We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery lifetime.

ABSTRACT. To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n ǫ vertices fixed. We answer this question in the affirmative with ǫ = 1/4. The previous best known bound was Ω( log n/ log log n). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least n/3 vertices fixed, while the best upper bound was O((n log n) 2/3 ). We answer a question of Spillner and Wolff [http://arxiv.org/abs/0709. 0170, 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3( √ n − 1) vertices fixed. Moreover, we improve the lower bound to n/2.

An abstract NP-hard covering problem is presented and fixed-parameter tractable algorithms for this problem are described. The running times of the algorithms are expressed in terms of three parameters: n, the number of elements to be covered, k, the number of sets allowed in the covering, and d, the combinatorial dimension of the problem. The first algorithm is deterministic and has a running time of O (k dk n). The second algorithm is also deterministic and has a running time of O (k d(k+1) + n d+1 ). The third is a MonteCarlo algorithm that runs in time O (k d(k+1) + c2 d k (d+1)/2 (d+1)/2 n log n) and is correct with probability 1 − n −c . Here, the O notation hides factors that are polynomial in d. These algorithms lead to fixed-parameter tractable algorithms for many geometric and non-geometric covering problems.

In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memoryconstrained algorithms. Given an algorithm A that runs in O(n) time using a stack of length Θ(n), we can modify it so that it runs in O(n 2 /2 s ) time using a workspace of O(s) variables (for any s ∈ o(log n)) or O(n log n/ log p) time using O(p log n/ log p) variables (for any 2 ≤ p ≤ n). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1-dimensional pyramid approximation of a 1-dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the best-known results for these problems in constant-workspace models (when they exist), and gives a trade-off between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memory-constrained algorithms.1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling

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