Finding good approximate vertex and edge partitions is NP-hard
Information Processing Letters
Randomized algorithms
Polynomial time approximation schemes for dense instances of NP-hard problems
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Approximating layout problems on random geometric graphs
Journal of Algorithms
Algorithms for VLSI Design Automation
Algorithms for VLSI Design Automation
Wireless sensor networks: a survey
Computer Networks: The International Journal of Computer and Telecommunications Networking
A Polylogarithmic Approximation of the Minimum Bisection
SIAM Journal on Computing
The bin-covering technique for thresholding random geometric graph properties
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Finding separator cuts in planar graphs within twice the optimal
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
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A random geometric graph ${\mathcal{G}}(n,r)$ is obtained by spreading n points uniformly at random in a unit square, and by associating a vertex to each point and an edge to each pair of points at Euclidian distance at most r. Such graphs are extensively used to model wireless ad-hoc networks, and in particular sensor networks. It is well known that, over a critical value of r, the graph is connected with high probability. In this paper we study the robustness of the connectivity of random geometric graphs in the supercritical phase, under deletion of edges. In particular, we show that, for a sufficiently large r, any cut which separates two components of Θ(n) vertices each contains Ω(n2r3) edges with high probability. We also present a simple algorithm that, again with high probability, computes one such cut of size O(n2r3). From these two results we derive a constant expected approximation algorithm for the β-balanced cut problem on random geometric graphs: find an edge cut of minimum size whose two sides contain at least βn vertices each.