to give a (1+ln2+ε)(1 + \ln{2} +\varepsilon)-approximation for the Steiner Ring Augmentation Problem (SRAP): given a cycle H=(V(H),E)H = (V(H),E) embedded in a larger graph G=(V,EL)G = (V, E \cup L) and a subset of terminals RV(H)R \subseteq V(H), choose a subset of links SLS \subseteq L of minimum cost so that (V,ES)(V, E \cup S) has 3 pairwise edge-disjoint paths between every pair of terminals. We show this yields a polynomial time algorithm with approximation ratio (1+ln2+ε)(1 + \ln{2} + \varepsilon) for 22-SCAP. We obtain an improved approximation guarantee of (1.5+ε)(1.5+\varepsilon) for SRAP in the case that R=V(H)R = V(H), which yields a (1.5+ε)(1.5+\varepsilon)-approximation for kk-SAG for any kk

    Connectivity and spanning trees of graphs

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    This dissertation focuses on connectivity, edge connectivity and edge-disjoint spanning trees in graphs and hypergraphs from the following aspects.;1. Eigenvalue aspect. Let lambda2(G) and tau( G) denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of tau(G), Cioaba and Wong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if lambda 2(G)) \u3c d -- 2k-1d+1 , then tau(G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. We propose a more general conjecture that for a graph G with minimum degree delta ≥ 2 k ≥ 4, if lambda2(G) \u3c delta -- 2k-1d+1 then tau(G) ≥ k. We prove the conjecture for k = 2, 3 and provide partial results for k ≥ 4. We also prove that for a graph G with minimum degree delta ≥ k ≥ 2, if lambda2( G) \u3c delta -- 2k-1d +1 , then the edge connectivity is at least k. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on tau(G) and edge connectivity.;2. Network reliability aspect. With graphs considered as natural models for many network design problems, edge connectivity kappa\u27(G) and maximum number of edge-disjoint spanning trees tau(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G. Let kappa\u27(G) = max{lcub}kappa\u27(H) : H is a subgraph of G{rcub}. We present: (i) For each integer k \u3e 0, a characterization for graphs G with the property that kappa\u27(G) ≤ k but for any additional edge e not in G, kappa\u27(G + e) ≥ k + 1. (ii) For any integer n \u3e 0, a characterization for graphs G with |V(G)| = n such that kappa\u27(G) = tau( G) with |E(G)| minimized.;3. Generalized connectivity. For an integer l ≥ 2, the l-connectivity kappal( G) of a graph G is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Let k ≥ 1, a graph G is called (k, l)-connected if kappa l(G) ≥ k. A graph G is called minimally (k, l)-connected if kappal(G) ≥ k but ∀e ∈ E( G), kappal(G -- e) ≤ k -- 1. A structural characterization for minimally (2, l)-connected graphs and some extremal results are obtained. These extend former results by Dirac and Plummer on minimally 2-connected graphs.;4. Degree sequence aspect. An integral sequence d = (d1, d2, ···, dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r- uniform hypergraph with degree sequence d. It is proved that an r-uniform hypergraphic sequence d = (d1, d2, ···, dn) has a k-edge-connected realization if and only if both di ≥ k for i = 1, 2, ···, n and i=1ndi≥ rn-1r-1 , which generalizes the formal result of Edmonds for graphs and that of Boonyasombat for hypergraphs.;5. Partition connectivity augmentation and preservation. Let k be a positive integer. A hypergraph H is k-partition-connected if for every partition P of V(H), there are at least k(| P| -- 1) hyperedges intersecting at least two classes of P. We determine the minimum number of hyperedges in a hypergraph whose addition makes the resulting hypergraph k-partition-connected. We also characterize the hyperedges of a k-partition-connected hypergraph whose removal will preserve k-partition-connectedness

    Minimum Input Selection for Structural Controllability

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    Given a linear system x˙=Ax\dot{x} = Ax, where AA is an n×nn \times n matrix with mm nonzero entries, we consider the problem of finding the smallest set of state variables to affect with an input so that the resulting system is structurally controllable. We further assume we are given a set of "forbidden state variables" FF which cannot be affected with an input and which we have to avoid in our selection. Our main result is that this problem can be solved deterministically in O(n+mn)O(n+m \sqrt{n}) operations

    Grad and classes with bounded expansion I. decompositions

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    We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, grad r(G). For these classes we prove the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. This generalizes and simplifies several earlier results (obtained for minor closed classes)

    Approximating subset kk-connectivity problems

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    A subset TVT \subseteq V of terminals is kk-connected to a root ss in a directed/undirected graph JJ if JJ has kk internally-disjoint vsvs-paths for every vTv \in T; TT is kk-connected in JJ if TT is kk-connected to every sTs \in T. We consider the {\sf Subset kk-Connectivity Augmentation} problem: given a graph G=(V,E)G=(V,E) with edge/node-costs, node subset TVT \subseteq V, and a subgraph J=(V,EJ)J=(V,E_J) of GG such that TT is kk-connected in JJ, find a minimum-cost augmenting edge-set FEEJF \subseteq E \setminus E_J such that TT is (k+1)(k+1)-connected in JFJ \cup F. The problem admits trivial ratio O(T2)O(|T|^2). We consider the case T>k|T|>k and prove that for directed/undirected graphs and edge/node-costs, a ρ\rho-approximation for {\sf Rooted Subset kk-Connectivity Augmentation} implies the following ratios for {\sf Subset kk-Connectivity Augmentation}: (i) b(ρ+k)+(3TTk)2H(3TTk)b(\rho+k) + {(\frac{3|T|}{|T|-k})}^2 H(\frac{3|T|}{|T|-k}); (ii) ρO(TTklogk)\rho \cdot O(\frac{|T|}{|T|-k} \log k), where b=1 for undirected graphs and b=2 for directed graphs, and H(k)H(k) is the kkth harmonic number. The best known values of ρ\rho on undirected graphs are min{T,O(k)}\min\{|T|,O(k)\} for edge-costs and min{T,O(klogT)}\min\{|T|,O(k \log |T|)\} for node-costs; for directed graphs ρ=T\rho=|T| for both versions. Our results imply that unless k=To(T)k=|T|-o(|T|), {\sf Subset kk-Connectivity Augmentation} admits the same ratios as the best known ones for the rooted version. This improves the ratios in \cite{N-focs,L}

    Non-Uniform Robust Network Design in Planar Graphs

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    Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on \textbf{bulk-robustness}, a model recently introduced~\cite{bulk} for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniques used in~\cite{bulk} to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.Comment: 17 pages, 2 figure
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