The P-complete Circuit Value Problem CVP, when restricted to monotone planar circuits MPCVP, is known to be in NC 3 , 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 AC 1 (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 AC 1 (LogDCFL). We re-examine the NC 3 algorithm for general MPCVP, and note that it is in AC 1 (LogCFL) = SAC 2 . 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.

We 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 L. We show that computing the permanent and determinant of tridiagonal matrices over Z is in GapNC 1 and is hard for NC 1 . 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 NP-hard over Q and that a certain restricted version is NP-complete. Restricting the problem further, we obtain variations which can be computed in PL and are hard for C = L.

For a Boolean function f : {0, 1} n → {0, 1} computed by a circuit C over a finite basis B, the energy complexity of C (denoted by EC B (C)) is the maximum over all inputs {0, 1} n the numbers of gates of the circuit C (excluding the inputs) that output a one. Energy Complexity of a Boolean function over a finite basis B denoted by EC B ( f ) def *

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The reversible pebble game is a combinatorial game played on rooted DAGs. This game was introduced by Bennett [1] motivated by applications in designing space efficient reversible algorithms. Recently, Siu Man Chan [2] showed that the reversible pebble game number of any DAG is the same as its Dymond-Tompa pebble number and Raz-Mckenzie pebble number.We show, as our main result, that for any rooted directed tree T , its reversible pebble game number is always just one more than the edge rank coloring number of the underlying undirected tree U of T . The most striking implication of this result is that the reversible pebble game number of a tree does not depend upon the direction of edges, a fact that does not hold in general for DAGs. It is known that given a DAG G as input, determining its reversible pebble game number is PSPACE-hard. Our result implies that the reversible pebble game number of trees can be computed in polynomial time as edge rank coloring number of trees can be computed in linear time ([8]).We also address the question of finding the number of steps required to optimally pebble various families of trees. It is known that trees can be pebbled in n O(log(n)) steps where n is the number of nodes in the tree. Using the equivalence between reversible pebble game and the Dymond-Tompa pebble game [2], we show that complete binary trees can be pebbled in n O(log log(n)) steps, a substantial improvement over the naive upper bound of n O(log(n)) .It remains open whether complete binary trees can be pebbled in polynomial number of steps (i.e., n k for some constant k). Towards this end, we show that almost optimal (i.e., within a factor of (1 + ǫ) for any constant ǫ > 0) pebblings of complete binary trees can be done in polynomial number of steps. * Sponsored by TCS Research FellowshipWe also show a time-space trade-off for reversible pebbling for families of bounded degree trees by a divide-and-conquer approach: for any constant ǫ > 0, such families can be pebbled using O(n ǫ ) pebbles in O(n) steps. This generalizes an analogous result of Královic[7] for chains.