Jayalal Sarma

Upper Bounds for Monotone Planar Circuit Value and Variants

Abstract.The P-complete Circuit Value Problem CVP, when restricted to monotone planar circuits MPCVP, is known to be in NC3, and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we re-examine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC1(LogDCFL), while monotone circuits with one-input-face planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC1(LogDCFL). We re-examine the NC3 algorithm for general MPCVP, and note that it is in AC1(LogCFL) = SAC2. Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar non-monotone circuits with polylogarithmic negation-height can be evaluated in NC.

On the Complexity of Matrix Rank and Rigidity

AbstractWe revisit a well studied linear algebraic problem, computing the rank and determinant of matrices, in order to obtain completeness results for small complexity classes. In particular, we prove that computing the rank of a class of diagonally dominant matrices is complete for

On Isomorphism Testing of Groups with Normal Hall Subgroups

\textsf{L}

Pebbling, Entropy, and Branching Program Size Lower Bounds

. We show that computing the permanent and determinant of tridiagonal matrices over ℤ is in

Depth Lower Bounds against Circuits with Sparse Orientation

\textsf {Gap} \textsf {NC}^{1}

Limiting negations in bounded treewidth and upward planar circuits

and is hard for

On the Complexity of L-reachability*

\textsf {NC}^{1}

Arithmetic Circuit Lower Bounds via Maximum-Rank of Partial Derivative Matrices

. We also initiate the study of computing the rigidity of a matrix: the number of entries that needs to be changed in order to bring the rank of a matrix below a given value. We show that some restricted versions of the problem characterize small complexity classes. We also look at a variant of rigidity where there is a bound on the amount of change allowed. Using ideas from the linear interval equations literature, we show that this problem is

Reversible Pebble Game on Trees

\textsf {NP}

Circuit Complexity of Properties of Graphs with Constant Planar Cutwidth

-hard over ℚ and that a certain restricted version is