Jirí Fiala

On distance constrained labeling of disk graphs

A disk graph is the intersection graph of a set of disks in the plane. For a k-tuple (p1 ..... pk) of positive integers, a distance constrained labeling of a graph G is an assignment of labels to the vertices of G such that the labels of any pair of vertices at graph distance i in G differ by at least Pi, for i = 1,...,k. In the case when k = 1 and p1 = 1, this gives a traditional coloring of G. We propose and analyze several online and offiine labeling algorithms for the class of disk graphs.

On-line coloring of geometric intersection graphs

This paper studies on-line coloring of geometric intersection graphs. It is shown that no deterministic on-line algorithm can achieve competitive ratio better than Ω (log n) for disk graphs and for square graphs with n vertices, even if the geometric representation is given as part of the input. Furthermore, it is proved that the standard First-fit heuristic achieves competitive ratio O(logn) for disk graphs and for square graphs and is thus best possible.

Scheduling of Independent Dedicated Multiprocessor Tasks

We study the off and on-line versions of the well known problem of scheduling a set of n independent multiprocessor tasks with pre-specified processor allocations on a set of identical processors in order to minimize the makespan. Recently, in [12], it has been proven that in the case when all tasks have unit processing time the problem cannot be approximated within a factor of m1/2 - ?, neither for some ? > 0, unless P= NP; nor for any ? > 0, unless NP=ZPP. For this special case we give a simple algorithm based on the classical first-fit technique. We analyze the algorithm for both tasks arrive over time and tasks arrive over list on-line scheduling versions, and show that its competitive ratio is bounded by 2?m and 2?m + 1, respectively. Here we also use some preliminary results on (vertex) coloring of k-tuple graphs. For the case of arbitrary processing times, we show that any algorithm which uses the first-fit technique cannot be better than m competitive. Then, by using our split-round technique, we give a 3?m-approximation algorithm for the off-line version of the problem. Finally, by using some ideas from [20], we adapt the algorithm to the on-line case, in the paradigm of tasks arriving over time in which the existence of a task is unknown until its release date, and show that its competitive ratio is bounded by 6?m. Due to the conducted experimental results, we conclude that our algorithms can perform well in practice.

Online and Offline Distance Constrained Labeling of Disk Graphs

A disk graph is the intersection graph of a set of disks in the plane. We consider the problem of assigning labels to vertices of a disk graph satisfying a sequence of distance constrains. Our objective is to minimize the distance between the smallest and the largest labels. We propose an on-line labeling algorithm on disk graphs, if the maximum and minimum diameters are bounded. We give the upper and lower bounds on its competitive ratio, and show that the algorithm is asymptotically optimal. In more detail we explore the case of distance constraints (2; 1), and present two off-line approximation algorithms. The last one we call robust, i.e. it does not require the disks representation and either outputs a feasible labeling, or answers the input is not a unit disk graph.

A Brooks-Type Theorem for the Generalized List T-Coloring

We study the notion of a generalized list

Graph Subcolorings: Complexity and Algorithms

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Geometric Systems of Disjoint Representatives

-coloring which is a common generalization of the channel assignment problem and the

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

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Generalized H-Coloring and H-Covering of Trees

-coloring. An instance of the generalized list

On Vertex- and Empty-Ply Proximity Drawings

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