Mustazee Rahman

Local algorithms for independent sets are half-optimal

We show that the largest density of factor of i.i.d. independent sets in the dd-regular tree is asymptotically at most (logd)/d(log⁡d)/d as d→∞d→∞. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random dd-regular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically 2(logd)/d2(log⁡d)/d. We prove analogous results for Poisson–Galton–Watson trees, which yield bounds for local algorithms on sparse Erdős–Renyi graphs.

Factor of IID Percolation on Trees

We study invariant percolation processes on the d-regular tree that are obtained as a factor of an iid process. We show that the density of any factor of iid site percolation process with finite clusters is asymptotically at most (log d)/d as d tends to infinity. This bound is asymptotically optimal as it can be realized by independent sets. One implication of the result is a (1/2)-factor approximation gap, asymptotically in d, for estimating the density of maximal induced forests in locally tree-like d-regular graphs via factor of iid processes.

Spot-Based Generations for Meta-Fibonacci Sequences

For many meta-Fibonacci sequences it is possible to identify a partition of the sequence into successive intervals (sometimes called blocks) with the property that the sequence behaves “similarly” in each block. This partition provides insights into the sequence properties. To date, for any given sequence, only ad hoc methods have been available to identify this partition. We apply a new concept—the spot-based generation sequence—to derive a general methodology for identifying this partition for a large class of meta-Fibonacci sequences. This class includes the Conolly and Conway sequences and many of their well-behaved variants, and even some highly chaotic sequences, such as Hofstadters famous Q-sequence.

Percolation with Small Clusters on Random Graphs

Consider the problem of determining the maximal induced subgraph in a random d-regular graph such that its components remain bounded as the size of the graph becomes arbitrarily large. We show, for asymptotically large d, that any such induced subgraph has size density at most

Geometry of Permutation Limits

2(\log d)/d

Random sorting networks: local statistics via random matrix laws

2(logd)/d with high probability. A matching lower bound is known for independent sets. We also prove the analogous result for sparse Erdős–Rényi graphs.

Suboptimality of local algorithms for a class of max-cut problems

The optimal power flow problem is concerned with finding a proper operating point for a power network while attempting to minimize a cost function and satisfy network constraints. We analyze the optimal power flow problem subject to contingency constraints and investigate the relationship between the cost of the optimal power flow problem and network topology. We find that when the network topology is that of a small world graph or a scale-free graph, the optimal power flow problem is robust in terms of satisfying contingency constraints.