Abstract: The enumeration of extensional acyclic digraphs, which have the property that the outneighbourhoods are pairwise distinct, was considered in a recent article of Policriti and Tomescu. Several asymptotic questions were left as open problems. In this article, we determine the asymptotic number of such digraphs and show that a number of distributional results can be carried over from ordinary acyclic digraphs. In particular, we consider the distribution of the number of sources, the maximum rank and the number of vertices of maximum rank, thereby also proving some conjectures made by Policriti and Tomescu.

Abstract: Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. We prove that the Walksat algorithm from Papadimitriou (FOCS 1991)/Schoning (FOCS 1999) finds a satisfying assignment of Φ in polynomial time w.h.p. if m/n ≤ ρ · 2k/k for a certain constant ρ > 0. This is an improvement by a factor of Θ(k) over the best previous analysis of Walksat from Coja-Oghlan, Feige, Frieze, Krivelevich, Vilenchik (SODA 2009).

Abstract: We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed walks of length k starting at a given vertex v. The first is a direct implication of a matrix inequality by Marcus and Newman and generalizes two inequalities by Lagarias et al. and Dress & Gutman. We then use an inequality of Blakley and Dixon to show the inequality wk2e+p ≤ w2e+pk · wk−12e which also generalizes the inequality by Dress and Gutman and also an inequality by Erdos and Simonovits. Both results can be translated directly into the corresponding forms using the higher order densities, which extends former results. In the second part, we provide a new family of lower bounds for the largest eigenvalue λ1 of the adjacency matrix based on closed walks and apply the before mentioned inequalities to show monotonicity in this and a related family of lower bounds of Nikiforov. This leads to generalized upper bounds for the energy of graphs. In the third part, we demonstrate that a further natural generalization of the inequality w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) is not valid for general graphs. We show that wa+b · wa+b+c ≤ wa · wa+2b+c does not hold even in very restricted cases like w1 · w2 ≤ w0 · w3 (i.e., d · w2 ≤ w3) in the context of bipartite or cycle free graphs. In contrast, we show that surprisingly this inequality is always satisfied for trees and show how to construct worst-case instances (regarding the difference of both sides of the inequality) for a given degree sequence. We also provide a proof for the inequality w1 · w4 ≤ w0 · w5 (i.e., d · w4 ≤ w5) for trees and conclude with a corresponding conjecture for longer walks.

Abstract: The rotor-router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph. Instead of distributing tokens to randomly chosen neighbors, the Propp machine deterministically serves the neighbors in a fixed order by associating to each vertex a "rotor-router" pointing to one of its neighbors. This paper investigates the discrepancy on a single vertex between the number of tokens in the rotor-router model and the expected number of tokens in a random walk, for finite graphs in general. We show that the discrepancy is bounded by O(mn) at any time for any initial configuration if the corresponding random walk is lazy and reversible, where n and m denote the numbers of nodes and edges, respectively. For a lower bound, we show examples of graphs and initial configurations for which the discrepancy on a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic bound, in terms of the number of nodes, for the discrepancy.

Abstract: MLS-3LIN problem is a problem of finding a most likely solution for a given system of perturbed 3LIN-equations under a certain planted solution model. This problem is essentially the same as MAX-3XORSAT problem. We investigate the average-case performance of message passing algorithms for this problem, where input instances are generated under the planted solution model with equation probability p and perturbation probability q. For some variant of a typical message passing algorithm, we prove that p = Θ(1/(n ln n)) is the threshold for the algorithm to work w.h.p. for any fixed constant q < 1/2.

Abstract: It is well known that, under some aperiodicity and irreducibility conditions, the number of occurrences of local patterns within a Markov chain (and, more generally, within the languages generated by weighted regular expressions/automata) follows a Gaussian distribution with both variance and mean in Θ(n). By contrast, when these conditions no longer hold, it has been observed that the limiting distribution may follow a whole diversity of distributions, including the uniform, power-law or even multimodal distribution, arising as trade-offs between structural properties of the regular expression and the weight/probabilities associated with its transitions/letters. However these cases only partially cover the full diversity of behaviors induced within regular expressions, and a characterization of attainable distributions remained to be provided. In this article, we constructively show that the limiting distribution of the simplest foreseeable motif (a single letter!) may already follow an arbitrarily complex continuous distribution (or cadlag process). We also give applications in random generation (Boltzmann sampling) and bioinformatics (parsimonious segmentation of DNA).

