Report On the Approximation of Unit Disk Graph Coordinates
نویسندگان
چکیده
In this paper we study a problem occuring in the context of geometric routing algorithms for mobile ad-hoc networks: Finding unit disk graph coordinates given a graph G = (V, E). Based on a proof that recognition of unit disk graphs is an NP-hard problem, we show that the problem of finding unit disk graph coordinates given a graph G = (V, E) is NPhard. Subsequently, we show that the proof does not extend to quasi unit disk graphs, a generalization of unit disk graphs. We give an exact formulation of the problem of finding unit disk graph coordinates in terms of a quadratic feasibility problem and explore different approximations in terms of linear programs, quadratic programs and semidefinite programs.
منابع مشابه
A new PTAS for Maximum Independent Sets in Unit Disk Graphs
A unit disk graph is an intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum 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).
متن کاملA Robust PTAS for Maximum Weight Independent Sets in Unit Disk Graphs
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 present...
متن کاملLocal 3-approximation algorithms for weighted dominating set and vertex cover in quasi unit-disk graphs
We present a simple 3-approximation algorithm for minimum-weight dominating set and minimum-weight vertex cover in unit-disk graphs and quasi unit-disk graphs in which each node knows its coordinates. The algorithm is local: the output of a node depends solely on the input within its constantradius neighbourhood. The local horizon of the algorithm is small, both in the worst case and on average.
متن کاملLocal PTAS for Independent Set and Vertex Cover in Location Aware Unit Disk Graphs
We present the first local approximation schemes for maximum independent set and minimum vertex cover in unit disk graphs. In the graph model we assume that each node knows its geographic coordinates in the plane (location aware nodes). Our algorithms are local in the sense that the status of each node v (whether or not v is in the computed set) depends only on the vertices which are a constant...
متن کاملApproximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs
Unit disk graphs are the intersection graphs of equal sized circles in the plane. In this paper, we consider the maximum independent set problems on unit disk graphs. When the given unit disk graph is de ned on a slab whose width is k, we propose an algorithm for nding a maximum independent set in O(n 4 d 2k= p 3 e ) time where n denotes the number of vertices. We also propose a (1 1=r)-approxi...
متن کامل