Terrain-Like Graphs: PTASs for Guarding Weakly-Visible Polygons and Terrains

  title={Terrain-Like Graphs: PTASs for Guarding Weakly-Visible Polygons and Terrains},
  author={Stav Ashur and Omrit Filtser and Matthew J. Katz and Rachel Saban},
A graph \(G = (V,E)\) is terrain-like if one can assign a unique integer from the range [1..|V|] to each vertex in V, such that, if both \(\{i,k\}\) and \(\{j,l\}\) are in E, for any \(i< j< k < l\), then so is \(\{i,l\}\). We present a local-search-based PTAS for minimum dominating set in terrain-like graphs. Then, we observe that, besides the visibility graphs of x-monotone terrains which are terrain-like, so are the visibility graphs of weakly-visible polygons and weakly-visible terrains… 

Terrain-like Graphs and the Median Genocchi Numbers

A graph with vertex set { 1 , . . . , n } is terrain-like if, for any edge pair { a, c } , { b, d } with a < b < c < d , the edge { a, d } also exists. Terrain-like graphs frequently appear in

A Fast Shortest Path Algorithm on Terrain-like Graphs

This paper devise a fast output-sensitive shortest path algorithm on a superclass of terrain visibility graphs called terrain-like graphs (including all induced subgraphs of terrain visible graphs), which runs in O ( d ∗ log Δ ) time.

Advancing Through Terrains

A fast output-sensitive shortest path algorithm on terrain-like graphs and a polynomial-time algorithm for \textsc{Dominating Set} on special terrain visibility graphs (called funnel visibility graphs) are devised.

Guarding Weakly-Visible Polygons with Half-Guards

A polynomial time approximation scheme for vertex guarding a weakly-visible polygon with half-guards is given and it is shown that even with many restrictions, the problem is NP-hardness.

A Constant-Factor Approximation Algorithm for Point Guarding an Art Gallery

This paper proposes an algorithm with a constant approximation factor for the point guarding problem where the location of guards is restricted to a grid and the running time depends on the number of cells of the grid.

Art Gallery Plus Single Specular-reflection

A variant of the Art Gallery problem in which the boundaries of P are replaced by single specularreflection edges, allowing the view rays to reflect once per collision with an edge, which allows the guards to see through the reflections, thereby viewing a larger portion of the polygon is studied.



The continuous 1.5D terrain guarding problem: Discretization, optimal solutions, and PTAS

This paper proposes several filtering techniques reducing the size of the discretization, allowing for an efficient IP-based algorithm that reliably provides optimal guard placements for terrains with up to $10^6$ vertices within minutes on a standard desktop computer.

Approximability of guarding weak visibility polygons

Terrain guarding is NP-hard

Using a reduction from PLANAR 3-SAT it is proved that the decision version of the minimum terrain guarding problem is NP-hard and complements recent positive approximability results for the optimization problem.

L-Graphs and Monotone L-Graphs

This paper gives a full characterization of monotone $\mathsf{L}$-embeddings by introducing a new class of graphs which they are called "non-jumping" graphs, and shows that outerplanar graphs, convex bipartite graphs, interval graphs, 3-leaf power graphs and complete graphs are subclasses of non-Jumping graphs.

Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards

Three polynomial-time algorithms with a constant approximation ratio for guarding an $n$-sided simple polygon $P$ using vertex guards are presented, settling the conjecture by Ghosh regarding the existence of constant-factor approximation algorithms for this problem.

Inapproximability Results for Guarding Polygons and Terrains

This paper proves that if the input polygon has no holes, there is a constant δ >0 such that no polynomial time algorithm can achieve an approximation ratio of 1+δ, for each of these guard problems, and shows inapproximability for the POINT GUARD problem for polygons with holes.

Improved Approximation Algorithms for Geometric Set Cover

It is shown that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset $R\subset S$ and nondecreasing function f(·), there is a decomposition of the complement ${Bbb U}\backslash\bigcup (R)$ into an expected at most f(|R|) regions, each region of a particular simple form.

Max point-tolerance graphs

Algorithms for Dominating Set in Disk Graphs: Breaking the logn Barrier - (Extended Abstract)

A randomized algorithm is given that obtains a dominating set whose weight is within a factor 2O(log* n) of a minimum cost solution, with high probability - the technique follows the framework proposed recently by Varadarajan.