Given a set of unit-disks in the plane with union area @math , what fraction of @math can be covered by selecting a pairwise disjoint subset of the disks? Rado conjectured 1 4 and proved @math . Motivated by the problem of channel-assignment for wireless access points, in which use of 3 channels is a standard practice, we consider a variant where the selected subset of disks must be 3-colourable with disks of the same colour pairwise-disjoint. For this variant of the problem, we conjecture that it is always possible to cover at least @math of the union area and prove @math . We also provide an @math algorithm to select a subset achieving a @math bound.
A more sophisticated formalization of the deployment problem allows disks assigned to the same channel to overlap but only counts the area where there is no interference---i.e. the area of the set of points covered by only one disk on some channel. In terms of colouring, the problem is to colour a subset of the given disks to maximize the area of @math for some colour, point @math is in exactly one disk of that colour @math . We call this the . @cite_1 proved that it is always possible to achieve approximately @math 1-covered area using only one colour. This model has also been considered with respect to two other optimization problems. For the problem of maximizing the 1-covered area using one colour, previous work has focused on approximation algorithms (though no proof yet exists, it is suspected that this problem is NP-hard). In @cite_1 , present a 5.83-approximation algorithm with polynomial runtime. In @cite_14 , show that the problem admits a polynomial time approximation scheme when the radius of the largest disk over the radius of the smallest disk is a constant.
@cite_14 @cite_1
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In the interference scheduling problem, one is given a set of n communication requests described by pairs of points from a metric space. The points correspond to devices in a wireless network. In the directed version of the problem, each pair of points consists of a dedicated sending and a dedicated receiving device. In the bidirectional version the devices within a pair shall be able to exchange signals in both directions. In both versions, each pair must be assigned a power level and a color such that the pairs in each color class (representing pairs communicating in the same time slot) can communicate simultaneously at the specified power levels. The feasibility of simultaneous communication within a color class is defined in terms of the Signal to Interference Plus Noise Ratio (SINR) that compares the strength of a signal at a receiver to the sum of the strengths of other signals. This is commonly referred to as the "physical model" and is the established way of modelling interference in the engineering community. The objective is to minimize the number of colors as this corresponds to the time needed to schedule all requests. We study oblivious power assignments in which the power value of a pair only depends on the distance between the points of this pair. We prove that oblivious power assignments cannot yield approximation ratios better than Ω(n) for the directed version of the problem, which is the worst possible performance guarantee as there is a straightforward algorithm that achieves an O(n)-approximation. For the bidirectional version, however, we can show the existence of a universally good oblivious power assignment: For any set of n bidirectional communication requests, the so-called "square root assignment" admits a coloring with at most polylog(n) times the minimal number of colors. The proof for the existence of this coloring is non-constructive. We complement it by an approximation algorithm for the coloring problem under the square root assignment. This way, we obtain the first polynomial time algorithm with approximation ratio polylog(n) for interference scheduling in the physical model. Given a set of n points and a set of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in is covered by at least one disk in or not and (ii) if so, then find a minimum cardinality subset such that the unit disks in cover all the points in . The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within , for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is . The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time . We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√n) colors in the worst case. We then modify this approach, to obtain an efficient deterministic algorithm that uses a maximum of Θ(log2 n) colors. Next, we present two randomized solutions. The first algorithm requires an expected number of at most O(log2 n) colors, and produces a coloring which is valid with high probability, and the second one, which is a variant of our efficient deterministic algorithm, requires an expected number of at most O(log n log log n) colors but always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case. The average-case behavior for disks, and cases involving other planar ranges, are still open. In this paper, we study the following disc covering problem: Given a set of discs of various radii on the plane and centers on the grid points, find a subset of discs to maximize the area covered by exactly one disc. Thisproblem originates from the application in digital halftoning, with the best known approximation factor being 5.83 (, 2004). We show that if their radii are between two positive constants, then there exists a polynomial time approximation scheme. Our techniques are based on the width-bounded geometric separator recently developed in Fu and Wang (2004), Fu (2006).