136 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    On Approximability of Steiner Tree in â„“p\ell_p-metrics

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    In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the ℓ1\ell_1-metric (and Hamming metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to approximate to factor of 96/95≈1.0196/95\approx 1.01 in the graph metric (and consequently ℓ∞\ell_\infty-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric! In this work, we prove that DST is APX-hard in every ℓp\ell_p-metric. We also prove that CST is APX-hard in the ℓ∞\ell_{\infty}-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in ℓp\ell_p-metrics. As an immediate consequence, this yields a 1.391.39-approximation polynomial time algorithm for CST in ℓp\ell_p-metrics.Comment: Abstract shortened due to arxiv's requirement

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Decomposing cubic graphs into isomorphic linear forests

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    A common problem in graph colouring seeks to decompose the edge set of a given graph into few similar and simple subgraphs, under certain divisibility conditions. In 1987 Wormald conjectured that the edges of every cubic graph on 4n4n vertices can be partitioned into two isomorphic linear forests. We prove this conjecture for large connected cubic graphs. Our proof uses a wide range of probabilistic tools in conjunction with intricate structural analysis, and introduces a variety of local recolouring techniques.Comment: 49 pages, many figure

    Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

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    A graph is OkO_k-free if it does not contain kk pairwise vertex-disjoint and non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in OkO_k-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that "sparse" (here, not containing large complete bipartite graphs as subgraphs) OkO_k-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of O2O_2-free graphs without K3,3K_{3,3}-subgraph and whose treewidth is (at least) logarithmic. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse OkO_k-free graphs, and that deciding the OkO_k-freeness of sparse graphs is polynomial time solvable.Comment: 28 pages, 6 figures. v3: improved complexity result

    SS-Packing Coloring of Cubic Halin Graphs

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    Given a non-decreasing sequence S=(s1,s2,…,sk)S = (s_{1}, s_{2}, \ldots , s_{k}) of positive integers, an SS-packing coloring of a graph GG is a partition of the vertex set of GG into kk subsets {V1,V2,…,Vk}\{V_{1}, V_{2}, \ldots , V_{k}\} such that for each 1≤i≤k1 \leq i \leq k, the distance between any two distinct vertices uu and vv in ViV_{i} is at least si+1s_{i} + 1. In this paper, we study the problem of SS-packing coloring of cubic Halin graphs, and we prove that every cubic Halin graph is (1,1,2,3)(1,1,2,3)-packing colorable. In addition, we prove that such graphs are (1,2,2,2,2,2)(1,2,2,2,2,2)-packing colorable.Comment: 9 page

    Twin-width VIII: delineation and win-wins

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    We introduce the notion of delineation. A graph class C\mathcal C is said delineated if for every hereditary closure D\mathcal D of a subclass of C\mathcal C, it holds that D\mathcal D has bounded twin-width if and only if D\mathcal D is monadically dependent. An effective strengthening of delineation for a class C\mathcal C implies that tractable FO model checking on C\mathcal C is perfectly understood: On hereditary closures D\mathcal D of subclasses of C\mathcal C, FO model checking is fixed-parameter tractable (FPT) exactly when D\mathcal D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we show that segment graphs, directed path graphs, and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that Kt,tK_{t,t}-free segment graphs, and axis-parallel HtH_t-free unit segment graphs have bounded twin-width, where HtH_t is the half-graph or ladder of height tt. In contrast, axis-parallel H4H_4-free two-lengthed segment graphs have unbounded twin-width. Our new results, combined with the known FPT algorithm for FO model checking on graphs given with O(1)O(1)-sequences, lead to win-win arguments. For instance, we derive FPT algorithms for kk-Ladder on visibility graphs of 1.5D terrains, and kk-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum
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