103 research outputs found

    Convex Polygons in Cartesian Products

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    We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors

    Induced Ramsey-type results and binary predicates for point sets

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    Let kk and pp be positive integers and let QQ be a finite point set in general position in the plane. We say that QQ is (k,p)(k,p)-Ramsey if there is a finite point set PP such that for every kk-coloring cc of (Pp)\binom{P}{p} there is a subset Q′Q' of PP such that Q′Q' and QQ have the same order type and (Q′p)\binom{Q'}{p} is monochromatic in cc. Ne\v{s}et\v{r}il and Valtr proved that for every k∈Nk \in \mathbb{N}, all point sets are (k,1)(k,1)-Ramsey. They also proved that for every k≥2k \ge 2 and p≥2p \ge 2, there are point sets that are not (k,p)(k,p)-Ramsey. As our main result, we introduce a new family of (k,2)(k,2)-Ramsey point sets, extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result to show that for every kk there is a point set PP such that no function Γ\Gamma that maps ordered pairs of distinct points from PP to a set of size kk can satisfy the following "local consistency" property: if Γ\Gamma attains the same values on two ordered triples of points from PP, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.Comment: 22 pages, 3 figures, final version, minor correction

    A Superlinear Lower Bound on the Number of 5-Holes

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    Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h_5(n) have been of order Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), obtaining the first superlinear lower bound on h_5(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted

    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(nlog⁥n)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(n2/3log⁥n)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(n2/3log⁥O(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

    Convex polygons in Cartesian products

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    We study several problems concerning convex polygons whose vertices lie in aCartesian product of two sets of n real numbers (for short, grid). First, we prove that everysuch grid contains Ω(log n) points in convex position and that this bound is tight up to aconstant factor. We generalize this result to d dimensions (for a fixed d ∈ N), and obtaina tight lower bound of Ω(logd−1 n) for the maximum number of points in convex positionin a d-dimensional grid. Second, we present polynomial-time algorithms for computing thelongest x- or y-monotone convex polygonal chain in a grid that contains no two points withthe same x- or y-coordinate. We show that the maximum size of a convex polygon with suchunique coordinates can be efficiently approximated up to a factor of 2. Finally, we presentexponential bounds on the maximum number of point sets in convex position in such grids,and for some restricted variants. These bounds are tight up to polynomial factors

    An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

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    What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn-(m + ? n/6 ?), for m ? n. We prove a new upper bound of mn-(m+n)+C for some constant C, which is optimal apart from the value of C

    An almost optimal bound on the number of intersections of two simple polygons

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    What is the maximum number of intersections of the boundaries of a simple mm-gon and a simple nn-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of mm and nn is even: If both mm and nn are even, then every pair of sides may cross and so the answer is mnmn. If exactly one polygon, say the nn-gon, has an odd number of sides, it can intersect each side of the mm-gon at most n−1n-1 times; hence there are at most mn−mmn-m intersections. It is not hard to construct examples that meet these bounds. If both mm and nn are odd, the best known construction has mn−(m+n)+3mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn−(m+⌈n6⌉)mn-(m + \lceil \frac{n}{6} \rceil), for m≥nm \ge n. We prove a new upper bound of mn−(m+n)+Cmn-(m+n)+C for some constant CC, which is optimal apart from the value of CC.Comment: 18 pages, 24 figures. To appear in the proceedings of the 36th International Symposium on Computational Geometry (SoCG 2020) in June 2016 (Eds. Sergio Cabello and Danny Chen
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