Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
On the 2-chain subgraph cover and related problems
Journal of Algorithms
Proper and unit tolerance graphs
Discrete Applied Mathematics - Special volume: Aridam VI and VII, Rutcor, New Brunswick, NJ, USA (1991 and 1992)
Trapezoid graphs and generalizations, geometry and algorithms
Discrete Applied Mathematics
Proper and unit bitolerance orders and graphs
Discrete Mathematics
Triangulating multitolerance graphs
Discrete Applied Mathematics
Coloring Algorithms for Tolerance Graphs: Reasoning and Scheduling with Interval Constraints
AISC '02/Calculemus '02 Proceedings of the Joint International Conferences on Artificial Intelligence, Automated Reasoning, and Symbolic Computation
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Journal of Graph Theory
Algorithm Design: Foundations, Analysis and Internet Examples
Algorithm Design: Foundations, Analysis and Internet Examples
A characterization of triangle-free tolerance graphs
Discrete Applied Mathematics
A New Intersection Model and Improved Algorithms for Tolerance Graphs
SIAM Journal on Discrete Mathematics
The recognition of triangle graphs
Theoretical Computer Science
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Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs has attracted many research efforts, mainly due to its interesting structure and its numerous applications, especially in DNA sequence analysis and resource allocation, among others. In one of the most natural generalizations of tolerance graphs, namely multitolerance graphs, two tolerances are allowed for each interval -- one from the left and one from the right side of the interval. Then, in its interior part, every interval tolerates the intersection with others by an amount that is a convex combination of its two border-tolerances. In the comparison of DNA sequences between different organisms, the natural interpretation of this model lies on the fact that, in some applications, we may want to treat several parts of the genomic sequences differently. That is, we may want to be more tolerant at some parts of the sequences than at others. These two tolerances for every interval -- together with their convex hull -- define an infinite number of the so called tolerance-intervals, which make the multitolerance model inconvenient to cope with. In this article we introduce the first non-trivial intersection model for multitolerance graphs, given by objects in the 3-dimensional space called trapezoepipeds. Apart from being important on its own, this new intersection model proves to be a powerful tool for designing efficient algorithms. Given a multitolerance graph with n vertices and m edges, we present algorithms that compute a minimum coloring and a maximum clique in optimal O(nlog n) time, and a maximum weight independent set in O(m + n log n) time. Moreover, our results imply an optimal O(n log n) time algorithm for the maximum weight independent set problem on tolerance graphs, thus closing the complexity gap for this problem. Additionally, by exploiting more the new 3D-intersection model, we completely classify multitolerance graphs in the hierarchy of perfect graphs.