ON THE DISCRETE UNIT DISK COVER PROBLEM

Given a set [Formula: see text] of n points and a set [Formula: see text] of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in [Formula: see text] is covered by at least one disk in [Formula: see text] or not and (ii) if so, then find a minimum cardinality subset [Formula: see text] such that the unit disks in [Formula: see text] cover all the points in [Formula: see text]. 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 [Formula: see text], 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 [Formula: see text]. The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time [Formula: see text].

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AN IMPROVED LINE-SEPARABLE ALGORITHM FOR DISCRETE UNIT DISK COVER

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830910000486 ◽

2010 ◽

Vol 02 (01) ◽

pp. 77-87 ◽

Cited By ~ 13

Author(s):

FRANCISCO CLAUDE ◽

GAUTAM K. DAS ◽

REZA DORRIGIV ◽

STEPHANE DUROCHER ◽

ROBERT FRASER ◽

...

Keyword(s):

Exact Solution ◽

Approximate Solution ◽

Unit Disk ◽

Time Algorithm ◽

Minimum Cardinality ◽

Practical Solution ◽

Cover Problem ◽

Unit Disks ◽

Set Of Points ◽

Disk Cover

Given a set [Formula: see text] of m unit disks and a set [Formula: see text] of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset [Formula: see text] to cover [Formula: see text]. This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by Carmi et al. [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n( log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of Carmi et al. [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem.

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Discrete unit square cover problem

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500726 ◽

2018 ◽

Vol 10 (06) ◽

pp. 1850072

Author(s):

Manjanna Basappa ◽

Gautam K. Das

Keyword(s):

Approximation Algorithm ◽

Graph Search ◽

Horizontal Strip ◽

Approximation Result ◽

Minimum Cardinality ◽

Plane Sweep ◽

Running Time ◽

Approximation Factor ◽

Unit Square ◽

Cover Problem

In this paper, we consider the discrete unit square cover (DUSC) problem as follows: given a set [Formula: see text] of [Formula: see text] points and a set [Formula: see text] of [Formula: see text] axis-aligned unit squares in [Formula: see text], the objective is (i) to check whether the union of the squares in [Formula: see text] covers all the points in [Formula: see text], and (ii) if the answer is yes, then select a minimum cardinality subset [Formula: see text] such that each point in [Formula: see text] is covered by at least one square in [Formula: see text]. For the DUSC problem:(i)we propose a [Formula: see text]-approximation algorithm, where [Formula: see text] is an integer parameter that defines a trade-off between the running time and the approximation factor of the algorithm. The running time of our proposed algorithm is [Formula: see text]. Our solution of the DUSC problem is based on a simple [Formula: see text]-approximation algorithm for the subproblem strip square cover (SSC) problem, where all the points in [Formula: see text] are lying within a horizontal strip of unit height.(ii)we also propose a 2-approximation algorithm, which runs in [Formula: see text] time. The 2-approximation algorithm is based on an algorithm for the subproblem SSC problem. The algorithm for the subproblem is developed using plane sweep and graph search traversal techniques. We also extend this algorithm to get 2-approximation result for the weighted DUSC problem where the squares are assigned weights, and the aim is to choose a subset [Formula: see text] such that each point in [Formula: see text] is covered by at least one square in [Formula: see text] and the sum of the weights of squares in [Formula: see text] is minimized.

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