Pritish Kamath

Arithmetic Circuits: A Chasm at Depth Three

We show that, over Q, if an n-variate polynomial of degree d = n^O(1) is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O(√(d log n log d log s))) (respectively of size exp(O(√(d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O(√(d log d))) computing the d × d determinant Det_d. It also means that if we can prove a lower bound of exp(omega(√(d log d))) on the size of any ΣΠΣ-circuit computing the d × d permanent Perm_d then we get super polynomial lower bounds for the size of any arithmetic branching program computing Perm_d. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Det_d or Perm_d, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)).

Approaching the Chasm at Depth Four

Agrawal-Vinay [AV08] and Koiran [Koi12] have recently shown that an exp(ω(√n log2 n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to super-polynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin. We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp(Ω(√n)).

Faster Algorithms for Alternating Refinement Relations

One central issue in the formal design and analysis of reactive systems is the notion of refinement that asks whether all behaviors of the implementation is allowed by the specification. The local interpretation of behavior leads to the notion of simulation. Alternating transition systems (ATSs) provide a general model for composite reactive systems, and the simulation relation for ATSs is known as alternating simulation. The simulation relation for fair transition systems is called fair simulation. In this work our main contributions are as follows: (1) We present an improved algorithm for fair simulation with Buchi fairness constraints; our algorithm requires O(n^3 * m) time as compared to the previous known O(n^6)-time algorithm, where n is the number of states and m is the number of transitions. (2) We present a game based algorithm for alternating simulation that requires O(m^2)-time as compared to the previous known O((n*m)^2)-time algorithm, where n is the number of states and m is the size of transition relation. (3) We present an iterative algorithm for alternating simulation that matches the time complexity of the game based algorithm, but is more space efficient than the game based algorithm.

Using dominances for solving the protein family identification problem

Identification of protein families is a computational biology challenge that needs efficient and reliable methods. Here we introduce the concept of dominance and propose a novel combined approach based on Distance Alignment Search Tool (DAST), which contains an exact algorithm with bounds. Our experiments show that this method successfully finds the most similar proteins in a set without solving all instances.

Arithmetic Circuits: A Chasm at Depth 3

We show that, over

Preservation under Substructures modulo Bounded Cores

{\mathbb Q}

Improved bounds for universal one-bit compressive sensing

, if an

Monotone circuit lower bounds from resolution

n

An exponential lower bound for homogeneous depth four arithmetic circuits with bounded bottom fanin.

-variate polynomial of degree

Arithmetic circuits: A chasm at depth three.

d = n^{O(1)}