Information Processing Letters
Discrete Applied Mathematics
Optimal node ranking of trees in linear time
Information Processing Letters
The role of elimination trees in sparse factorization
SIAM Journal on Matrix Analysis and Applications
Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
Approximating treewidth, pathwidth, frontsize, and shortest elimination tree
Journal of Algorithms
Discrete Mathematics
Generalized vertex-rankings of trees
Information Processing Letters
SIAM Journal on Discrete Mathematics
Vertex ranking of asteroidal triple-free graphs
Information Processing Letters
On the vertex ranking problem for trapezoid, circular-arc and other graphs
Discrete Applied Mathematics
Algorithms for generalized vertex-rankings of partial k-trees
Theoretical Computer Science - computing and combinatorics
On Minimum Edge Ranking Spanning Trees
MFCS '99 Proceedings of the 24th International Symposium on Mathematical Foundations of Computer Science
Efficient Parallel Query Processing by Graph Ranking
Fundamenta Informaticae
Edge ranking and searching in partial orders
Discrete Applied Mathematics
On Problems without Polynomial Kernels (Extended Abstract)
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Optimal vertex ranking of block graphs
Information and Computation
Vertex rankings of chordal graphs and weighted trees
Information Processing Letters
SOFSEM'05 Proceedings of the 31st international conference on Theory and Practice of Computer Science
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Vertex ranking has many practical applications, ranging from VLSI layout and sparse matrix factorization to parallel query processing and assembly of modular products.Much research has been done on vertex ranking and related problems, polynomial time algorithms are known for a wide variety of graph classes as well as NP-hardness has been shown for other graph classes. In this paper we propose an extension to vertex ranking. Vertex ranking has many applications in computing a parallel schedule, but there is the assumption that the amount of parallel capacity is unbounded. Many applications do have restricted capacity, such as the number of processors or machines. Therefore we introduce vertex ranking with capacity.In this paper we show that vertex ranking and vertex ranking with capacity do not have a polynomial sized kernel, unless all coNP-complete problems have distillation algorithms. Having to deal with the NP- hardness of both problems, we give, to our knowledge, the first O *(2 n ) time exact algorithm for vertex ranking and use this for devising an O *(2.5875 n ) time algorithm for vertex ranking with capacity. We also show that we can transform vertex rankings to vertex rankings with capacity, and use this for a polynomial time algorithm that transforms an f(n)-approximate vertex ranking to a vertex ranking with capacity of at most f(n) + 1 times the optimum size. Lastly, give an log(c) additive approximation for vertex ranking with capacity when restricted to trees and extend this to graphs of bounded treewidth.