Abstract: The hiring problem is a recent research problem, which has been introduced and studied first by Broder et al. [2] in 2008. It belongs to the category of on-line decision making under uncertainty. In such kind of research, the input is a sequence of instances and a decision must be taken for each instance depending on the instances seen so far while no information on the future is available. The hiring problem can be considered as a natural extension of the well-known secretary problem [4], where the employer is now looking for many candidates rather than only one (as it is the case for the secretary problem). Here the goal is to design some hiring strategy to meet the demands of the employer, which essentially are to obtain a good quality staff at a reasonable hiring rate, which is a main difference to the secretary problem, where an optimization policy, namely the demand of hiring the best candidate, occurs. Broder et al. introduced two so-called "Lake Wobegon strategies", namely "hiring above the current mean" and "hiring above the current median", applied for a continuous probabilistic model for the sequence of scores of the candidates. Archibald and Martinez [1] have reformulated the problem for a discrete model that considers the relative ranks amongst candidates as it is the case in the secretary problem. For this model in [1] the authors studied two general strategies, namely "hiring above the m-th best candidate", and "hiring above the median" (and other quantiles). In this work we give a detailed study of the "hiring above the median" strategy under this discrete model for the input sequence of scores of the candidates. This strategy processes the sequence of candidates as follows: hire the first interviewed candidate, and then any coming candidate is hired if and only if his rank is better than the rank of the median of the already hired staff, and discarded otherwise. Compared to the previous work of [1] we use a somewhat different recursive approach for a study of the "hiring above the median" strategy leading to rather explicit results. The key ingredients are to take into account the score of the median (the so-called threshold candidate) of the hired staff during the hiring process as well as to distinguish between two cases according to the parity of the size of the hiring set. Considering the transition probabilities during the hiring process yields, for fundamental hiring quantities, a system of linear recurrences that can be translated into a system of partial differential equations for the corresponding generating functions. In order to solve the PDEs appearing it turned out to be crucial to find suitable normalization factors of the studied recursive sequences, such that one of the corresponding generating functions itself reduces to a first order linear PDE. The exact solutions obtained for the differential equations yield to a rather detailed description of the exact probability distributions together with limiting distribution results for various hiring quantities, which might lead to a fairly good understanding of the "hiring above the median" hiring process. In particular we obtained results for the number of hired candidates, the score of the last hired candidate, the index of the last hired candidate, the distance between the last two hirings, the score of the best discarded candidate, and the number of hired candidates conditioned on the score of the first candidate. We also give the probability that a given score is getting hired in a sequence of n candidates.

Abstract: In 2001, Duchon, Flajolet, Louchard and Schaeffer introduced Boltzmann samplers, a radically novel way to efficiently generate huge random combinatorial objects without any preprocessing; the insight was that the probabilities can be obtained directly by evaluating the generating functions of combinatorials classes. Over the following decade, a vast array of papers has increased the formal expressiveness of these random samplers. Our paper introduces a new kind of sampler which generates multiplicative combinatorial structures, which enumerated by Dirichlet generating functions. Such classes, which are significantly harder to analyze than their additive counterparts, are at the intersection of combinatorics and analytic number theory. Indeed, one example we fully discuss is that of ordered factorizations. While we recycle many of the concepts of Boltzmann random sampling, our samplers no longer obey a Boltzmann distribution; we thus have coined a new name for them: Dirichlet samplers. These are very efficient as they can generate objects of size n in O((log n)2) worst-case time complexity. By providing a means by which to generate very large random multiplicative objects, our Dirichlet samplers can facilitate the investigation of these interesting, yet little studied structures. We also hope to illustrate some of our general ideas regarding the future direction for random sampling.

Abstract: Linear Search Problem has a venerable history, going back to R. Bellman ('63) and A. Beck ('64). They looked into the following question: An object is placed at a point P on the real line, according to a known probability distribution. A search plan (or trajectory is a sequence x = {xi} with ... x4 < x2 < 0 < x1 < x3 < ... (or ... x3 < x1 < 0 < x2 < x4 < ...). A search is performed by a searcher walking alternating to the points of the search plan, starting at 0, until the point P is found.

Abstract: Regarding so-called hairpin-loops as the building blocks of a RNA secondary structure, the order (as introduced by Waterman as a parameter on graphs in 1978) provides information on the (balanced) nesting-depth of hairpin-loops and thus on the overall complexity of the structure. Subsequent to Waterman's seminal work, Zucker et al. and Clote introduced a more realistic combinatorial model for RNA secondary structures, the so-called saturated secondary structures. Compared to the traditional model ofWaterman, unpaired nucleotides (vertices) which are in favorable position for a pairing do not exist, i.e. no base pair (edge) can be added without violating at least one restriction for the graphs. That way, one major shortcoming of the traditional model has been cleared. However, the resulting model gets much more challenging from a mathematical point of view. As a consequence, the current state of knowledge concerning saturated structures is limited to (1) their asymptotic number, (2) the expected number of base pairs, (3) the asymptotic normal density of states [4]. Nothing is known about the nature of the branching topology of saturated structures -- a topic that the current paper completely solves. In this paper we show how it is possible to attack saturated structures and especially how to analyze their order. This is of special interest since in the past it has been proven to be one of the most demanding parameters to address (for the traditional model it has been an open problem for more than 20 years to find asymptotic results for the number of structures of given order and similar). We show the expected order of RNA saturated secondary structures of size n is log4 n (1 + O (1/log2n)), if we select the saturated secondary structure uniformly at random. Furthermore, the order of saturated secondary structures is sharply concentrated around its mean. As a consequence saturated structures and structures in the traditional model behave the same with respect to the expected order. Thus we may conclude that the traditional model has already drawn the right picture and conclusions inferred from it with respect to the order (the overall shape) of a structure remain valid even if enforcing saturation (at least in expectation).

Abstract: We prove that the number of vertices of given degree in random planar maps satisfies a central limit theorem with mean and variance that are asymptotically linear in the number of edges. The proof relies on an analytic version of the quadratic method and singularity analysis of multivariate generating functions.