103 research outputs found
Convex Polygons in Cartesian Products
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
Let and be positive integers and let be a finite point set in
general position in the plane. We say that is -Ramsey if there is a
finite point set such that for every -coloring of
there is a subset of such that and have the same order type
and is monochromatic in . Ne\v{s}et\v{r}il and Valtr proved
that for every , all point sets are -Ramsey. They also
proved that for every and , there are point sets that are
not -Ramsey.
As our main result, we introduce a new family of -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 there is a point set such that no function
that maps ordered pairs of distinct points from to a set of size
can satisfy the following "local consistency" property: if attains
the same values on two ordered triples of points from , 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
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
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in 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 on string graphs while an algorithm running
in time 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 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
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 âŚ(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 â N), and obtaina tight lower bound of âŚ(logdâ1 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
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
What is the maximum number of intersections of the boundaries of a simple
-gon and a simple -gon, assuming general position? This is a basic
question in combinatorial geometry, and the answer is easy if at least one of
and is even: If both and are even, then every pair of sides may
cross and so the answer is . If exactly one polygon, say the -gon, has
an odd number of sides, it can intersect each side of the -gon at most
times; hence there are at most intersections. It is not hard to
construct examples that meet these bounds. If both and are odd, the
best known construction has intersections, and it is conjectured
that this is the maximum. However, the best known upper bound is only , for . We prove a new upper bound of
for some constant , which is optimal apart from the value of
.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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