Finding paths and deleting edges in directed acyclic graphs
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On paths avoiding forbidden pairs of vertices in a graph
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Optimization principles and application performance evaluation of a multithreaded GPU using CUDA
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Benchmarking GPUs to tune dense linear algebra
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On the complexity of paths avoiding forbidden pairs
Discrete Applied Mathematics
General context-free recognition in less than cubic time
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Predicting gene structures from multiple RT-PCR tests
WABI'09 Proceedings of the 9th international conference on Algorithms in bioinformatics
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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Let G=(V,E) be a directed acyclic graph with two distinguished vertices s,t, and let F be a set of forbidden pairs of vertices. We say that a path in G is safe, if it contains at most one vertex from each pair {u,v}@?F. Given G and F, the path avoiding forbidden pairs (PAFP) problem is to find a safe s-t path in G. We systematically study the complexity of different special cases of the PAFP problem defined by the mutual positions of forbidden pairs. Fix one topological ordering @? of vertices; we say that pairs {u,v} and {x,y} are disjoint, if u@?v@?x@?y, nested, if u@?x@?y@?v, and halving, if u@?x@?v@?y. The PAFP problem is known to be NP-hard in general or if no two pairs are disjoint; we prove that it remains NP-hard even when no two forbidden pairs are nested. On the other hand, if no two pairs are halving, the problem is known to be solvable in cubic time. We simplify and improve this result by showing an O(M(n)) time algorithm, where M(n) is the time to multiply two nxn boolean matrices.