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Register a new userA Constant-Factor Approximation for Multi-Covering with Disks
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Santanu Bhowmick
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We consider variants of the following multi-covering problem with disks. We are given two point sets $Y$ (servers) and $X$ (clients) in the plane, a coverage function $kappa :X rightarrow mathcal{N}$, and a constant $alpha geq 1$. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client $x in X$ be covered by at least $kappa(x)$ of the server disks. The objective function we wish to minimize is the sum of the $alpha$-th powers of the disk radii. We present a polynomial time algorithm for this problem achieving an $O(1)$ approximation.
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Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e. maintains the sorted ordering of the rectangle centers along both $x$-axis and $y$-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles(LADR). It was known that LADR is $mathbb{NP}$-hard, but only heuristics were known for it. We show that a certain decision version of LADR is $mathbb{APX}$-hard, and give a constant factor approximation for LADR.
The problem of vertex guarding a simple polygon was first studied by Subir K. Ghosh (1987), who presented a polynomial-time $O(log n)$-approximation algorithm for placing as few guards as possible at vertices of a simple $n$-gon $P$, such that every point in $P$ is visible to at least one of the guards. Ghosh also conjectured that this problem admits a polynomial-time algorithm with constant approximation ratio. Due to the centrality of guarding problems in the field of computational geometry, much effort has been invested throughout the years in trying to resolve this conjecture. Despite some progress (surveyed below), the conjecture remains unresolved to date. In this paper, we confirm the conjecture for the important case of weakly visible polygons, by presenting a $(2+varepsilon)$-approximation algorithm for guarding such a polygon using vertex guards. A simple polygon $P$ is weakly visible if it has an edge $e$, such that every point in $P$ is visible from some point on $e$. We also present a $(2+varepsilon)$-approximation algorithm for guarding a weakly visible polygon $P$, where guards may be placed anywhere on $P$s boundary (except in the interior of the edge $e$). Finally, we present a $3c$-approximation algorithm for vertex guarding a polygon $P$ that is weakly visible from a chord, given a subset $G$ of $P$s vertices that guards $P$s boundary whose size is bounded by $c$ times the size of a minimum such subset. Our algorithms are based on an in-depth analysis of the geometric properties of the regions that remain unguarded after placing guards at the vertices to guard the polygons boundary. It is plausible that our results will enable Bhattacharya et al. to complete their grand attempt to prove the original conjecture, as their approach is based on partitioning the underlying simple polygon into a hierarchy of weakly visible polygons.
We investigate a variety of problems of finding tours and cycle covers with minimum turn cost. Questions of this type have been studied in the past, with complexity and approximation results as well as open problems dating back to work by Arkin et al. in 2001. A wide spectrum of practical applications have renewed the interest in these questions, and spawned variants: for full coverage, every point has to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring an individual penalty. We make a number of contributions. We first show that finding a minimum-turn (full) cycle cover is NP-hard even in 2-dimensional grid graphs, solving the long-standing open Problem 53 in The Open Problems Project edited by Demaine, Mitchell and ORourke. We also prove NP-hardness of finding a subset cycle cover of minimum turn cost in thin grid graphs, for which Arkin et al. gave a polynomial-time algorithm for full coverage; this shows that their boundary techniques cannot be applied to compute exact solutions for subset and penalty variants. On the positive side, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants, making use of LP/IP techniques. For full coverage in more general grid graphs (e.g., hexagonal grids), our approximation factors are better than the combinatorial ones of Arkin et al. Our approach can also be extended to other geometric variants, such as scenarios with obstacles and linear combinations of turn and distance costs.
We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any $lambdageq 1$, the critical covering area $A^*(lambda)$ is the minimum value for which any set of disks with total area at least $A^*(lambda)$ can cover a rectangle of dimensions $lambdatimes 1$. We show that there is a threshold value $lambda_2 = sqrt{sqrt{7}/2 - 1/4} approx 1.035797ldots$, such that for $lambda<lambda_2$ the critical covering area $A^*(lambda)$ is $A^*(lambda)=3pileft(frac{lambda^2}{16} +frac{5}{32} + frac{9}{256lambda^2}right)$, and for $lambdageq lambda_2$, the critical area is $A^*(lambda)=pi(lambda^2+2)/4$; these values are tight. For the special case $lambda=1$, i.e., for covering a unit square, the critical covering area is $frac{195pi}{256}approx 2.39301ldots$. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.
In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clients) in an arbitrary metric space $(X cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $alpha geq 1$. Each server can have a single ball of arbitrary radius centered on it. Each client $x in X$ needs to be covered by at least $k$ such balls centered on servers. The objective function that we wish to minimize is the sum of the $alpha$-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the $t$-MMC, where we require the number of open servers to be at most some given integer $t$. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding $1$-covering problem, where the coverage demand of each client is $1$. Our reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for $1$-covering, we obtain the first constant approximations for the MMC and these generalizations.
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