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- Mustazee Rahman, Bálint Virág
- ArXiv
- 2014

We show that the largest density of factor of i.i.d. independent sets on the d-regular tree is asymptotically at most (log d)/d as d → ∞. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random dregular graphs have the same asymptotic density. In contrast, the density of… (More)

- Mustazee Rahman
- Graphs and Combinatorics
- 2016

We consider the problem of finding an induced subgraph in a random dregular graph such that its components have bounded size as the size of the graph gets arbitrarily large. We show that for any threshold τ , the largest size density of such an induced subgraph with component sizes bounded by τ is at most 2(log d)/d for asymptotically large d. A matching… (More)

- Barnaby Dalton, Mustazee Rahman, Stephen M. Tanny
- Experimental Mathematics
- 2011

- Abraham Isgur, Mustazee Rahman
- Electr. J. Comb.
- 2011

We study the recursions A(n) = A(n−a−A(n−b))+A(A(n−b)) where a ≥ 0, b ≥ 1 are integers and the superscript k denotes a k-fold composition, and also the recursion C(n) = C(n − s − C(n − 1)) + C(n − s − 2 − C(n − 3)) where s ≥ 0 is an interger. We prove that under suitable initial conditions the sequences A(n) and C(n) will be defined for all positive… (More)

- Mustazee Rahman
- 2015

- Mustazee Rahman
- 2015

A spanning tree in a connected graph G is a subgraph that contains every vertex of G and is itself a tree. Clearly, if G is a tree then it has only one spanning tree. Every connected graph contains at least one spanning tree: iteratively remove an edge from any cycle that is present until the graph contains no cycles. Counting spanning trees is a very… (More)

- Mustazee Rahman
- 2015

Sub–optimality of local algorithms on sparse random graphs Mustazee Rahman Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2015 This thesis studies local algorithms for solving combinatorial optimization problems on large, sparse random graphs. Local algorithms are randomized algorithms that run in parallel at each vertex of a… (More)

- Baha Alzalg, Catalina Anghel, +4 authors Chai Wah Wu
- ISCAS
- 2012

- Baha Alzalg, Catalina Anghel, +8 authors Mustazee Rahman
- 2011

The optimal power flow problem is concerned with finding a proper operating point for a power network while attempting to minimize some cost function and satisfy several network constraints. In this report we analyze the optimal power flow problem subject to contingency constraints, which demands that there be enough power to meet demands in the network in… (More)

- Mustazee Rahman
- SIAM J. Discrete Math.
- 2016

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 → ∞. This bound is asymptotically optimal as it can be realized by independent sets. One implication of the result… (More)