Nutan Limaye

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

We show here a 2^{Ω(√d·log N)} size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N = d³ in our case) with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Σ_i Π_j Q_ij, where the Qijs are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that Σi,j (Number of monomials of Q_ij) ≥ 2^{Ω(√d·log N)}. The above mentioned family, which we refer to as the NisanWigderson design-based family of polynomials, is in the complexity class VNP. Our work builds on the recent lower bound results [1], [2], [3], [4], [5] and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of [6] and the N^{Ω(log log N)} lower bound in the independent work of [7].

Planarity, Determinants, Permanents, and (Unique) Matchings

Viewing the computation of the determinant and the permanent of integer matrices as combinatorial problems on associated graphs, we explore the restrictiveness of planarity on their complexities and show that both problems remain as hard as in the general case, that is, GapL- and P- complete. On the other hand, both bipartite planarity and bimodal planarity bring the complexity of permanents down (but no further) to that of determinants. The permanent or the determinant modulo 2 is complete for ⊕L, and we show that parity of paths in a layered grid graph (which is bimodal planar) is also complete for this class. We also relate the complexity of grid graph reachability to that of testing existence/uniqueness of a perfect matching in a planar bipartite graph.

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.

Arithmetizing classes around NC 1 and L

The parallel complexity class NC1 has many equivalent models such as bounded width branching programs. Caussinus et.al[10] considered arithmetizations of two of these classes, #NC1 and #BWBP. We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata has the same power as #BWBP, while counting proof-trees in logarithmic width formulae has the same power as #NC1. We also consider polynomial-degree restrictions of SCi, denoted sSCi, and show that the Boolean class sSC1 lies between NC1 and L, whereas sSC0 equals NC1. On the other hand, #sSC0 contains #BWBP and is contained in FL, and #sSC1 contains #NC1 and is in SC2. We also investigate some closure properties of the newly defined arithmetic classes.

3-connected Planar Graph Isomorphism is in Log-space

We consider theisomorphism and canonization problem for3-connected planar graphs. The problem was known to be L -hard and in UL ∩ coUL (TW08). In this paper, we give a determin- istic log-space algorithm for 3-connected planar graph isomorphism and canonization. This gives an L -completeness result, thereby settling its complexity. The algorithm uses the notion of universal exploration sequences from (Kou02) and (Rei05). To our knowledge, this is a completely new approach to graph canonization.

Super-polynomial lower bounds for depth-4 homogeneous arithmetic formulas

We show that any depth-4 homogeneous arithmetic formula computing the Iterated Matrix Multiplication polynomial IMMn,d -- the (1, 1)-th entry of the product of d generic n × n matrices -- has size nΩ(log n), if d = Ω (log2 n). More-over, any depth-4 homogeneous formula computing the determinant polynomial Detn -- the determinant of a generic n × n matrix -- has size nΩ(log n).

Streaming algorithms for recognizing nearly well-parenthesized expressions

We study the streaming complexity of the membership problem of 1-turn-Dyck2 and Dyck2 when there are a few errors in the input string. 1-turn-Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x′ ∈ 1-turn-Dyck2 such that x is obtained by flipping at most k locations of x′ using: - O(k log n) space, O(k log n) randomness, and poly(k log n) time per item and with error at most 1/nΩ(1). - O(k1+e + log n) space for every 0 ≤ e ≤ 1, O(log n) randomness, O((logO(1) n + kO(1))) time per item, with error at most 1/8. Here, we also prove that any randomized one-pass algorithm that makes error at most k/n requires at least Ω(k log(n/k)) space to accept strings which are exactly k-away from strings in 1-turn-Dyck2 and to reject strings which are exactly k + 2-away from strings in 1-turn-Dyck2. Since 1-turn-Dyck2 and the Hamming Distance problem are closely related we also obtain new upper and lower bounds for this problem. Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x′ ∈ Dyck2 such that x is obtained from x′ by changing (in some restricted manner) at most k positions using: - O(k log n + √n log n) space, O(k log n) randomness, poly(k log n) time per element and with error at most 1/nΩ(1). - O(k1+e + √n log n) space for every 0 < e ≤ 1, O(log n) randomness, O((logO(1) n + kO(1))) time per element, with error at most 1/8.

Streaming algorithms for some problems in log-space

In this paper, we give streaming algorithms for some problems which are known to be in deterministic log-space, when the number of passes made on the input is unbounded If the input data is massive, the conventional deterministic log-space algorithms may not run efficiently We study the complexity of the problems when the number of passes is bounded. The first problem we consider is the membership testing problem for deterministic linear languages, DLIN Extending the recent work of Magniez et al.[11](to appear in STOC 2010), we study the use of fingerprinting technique for this problem We give the following streaming algorithms for the membership testing of DLIN s: a randomized one pass algorithm that uses O(logn) space (one-sided error, inverse polynomial error probability), and also a p-pass O(n/p)-space deterministic algorithm We also prove that there exists a language in DLIN, for which any p-pass deterministic algorithm for membership testing, requires Ω(n/p) space We also study the application of fingerprinting technique to visibly pushdown languages, VPL s. The other problem we consider is, given a degree sequence and a graph, checking whether the graph has the given degree sequence, Deg-Seq We prove that, any p-pass deterministic algorithm that takes as its input a degree sequence, followed by an adjacency list of a graph, requires Ω(n/p) space to decide Deg-Seq However, using randomness, for a more general input format: degree sequence, followed by a list of edges in any arbitrary order, Deg-Seq can be decided in O(logn) space We also give a p-pass, O(n/p)-space deterministic algorithm for Deg-Seq.

Arithmetizing Classes Around

AbstractThe parallel complexity class

\textsf{NC}

\textsf{NC}