164 research outputs found

    Helly-Type Theorems in Property Testing

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    Helly's theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If SS is a set of nn points in RdR^d, we say that SS is (k,G)(k,G)-clusterable if it can be partitioned into kk clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object GG. In this paper, as an application of Helly's theorem, by taking a constant size sample from SS, we present a testing algorithm for (k,G)(k,G)-clustering, i.e., to distinguish between two cases: when SS is (k,G)(k,G)-clusterable, and when it is ϵ\epsilon-far from being (k,G)(k,G)-clusterable. A set SS is ϵ\epsilon-far (0<ϵ1)(0<\epsilon\leq1) from being (k,G)(k,G)-clusterable if at least ϵn\epsilon n points need to be removed from SS to make it (k,G)(k,G)-clusterable. We solve this problem for k=1k=1 and when GG is a symmetric convex object. For k>1k>1, we solve a weaker version of this problem. Finally, as an application of our testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability

    Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem

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    The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in R^d , incomplete data objects correspond to affine subspaces (lines or Δ-flats).With this motivation we study the problem of finding the minimum intersection radius r(L) of a set of lines or Δ-flats L: the least r such that there is a ball of radius r intersecting every flat in L. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher dimensional flats, primarily because “distances” between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly’s theorem. This “intrinsic-dimension” Helly theorem states: for any family L of Δ-dimensional convex sets in a Hilbert space, there exist Δ + 2 sets L' ⊆ L such that r(L) ≤ 2r(L'). Based upon this we present an algorithm that computes a (1+ε)-core set L' ⊆ L, |L'| = O(Δ^4/ε), such that the ball centered at a point c with radius (1 +ε)r(L') intersects every element of L. The running time of the algorithm is O(n^(Δ+1)dpoly(Δ/ε)). For the case of lines or line segments (Δ = 1), the (expected) running time of the algorithm can be improved to O(ndpoly(1/ε)).We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space

    Tverberg-type theorems for intersecting by rays

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    In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set

    Byzantine Approximate Agreement on Graphs

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    Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors\u27 input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures

    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

    Tur\'an and Ramsey Properties of Subcube Intersection Graphs

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    The discrete cube {0,1}d\{0,1\}^d is a fundamental combinatorial structure. A subcube of {0,1}d\{0,1\}^d is a subset of 2k2^k of its points formed by fixing kk coordinates and allowing the remaining dkd-k to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no r+1r+1 of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no kk which have non-empty intersection and no ll which are pairwise disjoint? These questions are naturally expressed as Tur\'an and Ramsey type questions in intersection graphs of subcubes where the intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Tur\'an and Ramsey type problems are at the heart of extremal combinatorics and so these problems are mathematically natural. However, a second motivation is a connection with some questions in social choice theory arising from a simple model of agreement in a society. Specifically, if we have to make a binary choice on each of nn separate issues then it is reasonable to assume that the set of choices which are acceptable to an individual will be represented by a subcube. Consequently, the pattern of intersections within a family of subcubes will have implications for the level of agreement within a society. We pose a number of questions and conjectures relating directly to the Tur\'an and Ramsey problems as well as raising some further directions for study of subcube intersection graphs.Comment: 18 page
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