1,170 research outputs found

    ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs

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    We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{?(?k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(?k)}(n+m)^?(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{?(?k)}(n+m)^?(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{?(?klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width ?(?k)

    Eth-tight algorithms for long path and cycle on unit disk graphs

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    We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2O(√k)(n + m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2o(√k)(n + m)O(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2O(√k)(n + m)O(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2O(√k log k)(n + m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width O(√k). © Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020)

    Hyperbolic intersection graphs and (quasi)-polynomial time

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    We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in dd-dimensional hyperbolic space, which we denote by Hd\mathbb{H}^d. Using a new separator theorem, we show that unit ball graphs in Hd\mathbb{H}^d enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in 2O(n11/(d1))2^{O(n^{1-1/(d-1)})} time for any fixed d3d\geq 3, while the same problems need 2O(n11/d)2^{O(n^{1-1/d})} time in Rd\mathbb{R}^d. We also show that these algorithms in Hd\mathbb{H}^d are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in H2\mathbb{H}^2, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial (nO(logn)n^{O(\log n)}) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and 33-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in H2\mathbb{H}^2 have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require 2Ω(n)2^{\Omega(\sqrt{n})} time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching nΩ(logn)n^{\Omega(\log n)} lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

    Optimality program in segment and string graphs

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    Planar graphs are known to allow subexponential algorithms running in time 2O(n)2^{O(\sqrt n)} or 2O(nlogn)2^{O(\sqrt n \log n)} for most of the paradigmatic problems, while the brute-force time 2Θ(n)2^{\Theta(n)} is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in 2O(n2/3logn)2^{O(n^{2/3}\log n)} by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time 2O(n2/3logO(1)n)2^{O(n^{2/3}\log^{O(1)}n)} on string graphs while an algorithm running in time 2o(n)2^{o(n)} for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of 2o(n2/3)2^{o(n^{2/3})} which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure

    Feedback Vertex Set on Geometric Intersection Graphs

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    In this paper, we present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time 2^O(?k)(n+m), where n and m denote the numbers of vertices and edges, respectively. This improves the 2^O(?klog k) n^O(1)-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis. Also, our algorithm can be extended to handle geometric intersection graphs of similarly sized fat objects without increasing the running time

    Syntactic Separation of Subset Satisfiability Problems

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    Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems

    A quasi-polynomial algorithm for well-spaced hyperbolic TSP

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    We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature 1-1. Let α\alpha denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an nO(log2n)max(1,1/α)n^{O(\log^2 n)\max(1,1/\alpha)} algorithm for Hyperbolic TSP. This is quasi-polynomial time if α\alpha is at least some absolute constant, and it grows to nO(n)n^{O(\sqrt{n})} as α\alpha decreases to log2n/n\log^2 n/\sqrt{n}. (For even smaller values of α\alpha, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of nO(n)n^{O(\sqrt{n})}.)Comment: SoCG 202

    Clique-Based Separators for Geometric Intersection Graphs

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    Let F be a set of n objects in the plane and let G^x(F) be its intersection graph. A balanced clique-based separator of G^x(F) is a set S consisting of cliques whose removal partitions G^x(F) into components of size at most δn, for some fixed constant δ < 1. The weight of a clique-based separator is defined as ∑_{C ∈ S} log (|C|+1). Recently De Berg et al. (SICOMP 2020) proved that if S consists of convex fat objects, then G^x(F) admits a balanced clique-based separator of weight O(√n). We extend this result in several directions, obtaining the following results. - Map graphs admit a balanced clique-based separator of weight O(√n), which is tight in the worst case. - Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3} log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(√n log n). - Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3} log n). - Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(√n + r log(n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for MAXIMUM INDEPENDENT SET (and, hence, VERTEX COVER), for FEEDBACK VERTEX SET, and for q-Coloring for constant q in these graph classes.ISSN:1868-896
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