arXiv subjects

Saeed Mehrabi

Publications and source records attributed to Saeed Mehrabi.

31 records · Page 2Linked to original sources

arXiv2017

Approximating Weighted Duo-Preservation in Comparative Genomics

Motivated by comparative genomics, Chen et al. [9] introduced the Maximum Duo-preservation String Mapping (MDSM) problem in which we are given two strings $s_1$ and $s_2$ from the same alphabet and the goal is to find a mapping $π$ between them so as to maximize the number of duos preserved. A duo is any two consecutive characters in a string and it is preserved in the mapping if its two consecutive characters in $s_1$ are mapped to same two consecutive characters in $s_2$. The MDSM problem is known to be NP-hard and there are approximation algorithms for this problem [3, 5, 13], but all of them consider only the "unweighted" version of the problem in the sense that a duo from $s_1$ is preserved by mapping to any same duo in $s_2$ regardless of their positions in the respective strings. However, it is well-desired in comparative genomics to find mappings that consider preserving duos that are "closer" to each other under some distance measure [19]. In this paper, we introduce a generalized version of the problem, called the Maximum-Weight Duo-preservation String Mapping (MWDSM) problem that captures both duos-preservation and duos-distance measures in the sense that mapping a duo from $s_1$ to each preserved duo in $s_2$ has a weight, indicating the "closeness" of the two duos. The objective of the MWDSM problem is to find a mapping so as to maximize the total weight of preserved duos. In this paper, we give a polynomial-time 6-approximation algorithm for this problem.

cs.DS↗

arXiv2017

Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.

cs.CG↗

arXiv2017

On Guarding Orthogonal Polygons with Bounded Treewidth

There exist many variants of guarding an orthogonal polygon in an orthogonal fashion: sometimes a guard can see an entire rectangle, or along a staircase, or along an orthogonal path with at most $k$ bends. In this paper, we study all these guarding models in the special case of orthogonal polygons that have bounded treewidth in some sense. Exploiting algorithms for graphs of bounded treewidth, we show that the problem of finding the minimum number of guards in these models becomes linear-time solvable in orthogonal polygons of bounded treewidth.

cs.CG↗

arXiv2017

Approximation Algorithms for Independence and Domination on B$_1$-VPG and B$_1$-EPG Graphs

A graph $G$ is called B$_k$-VPG (resp., B$_k$-EPG), for some constant $k\geq 0$, if it has a string representation on a grid such that each vertex is an orthogonal path with at most $k$ bends and two vertices are adjacent in $G$ if and only if the corresponding strings intersect (resp., the corresponding strings share at least one grid edge). If two adjacent strings of a B$_k$-VPG graph intersect exactly once, then the graph is called a one-string B$_k$-VPG graph. In this paper, we study the Maximum Independent Set and Minimum Dominating Set problems on B$_1$-VPG and B$_1$-EPG graphs. We first give a simple $O(\log n)$-approximation algorithm for the Maximum Independent Set problem on B$_1$-VPG graphs, improving the previous $O((\log n)^2)$-approximation algorithm of Lahiri et al. (COCOA 2015). Then, we consider the Minimum Dominating Set problem. We give an $O(1)$-approximation algorithm for this problem on one-string B$_1$-VPG graphs, providing the first constant-factor approximation algorithm for this problem. Moreover, we show that the Minimum Dominating Set problem is APX-hard on B$_1$-EPG graphs, ruling out the possibility of a PTAS unless P=NP. Finally, we give constant-factor approximation algorithms for this problem on two non-trivial subclasses of B$_1$-EPG graphs. To our knowledge, these are the first results for the Minimum Dominating Set problem on B$_1$-EPG graphs, partially answering a question posed by Epstein et al. (WADS 2013).

cs.CG↗

arXiv2016

1-Bend RAC Drawings of 1-Planar Graphs

A graph is 1-planar if it has a drawing where each edge is crossed at most once. A drawing is RAC (Right Angle Crossing) if the edges cross only at right angles. The relationships between 1-planar graphs and RAC drawings have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC drawable and graphs that have a straight-line RAC drawing but that are not 1-planar. Also, straight-line RAC drawings always exist for IC-planar graphs, a subclass of 1-planar graphs. One of the main questions still open is whether every 1-planar graph has a RAC drawing with at most one bend per edge. We positively answer this question.

cs.CG↗

arXiv2016

Sliding k-Transmitters: Hardness and Approximation

A sliding k-transmitter in an orthogonal polygon P is a mobile guard that travels back and forth along an orthogonal line segment s inside P. It can see a point p in P if the perpendicular from p onto s intersects the boundary of P at most k times. We show that guarding an orthogonal polygon P with the minimum number of k-transmitters is NP-hard, for any fixed k>0, even if P is simple and monotone. Moreover, we give an O(1)-approximation algorithm for this problem.

