Nikhil Balaji

Low-Depth Uniform Threshold Circuits and the Bit-Complexity of Straight Line Programs

We present improved uniform TC 0 circuits for division, matrix powering, and related problems, where the improvement is in terms of “majority depth” (as studied by Maciel and Therien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy.

Counting Euler Tours in Undirected Bounded Treewidth Graphs

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a

Bounded Treewidth and Space-Efficient Linear Algebra

\#SAC^1

Complexity of Restricted Variants of Skolem and Related Problems

upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize

Skew Circuits of Small Width

\#SAC^1

Collapsing Exact Arithmetic Hierarchies

) is relatively easy, establishing a uniform

Collapsing Exact Arithmetic Hierarchies.

\#SAC^1

Skew circuits of small width

bound needs a careful use of polynomial interpolation.

Graph Properties in Node-Query Setting: Effect of Breaking Symmetry

Motivated by a recent result of Elberfeld, Jakoby and Tantau [EJT10] showing that \(\mathsf {MSO}\) properties are Logspace computable on graphs of bounded treewidth, we consider the complexity of computing the determinant of the adjacency matrix of a bounded treewidth graph and as our main result prove that it is in Logspace. It is important to notice that the determinant is neither an \(\mathsf {MSO}\)-property nor counts the number of solutions of an \(\mathsf {MSO}\)-predicate. This technique yields Logspace algorithms for counting the number of spanning arborescences and directed Euler tours in bounded treewidth digraphs.

An Almost Cubic Lower Bound for ΣΠΣ Circuits Computing a Polynomial in VP.

Given a linear recurrence sequence (LRS), the Skolem problem, asks whether it ever becomes zero. The decidability of this problem has been open for several decades. Currently decidability is known only for LRS of order upto 4. For arbitrary orders (i.e., number of terms the n-th depends on), the only known complexity result is NP-hardness by a result of Blondel and Portier from 2002. In this paper, we give a different proof of this hardness result, which is arguably simpler and pinpoints the source of hardness. To demonstrate this, we identify a subclass of LRS for which the Skolem problem is in fact NP-complete. We show the generic nature of our lower-bound technique by adapting it to show stronger lower bounds of a related problem that encompasses many known decision problems on linear recurrent sequences.