The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension

Giannopoulos, Panos; Knauer, Christian; Rote, Günter; Werner, Daniel

doi:10.1007/978-3-642-11269-0_16

Cited by 14 publications

(13 citation statements)

References 17 publications

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“…By establishing W [1]-hardness in the dimension, this section shows that it is not possible. Our proofs are inspired by, but extends quite significantly from, that of [30,Section 5]. First, the source problem is as follows.…”

Section: W[1]-hard In the Dimensionmentioning

confidence: 99%

Robust Fitting in Computer Vision: Easy or Hard?

2019

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Robust model fitting plays a vital role in computer vision, and research into algorithms for robust fitting continues to be active. Arguably the most popular paradigm for robust fitting in computer vision is consensus maximisation, which strives to find the model parameters that maximise the number of inliers. Despite the significant developments in algorithms for consensus maximisation, there has been a lack of fundamental analysis of the problem in the computer vision literature. In particular, whether consensus maximisation is "tractable" remains a question that has not been rigorously dealt with, thus making it difficult to assess and compare the performance of proposed algorithms, relative to what is theoretically achievable. To shed light on these issues, we present several computational hardness results for consensus maximisation. Our results underline the fundamental intractability of the problem, and resolve several ambiguities existing in the literature.

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Section: W[1]-hard In the Dimensionmentioning

confidence: 99%

Robust Fitting in Computer Vision: Easy or Hard?

2019

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“…For k = 1, the problem can be solved in linear time [30], but the problem becomes NP-hard even for k = 2 [31] and only algorithms with running time of the form n O(d) is known [2]. In recent years, the framework of W[1]-hardness has been used to give evidence that for several problems (including Euclidean 2center), the exponent of n has to depend on the dimension d; in fact, for many of these problems tight lower bounds have been given that show that no n o(d) algorithm is possible under standard complexity assumptions [13,14,7,23,22,12,6,8].…”

Section: Introductionmentioning

confidence: 99%

The limited blessing of low dimensionality

Marx

Sidiropoulos

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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We are studying d-dimensional geometric problems that have algorithms with 1−1/d appearing in the exponent of the running time, for example, in the form of 2 n 1−1/d or n k 1−1/d . This means that these algorithms perform somewhat better in low dimensions, but the running time is almost the same for all large values d of the dimension. Our main result is showing that for some of these problems the dependence on 1 − 1/d is best possible under a standard complexity assumption. We show that, assuming the Exponential Time Hypothesis,• d-dimensional Euclidean TSP on n points cannot be solved in time 2 O(n 1−1/d− ) for any > 0, and• the problem of finding a set of k pairwise nonintersecting d-dimensional unit balls/axis parallel unit cubes cannot be solved in time f (k)n o(k 1−1/d ) for any computable function f .These lower bounds essentially match the known algorithms for these problems. To obtain these results, we first prove lower bounds on the complexity of Constraint Satisfaction Problems (CSPs) whose constraint graphs are d-dimensional grids. We state the complexity results on CSPs in a way to make them convenient starting points for problem-specific reductions to particular d-dimensional geometric problems and to be reusable in the future for further results of similar flavor.Theory 1 This variant of TSP is also known as path-TSP. Our lower bound also holds for tour-TSP, i.e. the variant where one seeks to find a cycle visiting all vertices. In order to simplify the discussion, we restrict our attention to path-TSP for now; on Section 4 we explain how our proof can be modified to obtain the same lower bound for cycle-TSP.

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“…Reduction from Hitting Set on interval graphs to a restricted version of the art gallery problem. problems parameterized by the dimension d has been addressed [17]; it was shown that, assuming the ETH, algorithms running in time n O(d) are essentially optimal (with n being the size of the instance). Extracting from a finite set of points of R 3 the largest subset in convex position and whose convex-hull interior is empty is W[1]-hard [16].…”

Section: Figurementioning

confidence: 99%

Parameterized Hardness of Art Gallery Problems

Bonnet

Miltzow

2020

ACM Trans. Algorithms

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Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S. The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard. For both variants, we rule out a f (k)n o(k/ log k) algorithm, for any computable function f , where k := |S| is the number of guards, unless the Exponential Time Hypothesis fails. These lower bounds almost match the n O(k) algorithms that exist for both problems.

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The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension

Cited by 14 publications

References 17 publications

Robust Fitting in Computer Vision: Easy or Hard?

Robust Fitting in Computer Vision: Easy or Hard?

The limited blessing of low dimensionality

Parameterized Hardness of Art Gallery Problems

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