cs.CG↗

arXiv2016

Unique Set Cover on Unit Disks and Unit Squares

We study the Unique Set Cover problem on unit disks and unit squares. For a given set $P$ of $n$ points and a set $D$ of $m$ geometric objects both in the plane, the objective of the Unique Set Cover problem is to select a subset $D'\subseteq D$ of objects such that every point in $P$ is covered by at least one object in $D'$ and the number of points covered uniquely is maximized, where a point is covered uniquely if the point is covered by exactly one object in $D'$. In this paper, (i) we show that the Unique Set Cover is NP-hard on both unit disks and unit squares, and (ii) we give a PTAS for this problem on unit squares by applying the mod-one approach of Chan and Hu (Comput. Geom. 48(5), 2015).

cs.CG↗

arXiv2016

On Guarding Orthogonal Polygons with Sliding Cameras

A sliding camera inside an orthogonal polygon $P$ is a point guard that travels back and forth along an orthogonal line segment $γ$ in $P$. The sliding camera $g$ can see a point $p$ in $P$ if the perpendicular from $p$ onto $γ$ is inside $P$. In this paper, we give the first constant-factor approximation algorithm for the problem of guarding $P$ with the minimum number of sliding cameras. Next, we show that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth. Finally, we study art gallery theorems for sliding cameras, thus, give upper and lower bounds in terms of the number of guards needed relative to the number of vertices $n$.

cs.CG↗

arXiv2016

On $r$-Guarding Thin Orthogonal Polygons

Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It was known that finding the minimum set of $r$-guards is polynomial for tree polygons, but the run-time was $\tilde{O}(n^{17})$. We show here that with a different approach the running time becomes linear, answering a question posed by Biedl et al. (SoCG 2011). Furthermore, the approach is much more general, allowing to specify subsets of points to guard and guards to use, and it generalizes to polygons with $h$ holes or thickness $K$, becoming fixed-parameter tractable in $h+K$.

cs.CG↗

arXiv2015

Guarding the Vertices of an Orthogonal Terrain using Vertex Guards

A terrain T is an x-monotone polygonal chain in the plane; T is orthogonal if each edge of T is either horizontal or vertical. In this paper, we give an exact algorithm for the problem of guarding the convex vertices of an orthogonal terrain with the minimum number of reflex vertices.

cs.CG↗

arXiv2015

A Note on Approximating 2-Transmitters

A k-transmitter in a simple orthogonal polygon P is a mobile guard that travels back and forth along an orthogonal line segment s inside P. The k-transmitter can see a point p in P if there exists a point q on s such that the line segment pq is normal to s and pq intersects the boundary of P in at most k points. In this paper, we give a 2-approximation algorithm for the problem of guarding a monotone orthogonal polygon with the minimum number of 2-transmitters.

cs.CG↗

arXiv2013

A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras

Consider a sliding camera that travels back and forth along an orthogonal line segment $s$ inside an orthogonal polygon $P$ with $n$ vertices. The camera can see a point $p$ inside $P$ if and only if there exists a line segment containing $p$ that crosses $s$ at a right angle and is completely contained in $P$. In the minimum sliding cameras (MSC) problem, the objective is to guard $P$ with the minimum number of sliding cameras. In this paper, we give an $O(n^{5/2})$-time $(7/2)$-approximation algorithm to the MSC problem on any simple orthogonal polygon with $n$ vertices, answering a question posed by Katz and Morgenstern (2011). To the best of our knowledge, this is the first constant-factor approximation algorithm for this problem.

cs.CG↗

arXiv2013

New Hardness Results for Guarding Orthogonal Polygons with Sliding Cameras

Let $P$ be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment $s\in P$ as its \emph{trajectory}. The camera can see a point $p\in P$ if there exists a point $q\in s$ such that $pq$ is a line segment normal to $s$ that is completely inside $P$. In the \emph{minimum-cardinality sliding cameras problem}, the objective is to find a set $S$ of sliding cameras of minimum cardinality to guard $P$ (i.e., every point in $P$ can be seen by some sliding camera) while in the \emph{minimum-length sliding cameras problem} the goal is to find such a set $S$ so as to minimize the total length of trajectories along which the cameras in $S$ travel. In this paper, we first settle the complexity of the minimum-length sliding cameras problem by showing that it is polynomial tractable even for orthogonal polygons with holes, answering a question asked by Katz and Morgenstern (2011). We next show that the minimum-cardinality sliding cameras problem is \textsc{NP}-hard when $P$ is allowed to have holes, which partially answers another question asked by Katz and Morgenstern (2011).

cs.CG↗