Parameterized Complexity of Two-Interval Pattern Problem

A 2-interval is the union of two disjoint intervals on the real line. Two 2-intervals D? and D? are disjoint if their intersection is empty (i.e., no interval of D? intersects any interval of D?). There can be three different relations between two disjoint 2-intervals; namely, preceding (<), nested (?) and crossing (?). Two 2-intervals D? and D? are called R-comparable for some R?{<,?,?}, if either D?RD? or D?RD?. A set ? of disjoint 2-intervals is ?-comparable, for some ??{<,?,?} and ???, if every pair of 2-intervals in ? are R-comparable for some R??. Given a set of 2-intervals and some ??{<,?,?}, the objective of the {2-interval pattern problem} is to find a largest subset of 2-intervals that is ?-comparable. The 2-interval pattern problem is known to be W[1]-hard when |?|=3 and NP-hard when |?|=2 (except for ?={<,?}, which is solvable in quadratic time). In this paper, we fully settle the parameterized complexity of the problem by showing that it is W[1]-hard for both ?={?,?} and ?={<,?} (when parameterized by the size of an optimal solution). This answers the open question posed by Vialette [Encyclopedia of Algorithms, 2008]

TRStalker: an efficient heuristic for finding fuzzy tandem repeats

Motivation: Genomes in higher eukaryotic organisms contain a substantial amount of repeated sequences. Tandem Repeats (TRs) constitute a large class of repetitive sequences that are originated via phenomena such as replication slippage and are characterized by close spatial contiguity. They play an important role in several molecular regulatory mechanisms, and also in several diseases (e.g. in the group of trinucleotide repeat disorders). While for TRs with a low or medium level of divergence the current methods are rather effective, the problem of detecting TRs with higher divergence (fuzzy TRs) is still open. The detection of fuzzy TRs is propaedeutic to enriching our view of their role in regulatory mechanisms and diseases. Fuzzy TRs are also important as tools to shed light on the evolutionary history of the genome, where higher divergence correlates with more remote duplication events