Graph layout problems are a particular class of combinatorial optimization problems whose goal is to find a linear layout of an input graph in such way that a certain objective cost is optimized. This survey considers their motivation, complexity, approximation properties, upper and lower bounds, heuristics and probabilistic analysis on random graphs. The result is a complete view of the current state of the art with respect to layout problems from an algorithmic point of view.

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Citation for published item:h¡ % zD tF nd qold ergD vFeF nd wertziosD qFfF nd i her yD hF nd ern D wF nd pir kisD FqF @PHIRA 9epproxim ting (x tion pro ilities in the gener lized wor n pro essF9D elgorithmi FD TW @IAF ppF UVEWIFFurther information on publisher's website:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00453-012-9722-7.Additional information:A preliminary version of this work appeared in Proceedings of the ACM SIAM Symposium on Discrete Algorithms (SODA), pp. 954 960, 2012. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO• the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312-316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned "fitness" value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness r > 0 placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when r 1) and of extinction (for all r > 0). * A preliminary version of this work appeared in