Polynomial - time approximation schemes for geometric graphs

Lecture Notes in Computer Science, 2004

A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1 + ε)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.

Information Processing Letters, 2007

The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an efficient algorithm to obtain an outerplanar subgraph of a graph of disk dimension k by removing at most 2k −2 vertices. This reduction is used to obtain linear-time exact and approximation algorithms for problems on graphs of fixed disk dimension. In particular, a linear-time 3-approximation algorithm is presented for the pathwidth problem on graphs of fixed disk dimension. This approximation ratio was previously known only for outerplanar graphs (graphs of disk dimension one).

Lecture Notes in Computer Science, 2006

We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. The algorithm accepts any undirected graph as input, and is robust in the sense that for instances not reflecting unit disk graphs, it either returns a (1 + ε)-approximate minimum dominating set, or a certificate showing that the input graph is no unit disk graph. The given PTAS can easily be adapted to other classes of related geometric intersection graphs.

Algorithmica, 2007

The minimum disc cover can be used to construct a dominating set on the fly for energy-efficient communications in mobile ad hoc networks, but the approach used to compute the minimum disc cover proposed in previous studies is computationally relatively expensive. In this paper, we show that the disc cover problem is in fact a special case of the general α-hull problem. In spite of being a special case, the disc cover problem is not easier than the general α-hull problem. In addition to applying the existing α-hull algorithm to solve the disc cover problem, we present a simple, yet optimal divideand-conquer algorithm that constructs the minimum disc cover for arbitrary cases, including those degenerate cases where the α-hull approach would fail.

Proceedings of the …, 2006

Journal of …, 1998

Theoretical Computer Science, 2014

A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their applicability in wireless ad-hoc networks. Because the minimum dominating set problem for unit disk graphs is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTASs. However, since the proposal of a linear-time 5-approximation algorithm in 1995, the lack of efficient algorithms attaining better approximation factors has aroused attention. We introduce an O(n + m) algorithm that takes the usual adjacency representation of the graph as input and outputs a 44/9-approximation. This approximation factor is also attained by a second algorithm, which takes the geometric representation of the graph as input and runs in O(n log n) time regardless of the number of edges. Additionally, we propose a 43/9approximation which can be obtained in O(n 2 m) time given only the graph's adjacency representation. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.

SIAM Journal on Computing, 2011

Let P be a simple polygon, and let Q be a set of points in P . We present an almost-linear time algorithm for computing a minimum cover of Q by disks that are contained in P . We generalize the algorithm above, so that it can compute a minimum cover of Q by homothets of a fixed compact convex set of constant description complexity O that are contained in P . This improves previous results of Katz and Morgenstern . We also consider the disk-cover problem when Q is contained in a (not too wide) annulus, and present an O(|Q| log |Q|) algorithm for this case.

Let P be a set of points in the plane. The goal is to place two unit disks in the plane such that the number of points from P covered by the disks is maximized. In addition, the distance between the centers of the two disks should not exceed a specified constant R c ≥ 0. We propose two algorithms to solve this problem. The first algorithm is a simple exhaustive algorithm which runs in O(n 4) time. We then improve this algorithm by a constructing connectivity region and building a segment tree to compute two optimal disks. The resulting algorithm has O(n 3 log n) time complexity.