4 research outputs found
Longest Common Separable Pattern between Permutations
In this article, we study the problem of finding the longest common separable
pattern between several permutations. We give a polynomial-time algorithm when
the number of input permutations is fixed and show that the problem is NP-hard
for an arbitrary number of input permutations even if these permutations are
separable. On the other hand, we show that the NP-hard problem of finding the
longest common pattern between two permutations cannot be approximated better
than within a ratio of (where is the size of an optimal
solution) when taking common patterns belonging to pattern-avoiding classes of
permutations.Comment: 15 page
Recognizing Unit Multiple Intervals Is Hard
Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A d-interval is the union of d intervals on the real line, and a graph is a d-interval graph if it is the intersection graph of d-intervals. In particular, it is a unit d-interval graph if it admits a d-interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit d-interval graphs for any d ⩾ 2, which does not follow directly in graph recognition problems -as an example, it took almost 20 years to close the gap between d = 2 and d > 2 for the recognition of d-track interval graphs. Our result has several implications, including that recognizing (x, …, x) d-interval graphs and depth r unit 2-interval graphs is NP-complete for every x ⩾ 11 and every r ⩾ 4
Graph Theory Based Methodology for Comparing
Introduction If the genome is mostly invariant in each cell of an organism, genes can have dierent expression pattern related to the environmental conditions or developmental programs. For one genome, dierent functional states can be observed, which depend on complex interaction networks between genes. Transcriptome analysis allows the identi cation of genes with correlated expression pro les, thus giving clues to the structure and the organization of these networks. Today, DNA microarrays are among tools providing direct access to transcriptome analysis. This technology gives the opportunity to observe simultaneously the expression of several thousand genes within a cell, so measuring transcriptome changes in various cellular states. In order to understand genome functioning and to establish functional catalogues, the next major step is to compare sets of genes with similar expression pro les across organisms. To achieve this, we use a graph theory based approach. Methodolog