1,009 research outputs found

    Computational Complexity of the ?-Ham-Sandwich Problem

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    ?_d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the ?-Ham-Sandwich theorem. They gave an algorithm to find the hyperplane in time O(n (log n)^{d-3}), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearnley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called Unique End-of-Potential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the ?-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P-matrix linear complementarity problem

    K1,3K_{1,3}-covering red and blue points in the plane

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    We say that a finite set of red and blue points in the plane in general position can be K1,3K_{1,3}-covered if the set can be partitioned into subsets of size 44, with 33 points of one color and 11 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set RR of rr red points and a set BB of bb blue points in the plane in general position, how many points of RâˆȘBR\cup B can be K1,3K_{1,3}-covered? and we prove the following results: (1) If r=3g+hr=3g+h and b=3h+gb=3h+g, for some non-negative integers gg and hh, then there are point sets RâˆȘBR\cup B, like {1,3}\{1,3\}-equitable sets (i.e., r=3br=3b or b=3rb=3r) and linearly separable sets, that can be K1,3K_{1,3}-covered. (2) If r=3g+hr=3g+h, b=3h+gb=3h+g and the points in RâˆȘBR\cup B are in convex position, then at least r+b−4r+b-4 points can be K1,3K_{1,3}-covered, and this bound is tight. (3) There are arbitrarily large point sets RâˆȘBR\cup B in general position, with r=b+1r=b+1, such that at most r+b−5r+b-5 points can be K1,3K_{1,3}-covered. (4) If b≀r≀3bb\le r\le 3b, then at least 89(r+b−8)\frac{8}{9}(r+b-8) points of RâˆȘBR\cup B can be K1,3K_{1,3}-covered. For r>3br>3b, there are too many red points and at least r−3br-3b of them will remain uncovered in any K1,3K_{1,3}-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl

    Distance-Sensitive Planar Point Location

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    Let S\mathcal{S} be a connected planar polygonal subdivision with nn edges that we want to preprocess for point-location queries, and where we are given the probability Îłi\gamma_i that the query point lies in a polygon PiP_i of S\mathcal{S}. We show how to preprocess S\mathcal{S} such that the query time for a point~p∈Pip\in P_i depends on~Îłi\gamma_i and, in addition, on the distance from pp to the boundary of~PiP_i---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time O(min⁥(log⁥n,1+log⁥area(Pi)ÎłiΔp2))O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{\gamma_i \Delta_{p}^2}\right)\right), where Δp\Delta_{p} is the shortest Euclidean distance of the query point~pp to the boundary of PiP_i. Our structure uses O(n)O(n) space and O(nlog⁥n)O(n \log n) preprocessing time. It is based on a decomposition of the regions of S\mathcal{S} into convex quadrilaterals and triangles with the following property: for any point p∈Pip\in P_i, the quadrilateral or triangle containing~pp has area Ω(Δp2)\Omega(\Delta_{p}^2). For the special case where S\mathcal{S} is a subdivision of the unit square and Îłi=area(Pi)\gamma_i=\mathrm{area}(P_i), we present a simpler solution that achieves a query time of O(min⁥(log⁥n,log⁥1Δp2))O\left(\min \left(\log n, \log \frac{1}{\Delta_{p}^2}\right)\right). The latter solution can be extended to convex subdivisions in three dimensions

    The Complexity of Sharing a Pizza

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    Assume you have a 2-dimensional pizza with 2n ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need? It was recently shown using topological methods that n cuts always suffice. In this work, we study the computational complexity of finding such n cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets n cuts suffice, which does not use any topological methods. We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated

    Ham Sandwich is Equivalent to Borsuk-Ulam

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    The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R^n can always be simultaneously bisected by an (n-1)-dimensional hyperplane. In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and the Ham Sandwich theorem. The main technical result we show towards establishing the equivalence is the following: For every odd polynomial restricted to the hypersphere f:S^n->R, there exists a compact set A in R^{n+1}, such that for every x in S^n we have f(x)=vol(A cap H^+) - vol(A cap H^-), where H is the oriented hyperplane containing the origin with x as the normal. A noteworthy aspect of the proof of the above result is the use of hyperspherical harmonics. Finally, using the above result we prove that there exist constants n_0, epsilon_0>0 such that for every n>= n_0 and epsilon <= epsilon_0/sqrt{48n}, any query algorithm to find an epsilon-bisecting (n-1)-dimensional hyperplane of n compact set in [-n^4.51,n^4.51]^n, even with success probability 2^-Omega(n), requires 2^Omega(n) queries

    On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems

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    We study several canonical decision problems arising from some well-known theorems from combinatorial geometry. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the hs cut problem are W[1]-hard (and NP-hard) if the dimension is part of the input. This is done by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Our reductions also imply that the problems we consider cannot be solved in time n^{o(d)} (where n is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false. The technique of embedding d-Sum into a geometric setting is conceptually much simpler than direct fpt-reductions from purely combinatorial W[1]-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems
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