Abstract: We 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 approximationschemes (FPRAS) for the probability of fixation (when r≥1) and of extinction (for all r>0).

Abstract: We examine the problem of assigning n independent jobs to m unrelated parallel machines, so that each job is processed without interruption on one of the machines, and at any time, every machine processes at most one job. We focus on the case where m is a fixed constant, and present a new rounding approach that yields approximationschemes for multi-objective minimum makespan scheduling with a fixed number of linear cost constraints. The same approach gives approximationschemes for covering problems like maximizing the minimum load on any machine, and for assigning specific or equal loads to the machines.

Abstract: We study geometric versions of the min-size k-clustering
problem, a clustering problem which generalizes clustering to minimize
the sum of cluster radii and has important applications. We prove that
the problem can be solved in polynomial time when the points to be clus-
tered are located on a line. For Euclidean spaces of higher dimensions,
we show that the problem is NP-hard and present polynomial time ap-
proximation schemes. The latter result yields an improved approximation
algorithm for the related problem of k-clustering to minimize the sum of
cluster diameters.

Abstract: A constraint network is arc consistent if any value of any of its variables is compatible with at
least one value of any other variable. The Arc Consistency Problem (ACP) consists in filtering out values of
the variables of a given network to obtain one that is arc consistent, without eliminating any solution. ACP is
known to be inherently sequential, or P-complete, so in this paper we examine some weaker versions of it and
their parallel complexity. We propose several natural approximationschemes for ACP and show that they are also
P-complete. In an attempt to overcome these negative results, we turn our attention to the problem of filtering
out values from the variables so that each value in the resulting network is compatible with at least one value of
not necessarily all, but a constant fraction of the other variables. We call such a network partially arc consistent.
We give a parallel algorithm that, for any constraint network, outputs a partially arc consistent subnetwork of it in
sublinear (O.pn log n/) parallel time using O.n2/ processors. This is the first (to our knowledge) sublinear-time
parallel algorithm with polynomially many processors that guarantees that in the resulting network every value is
compatible with at least one value in at least a constant fraction of the remaining variables. Finally, we generalize
the notion of partiality to the k-consistency problem.