The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the paper. Strong amplification is a rather surprising property -it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name "strongly amplifying" comes from the fact that this selective advantage is "amplified". Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families -superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this paper, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family "megastars". When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly n −1/2 (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tightthere is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counter-example which clarifies the literature concerning the isothermal theorem of Lieberman et al.

In this paper, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of Müller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). While all these problems have simple algorithms over which it is conjectured that no polynomial improvement is possible, our reductions would remain interesting even if these conjectures were proved; they have only polylogarithmic overhead, and can therefore be applied to subpolynomial improvements such as the n 3 / exp(Θ( √ log n))-time algorithm for the Negative-Weight Triangle problem due to Williams (STOC 2014). Our framework is also general enough to apply to versions of the problems for which more efficient algorithms are known. For example, the Orthogonal Vectors problem over GF(m) d for constant m can be solved in time n · poly(d) by a result of Williams and Yu (SODA 2014); our result implies that we can approximately count the number of orthogonal pairs with essentially the same running time.We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some 1 < c < 2 and all k there is an O(c n )-time algorithm for k-SAT. Then we prove * We thank Rahul Santhanam and Ryan Williams for some valuable discussions. that for all k, there is an O((c + o(1)) n )-time algorithm for approximate #k-SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly-weaker statement that there is no algorithm to approximate #3-SAT to within a factor of 1 + ε in time 2 o(n) /ε 2 (taking ε > 0 as part of the input).

Abstract. We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ = (1/2+α)n. For any constant α > 0, we give an optimal answer in the following sense: let reg even (n, δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals reg even (n, δ)/2. The value of reg even (n, δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

Abstract. A conjecture of Thomassen from 1982 states that for every k there is an f (k) so that every strongly f (k)-connected tournament contains k edge-disjoint Hamilton cycles. A classical theorem of Camion, that every strongly connected tournament contains a Hamilton cycle, implies that f (1) = 1. So far, even the existence of f (2) was open. In this paper, we prove Thomassen's conjecture by showing that f (k) = O(k 2 log 2 k). This is best possible up to the logarithmic factor. As a tool, we show that every strongly 10 4 k log k-connected tournament is k-linked (which improves a previous exponential bound). The proof of the latter is based on a fundamental result of Ajtai, Komlós and Szemerédi on asymptotically optimal sorting networks.

We study the Moran process as adapted by Lieberman, Hauert and Nowak. This is a model of an evolving population on a graph or digraph where certain individuals, called "mutants" have fitness r and other individuals, called "non-mutants" have fitness 1. We focus on the situation where the mutation is advantageous, in the sense that r > 1. A family of digraphs is said to be strongly amplifying if the extinction probability tends to 0 when the Moran process is run on digraphs in this family. The most-amplifying known family of digraphs is the family of megastars of Galanis et al. We show that this family is optimal, up to logarithmic factors, since every strongly-connected n-vertex digraph has extinction probability Ω(n −1/2 ). Next, we show that there is an infinite family of undirected graphs, called dense incubators, whose extinction probability is O(n −1/3 ). We show that this is optimal, up to constant factors. Finally, we introduce sparse incubators, for varying edge density, and show that the extinction probability of these graphs is O(n/m), where m is the number of edges. Again, we show that this is optimal, up to constant factors. *