104 research outputs found

    (Total) Vector Domination for Graphs with Bounded Branchwidth

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    Given a graph G=(V,E)G=(V,E) of order nn and an nn-dimensional non-negative vector d=(d(1),d(2),,d(n))d=(d(1),d(2),\ldots,d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum SVS\subseteq V such that every vertex vv in VSV\setminus S (resp., in VV) has at least d(v)d(v) neighbors in SS. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the kk-tuple dominating set problem (this kk is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respectto kk, where kk is the size of solution.Comment: 16 page

    Maximum matching width: new characterizations and a fast algorithm for dominating set

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    We give alternative definitions for maximum matching width, e.g. a graph GG has mmw(G)k\operatorname{mmw}(G) \leq k if and only if it is a subgraph of a chordal graph HH and for every maximal clique XX of HH there exists A,B,CXA,B,C \subseteq X with ABC=XA \cup B \cup C=X and A,B,Ck|A|,|B|,|C| \leq k such that any subset of XX that is a minimal separator of HH is a subset of either A,BA, B or CC. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing on nodes and on edges. Based on this we prove that given a graph GG and a branch decomposition of mm-width kk we can solve Dominating Set in time O(8k)O^*({8^k}), thereby beating O(3tw(G))O^*(3^{\operatorname{tw}(G)}) whenever tw(G)>log38×k1.893k\operatorname{tw}(G) > \log_3{8} \times k \approx 1.893 k. Note that mmw(G)tw(G)+13mmw(G)\operatorname{mmw}(G) \leq \operatorname{tw}(G)+1 \leq 3 \operatorname{mmw}(G) and these inequalities are tight. Given only the graph GG and using the best known algorithms to find decompositions, maximum matching width will be better for solving Dominating Set whenever tw(G)>1.549×mmw(G)\operatorname{tw}(G) > 1.549 \times \operatorname{mmw}(G)

    Variants of Plane Diameter Completion

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    The {\sc Plane Diameter Completion} problem asks, given a plane graph GG and a positive integer dd, if it is a spanning subgraph of a plane graph HH that has diameter at most dd. We examine two variants of this problem where the input comes with another parameter kk. In the first variant, called BPDC, kk upper bounds the total number of edges to be added and in the second, called BFPDC, kk upper bounds the number of additional edges per face. We prove that both problems are {\sf NP}-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when k=1k=1 on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in O(n3)+22O((kd)2logd)nO(n^{3})+2^{2^{O((kd)^2\log d)}}\cdot n steps.Comment: Accepted in IPEC 201

    Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n1+ϵ) time

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    AbstractWe give constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs. For a planar graph G with n vertices, let bw(G) be the branchwidth of G and gm(G) the largest integer g such that G has a g×g grid as a minor. Let c≥1 be a fixed integer and α,β arbitrary constants satisfying α>c+1 and β>2c+1. We give an algorithm which constructs in O(n1+1clogn) time a branch-decomposition of G with width at most αbw(G). We also give an algorithm which constructs a g×g grid minor of G with g≥gm(G)β in O(n1+1clogn) time. The constants hidden in the Big-O notations are proportional to cα−(c+1) and cβ−(2c+1), respectively

    Parameterization of Tensor Network Contraction

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    We present a conceptually clear and algorithmically useful framework for parameterizing the costs of tensor network contraction. Our framework is completely general, applying to tensor networks with arbitrary bond dimensions, open legs, and hyperedges. The fundamental objects of our framework are rooted and unrooted contraction trees, which represent classes of contraction orders. Properties of a contraction tree correspond directly and precisely to the time and space costs of tensor network contraction. The properties of rooted contraction trees give the costs of parallelized contraction algorithms. We show how contraction trees relate to existing tree-like objects in the graph theory literature, bringing to bear a wide range of graph algorithms and tools to tensor network contraction. Independent of tensor networks, we show that the edge congestion of a graph is almost equal to the branchwidth of its line graph

    Approximating branchwidth on parametric extensions of planarity

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    The \textsl{branchwidth} of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph classes, further than planar graphs as follows: Let H0H_{0} be a graph embeddedable in the projective plane and H1H_{1} be a graph embeddedable in the torus. We prove that every {H0,H1}\{H_{0},H_{1}\}-minor free graph GG contains a subgraph GG' where the difference between the branchwidth of GG and the branchwidth of GG' is bounded by some constant, depending only on H0H_{0} and H1H_{1}. Moreover, the graph GG' admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving an EPTAS for branchwidth: For {H0,H1}\{H_{0},H_{1}\}-minor free graphs, there is a function f ⁣:NNf\colon\mathbb{N}\to\mathbb{N} and a (1+ϵ)(1+\epsilon)-approximation algorithm for branchwidth, running in time O(n3+f(1ϵ)n),\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n), for every ϵ>0\epsilon>0
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