Exact Algorithms for Terrain Guarding

Ashok, Pradeesha; Fomin, Fedor V.; Kolay, Sudeshna; Saurabh, Saket; Zehavi, Meirav

doi:10.1145/3186897

Cited by 15 publications

(8 citation statements)

References 22 publications

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“…In addition, we note that Gabizon et al [24] also studied the Degree-Bounded Spanning Tree problem: given a graph G and an integer d, decide whether G has a spanning tree of 6 In addition, we show that this bound is essentially tight. Finally, let us remark that k-Path (on both directed and undirected graph) and p-Set q-Packing are both among the most extensively studied problems in Parameterized Complexity.…”

Section: Introductionmentioning

confidence: 72%

On r-Simple k-Path and Related Problems Parameterized by k/r

Gutin

Wahlström

Zehavi

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Abasi et al. (2014) introduced the following two problems. In the r-Simple k-Path problem, given a digraph G on n vertices and positive integers r, k, decide whether G has an r-simple k-path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k. In the (r, k)-Monomial Detection problem, given an arithmetic circuit that succinctly encodes some polynomial P on n variables and positive integers k, r, decide whether P has a monomial of total degree k where the degree of each variable is at most r. Abasi et al. obtained randomized algorithms of running time 4 (k/r) log r · n O(1) for both problems. Gabizon et al. (2015) designed deterministic 2 O((k/r) log r) · n O(1) -time algorithms for both problems (however, for the (r, k)-Monomial Detection problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. In the p-Set (r, q)-Packing problem, given a universe V , positive integers p, q, r, and a collection H of sets of size p whose elements belong to V , decide whether there exists a subcollection H of H of size q where each element occurs in at most r sets of H . Gabizon et al. obtained a deterministic 2 O((pq/r) log r) · n O(1) -time algorithm for p-Set (r, q)-Packing.The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by the product of two parameters, that is, k/r and log r, where k = pq for p-Set (r, q)-Packing. Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (r, k)-Monomial Detection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2 o((k/r) log r) · (n + log k) O(1) -time algorithm for (r, k)-Monomial Detection, i.e. (r, k)-Monomial Detection is highly unlikely to be single-exponentially FPT when parameterized by k/r alone. The question remains open for r-Simple k-Path and p-Set (r, q)-Packing.We consider the question from a wider perspective: are the above problems FPT when parameterized by k/r only, i.e. whether there exists a computable function f such that the problems admit a f (k/r)(n + log k) O(1) -time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by k/r alone. We resolve the wider question by (a) obtaining a 2 O((k/r) 2 log(k/r)) · (n + log k) O(1) -time algorithm for r-Simple k-Path on digraphs and a 2 O(k/r) · (n + log k) O(1) -time algorithm for r-Simple k-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that p-Set (r, q)-Packing is FPT (in contrast, we prove that p-Multiset (r, q)-Packing is W[1]-hard), and (c) proving that (r, k)-Monomial Detection is para-NP-hard even if only two distinct variables are in polynomial P and the circuit is non-canceling. For the special case of (r, k)-Monomial De...

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Section: Introductionmentioning

confidence: 72%

On r-Simple k-Path and Related Problems Parameterized by k/r

Gutin

Wahlström

Zehavi

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In contrast, this not the case for terrain visibility graphs since the two terrain visibility graphs in Figures 4 and 5 both have the ordered degree sequence (7,4,3,4,5,7,4,4,4,6,4) and are not isomorphic (since the unique degree-3 vertex has a degree-7 neighbor in one graph but not in the other). A terrain visibility graph G 1 with ordered degree sequence (7,4,3,4,5,7,4,4,4,6,4). Vertices have unit-spaced x-coordinates.…”

Section: Degree Sequencesmentioning

confidence: 93%

“…Note that in the drawing the y-axis is scaled down. Figure 5: A terrain visibility graph G 2 with ordered degree sequence (7,4,3,4,5,7,4,4,4,6,4). Vertices have unit-spaced x-coordinates.…”

Section: Degree Sequencesmentioning

confidence: 99%

“…Notably, the Terrain Guarding problem, that is, selecting k terrain points that guard the whole terrain (which is closely related to Dominating Set on terrain visibility graphs) has been extensively studied in the literature and is known to be NP-hard [23] even on orthogonal terrains [7]. It has recently been studied from a parameterized perspective [5].…”

Section: Introductionmentioning

confidence: 99%

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Advancing Through Terrains

Froese,

Renken

2019

Preprint

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We study terrain visibility graphs, a well-known graph class closely related to polygon visibility graphs in computational geometry, for which a precise graphtheoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted attention in the context of time series analysis with various practical applications in areas such as physics, geography and medical sciences.We make progress in understanding terrain visibility graphs by providing several graph-theoretic results. For example, we show that they cannot contain antiholes of size larger than five. Moreover, we obtain two algorithmic results. We devise a fast output-sensitive shortest path algorithm on arbitrary induced subgraphs of terrain visibility graphs and a polynomial-time algorithm for Dominating Set on special terrain visibility graphs (called funnel visibility graphs).

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“…Many studies have referred to applications of 1.5D terrain guarding in real world [1,2,3]. The examples include guarding or covering a road with security cameras or lights and using line-of-sight transmission networks for radio broadcasting.…”

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confidence: 99%

Continuous Terrain Guarding with Two-Sided Guards

Lai,

Hsiang

2018

Preprint

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Herein, we consider the continuous 1.5dimensional(1.5D) terrain guarding problem with two-sided guarding. We provide an x-monotone chain T and determine the minimal number of vertex guards such that all points of T have been two-sided guarded. A point p is two-sided guarded if there exist two vertices v i (left of p) and (right of p) that both see p. A vertex v i sees a point p on T if the line segment connecting v i to p is on or above T . We demonstrate that the continuous 1.5D terrain guarding problem can be transformed to the discrete terrain guarding problem with a finite point set X and that if X is two-sided guarded, then T is also two-sided guarded. Through this transformation, we achieve an optimal algorithm that solves the continuous 1.5D terrain guarding problem under two-sided guarding.

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Exact Algorithms for Terrain Guarding

Cited by 15 publications

References 22 publications

On r-Simple k-Path and Related Problems Parameterized by k/r

On r-Simple k-Path and Related Problems Parameterized by k/r

Advancing Through Terrains

Continuous Terrain Guarding with Two-Sided Guards

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