Topological Art in Simple Galleries

Bertschinger, Daniel; El, Maalouly, Nicolas; Miltzow, Tillmann; Schnider, Patrick; Weber, Simon

doi:10.1137/1.9781611977066.8

Cited by 4 publications

(15 citation statements)

References 33 publications

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“…In this work we have settled the open question of Bertschinger et al [2] by showing that solution spaces to the art gallery problem can capture the topology of any semi-algebraic set up to homeomorphism. In doing so, we have introduced a new form of copying gadget that enables simpler arguments about the structure of valid solutions to art gallery problem instances.…”

Section: Discussionmentioning

confidence: 96%

“…In addition to yielding a strictly stronger result than that of Bertschinger et al [2], our construction is structurally simpler: the art gallery always consists of a single rectangular room with two types of convex slits repeatedly cut out of the edges. As an example, we explicitly draw an art gallery for the Möbius strip (to the best resolution that can reasonably fit on a page) in Figure 5.…”

Section: • the 2-holed Torusmentioning

confidence: 91%

“…Theorem 2.1 (Bertschinger et al [2], Lemma 3). Any compact semi-algebraic set is homeomorphic to the geometric realization of an abstract cubical complex.…”

Section: Hironaka's Theoremmentioning

confidence: 99%

“…As in Bertschinger et al [2], the starting point for our reduction is a theorem by Hironaka [4] that any compact semialgebraic set can be triangulated as a cubical complex. We begin by reviewing this result.…”

Section: General Constructionmentioning

confidence: 99%

“…As a consequence of their reduction from ETR to the art gallery problem, Abrahamsen et al [1] show, for any compact semi-algebraic set S, how to construct an art gallery P such that V (P ) surjects continuously onto S. However, they do not show that the mapping is injective, so this fails to establish universality. In a recent paper, Bertschinger, El Maalouly, Miltzow, Schnider, and Weber [2] show that any compact semi-algebraic set S is homotopy-equivalent to V (P ) for some polygon P . They leave as an open question whether P can be constructed so that V (P ) is not just homotopy-equivalent, but homeomorphic to S, with only the following list of spaces shown to be captured up to homeomorphism:…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Topological Universality of the Art Gallery Problem

Stade¹,

Tucker-Foltz²

2022

Preprint

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We prove that any compact semi-algebraic set is homeomorphic to the solution space of some art gallery problem. Previous works have established similar universality theorems, but holding only up to homotopy equivalence, rather than homeomorphism, and prior to this work, the existence of art galleries even for simple spaces such as the Möbius strip or the three-holed torus were unknown. Our construction relies on an elegant and versatile gadget to copy guard positions with minimal overhead, and is thus simpler than previous constructions, consisting of a single rectangular room with convex slits cut out from the edges. We additionally show that both the orientable and non-orientable surfaces of genus n require galleries with only O(n) vertices.

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Section: Discussionmentioning

confidence: 96%

Section: • the 2-holed Torusmentioning

confidence: 91%

“…Theorem 2.1 (Bertschinger et al [2], Lemma 3). Any compact semi-algebraic set is homeomorphic to the geometric realization of an abstract cubical complex.…”

Section: Hironaka's Theoremmentioning

confidence: 99%

Section: General Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Topological Universality of the Art Gallery Problem

Stade¹,

Tucker-Foltz²

2022

Preprint

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The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

Froese¹,

Hertrich

Niedermeier

2022

jair

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Understanding the computational complexity of training simple neural networks with rectified linear units (ReLUs) has recently been a subject of intensive research. Closing gaps and complementing results from the literature, we present several results on the parameterized complexity of training two-layer ReLU networks with respect to various loss functions. After a brief discussion of other parameters, we focus on analyzing the influence of the dimension d of the training data on the computational complexity. We provide running time lower bounds in terms of W[1]-hardness for parameter d and prove that known brute-force strategies are essentially optimal (assuming the Exponential Time Hypothesis). In comparison with previous work, our results hold for a broad(er) range of loss functions, including lp-loss for all p ∈ [0, ∞]. In particular, we improve a known polynomial-time algorithm for constant d and convex loss functions to a more general class of loss functions, matching our running time lower bounds also in these cases.

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A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Hengeveld,

Miltzow

2024

Discrete Mathematics &Amp; Theoretical Computer Science

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Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.

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Topological Art in Simple Galleries

Cited by 4 publications

References 33 publications

Topological Universality of the Art Gallery Problem

Topological Universality of the Art Gallery Problem

The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

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