Uniform Random Model Research Articles

Phase Transitions for Scale-Free SAT Formulas

Recently, a number of non-uniform random satisfiability models have been proposed that are closer to practical satisfiability problems in some characteristics. In contrast to uniform random Boolean formulas, scale-free formulas have a variable occurrence distribution that follows a power law. It has been conjectured that such a distribution is a more accurate model for some industrial instances than the uniform random model. Though it seems that there is already an awareness of a threshold phenomenon in such models, there is still a complete picture lacking. In contrast to the uniform model, the critical density threshold does not lie at a single point, but instead exhibits a functional dependency on the power-law exponent. For scale-free formulas with clauses of length k=2, we give a lower bound on the phase transition threshold as a function of the scaling parameter. We also perform computational studies that suggest our bound is tight and investigate the critical density for formulas with higher clause lengths. Similar to the uniform model, on formulas with k>=3, we find that the phase transition regime corresponds to a set of formulas that are difficult to solve by backtracking search.

Eigenvalues and energy in threshold graphs

Assuming a uniform random model of selecting creation sequences, we show that almost every connected threshold graph has more negative than positive eigenvalues. We show that no threshold graphs have eigenvalues in (−1,0). Sequences of equienergetic graphs are given including the striking result that for all n≥3, there exist n−1 threshold graphs of order n2, pairwise noncospectral, each equienergetic to Kn2. For all n>8, we find an n-vertex hyperenergetic threshold graph.

New results on optimizing rooted triplets consistency

A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets consistency problem ( MaxRTC ) and the minimum rooted triplets inconsistency problem ( MinRTI ) in which the input is a set R of rooted triplets, and where the objectives are to find a largest cardinality subset of R which is consistent and a smallest cardinality subset of R whose removal from R results in a consistent set, respectively. We first show that a simple modification to Wu’s Best-Pair-Merge-First heuristic Wu (2004) [38] results in a bottom-up-based 3-approximation algorithm for MaxRTC . We then demonstrate how any approximation algorithm for MinRTI could be used to approximate MaxRTC , and thus obtain the first polynomial-time approximation algorithm for MaxRTC with approximation ratio less than 3. Next, we prove that for a set of rooted triplets generated under a uniform random model, the maximum fraction of triplets which can be consistent with any phylogenetic tree is approximately one third. We then provide a deterministic construction of a triplet set having a similar property which is subsequently used to prove that both MaxRTC and MinRTI are NP -hard even if restricted to minimally dense instances. Finally, we prove that unless P = NP , MinRTI cannot be approximated within a ratio of c ⋅ ln n for some constant c > 0 in polynomial time, where n denotes the cardinality of the leaf label set of R .

CORRELATED RANDOM WALK EQUATIONS OF ANIMAL DISPERSAL RESOLVED BY SIMULATION

Animal movement and dispersal can be described as a correlated random walk dependent on three parameters: number of steps, step size, and distribution of random turning angles. Equations of Kareiva and Shigesada use the parameters to predict the mean square displacement distance (MSDD), but this is less meaningful than the mean dispersal distance (MDD) about which the population would be distributed. I found that the MDD can be estimated by multiplying the square root of the MSDD by a three-dimensional surface correction factor obtained from simulations. The correction factors ranged from 0.89 to 1 depending on the number of steps and the variation in random turns, expressed as the standard deviation of the turning angles (SDA) about 0° (straight ahead). Corrected equations were used to predict MDDs for bark beetles, butterflies, ants, and beetles (based on parameters from the literature) and the nematode Steinernema carpocapsae (Weiser). Another equation from the literature finds the MDD directly, and this agreed with the MDD obtained by simulation at some combinations of SDA and numbers of steps. However, the equation has an error that increases as a power function when the standard deviation of turning angles becomes smaller (e.g., <6° at 1000 steps or <13° at 250 steps). Lower numbers of steps also increase the error. Equivalent values of AMT (angle of maximum turn) in uniform random models and of SDA in normal random models were found that allowed these two models to yield similar MDD values. The step size and turning angle variation of animal paths during dispersal and host and mate searching were investigated and found to be correlated; thus, use of different measured step sizes gives consistent estimates of the MDD.