Somenath Biswas

Primality and identity testing via Chinese remaindering

We give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over Zn holds.We also give new randomized algorithms for testing if a multivariate polynomial, over a finite field or over rationals, is identically zero. The first of these algorithms also works over Zn for any n. The running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter. These algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods, for example, the Schwartz--Zippel test [Schwartz 1980; Zippel 1979], Chen--Kao and Lewin--Vadhan tests [Chen and Kao 1997; Lewin and Vadhan 1998].

Evolution and similarity evaluation of protein structures in contact map space

Prediction of fold from amino acid sequence of a protein has been an active area of research in the past few years, but the limited accuracy of existing techniques emphasizes the need to develop newer approaches to tackle this task. In this study, we use contact map prediction as an intermediate step in fold prediction from sequence. Contact map is a reduced graph‐theoretic representation of proteins that models the local and global inter‐residue contacts in the structure. We start with a population of random contact maps for the protein sequence and “evolve” the population to a “high‐feasibility” configuration using a genetic algorithm. A neural network is employed to assess the feasibility of contact maps based on their 4 physically relevant properties. We also introduce 5 parameters, based on algebraic graph theory and physical considerations, that can be used to judge the structural similarity between proteins through contact maps. To predict the fold of a given amino acid sequence, we predict a contact map that will sufficiently approximate the structure of the corresponding protein. Then we assess the similarity of this contact map with the representative contact map of each fold; the fold that corresponds to the closest match is our predicted fold for the input sequence. We have found that our feasibility measure is able to differentiate between feasible and infeasible contact maps. Further, this novel approach is able to predict the folds from sequences significantly better than a random predictor. Proteins 2005.

Computing single source shortest paths using single-objective fitness

Runtime analysis of evolutionary algorithms has become an important part in the theoretical analysis of randomized search heuristics. The first combinatorial problem where rigorous runtime results have been achieved is the well-known single source shortest path (SSSP) problem. Scharnow, Tinnefeld and Wegener [PPSN 2002, J. Math. Model. Alg. 2004] proposed a multi-objective approach which solves the problem in expected polynomial time. They also suggest a related single-objective fitness function. However, it was left open whether this does solve the problem efficiently, and, in a broader context, whether multi-objective fitness functions for problems like the SSSP yield more efficient evolutionary algorithms. In this paper, we show that the single objective approach yields an efficient (1+1) EA with runtime bounds very close to those of the multi-objective approach.

Approximation Algorithms for 3-D Commom Substructure Identification in Drug and Protein Molecules

Identifying the common 3-D substructure between two drug or protein molecules is an important problem in synthetic drug design and molecular biology. This problem can be represented as the following geometric pattern matching problem: given two point sets A and B in three-dimensions, and a real number Ɛ > 0, find the maximum cardinality subset S ⊆ A for which there is an isometry I, such that each point of I(S) is within Ɛ distance of a distinct point of B. Since it is difficult to solve this problem exactly, in this paper we have proposed several approximation algorithms with guaranteed approximation ratio. Our algorithms can be classified into two groups. In the first we extend the notion of partial decision algorithms for Ɛ-congruence of point sets in 2-D in order to approximate the size of S. All the algorithms in this class exactly satisfy the constraint imposed by Ɛ. In the second class of algorithms this constraint is satisfied only approximately. In the latter case, we improve the known approximation ratio for this class of algorithms, while keeping the time complexity unchanged. For the existing approximation ratio, we propose algorithms with substantially better running times. We also suggest several improvements of our basic algorithms, all of which have a running time of O(n8:5). These improvements consist of using randomization, and/or an approximate maximum matching scheme for bipartite graphs.

Polynomial isomorphism of 1-L-complete sets

Let C be any complexity class closed under log-lin reductions. It is shown that all complete sets for C under 1-L reductions are polynomial time isomorphic to one other. It is indicated how to generalize the result to reductions computed by finite-crossing machines.<<ETX>>

Necessary and sufficient conditions for success of the metropolis algorithm for optimization

This paper focusses on the performance of the Metropolis algorithm when employed for solving combinatorial optimization problems. One finds in the literature two notions of success for the Metropolis algorithm in the context of such problems. First, we show that both these notions are equivalent. Next, we provide two characterizations, or in other words, necessary and sufficient conditions, for the success of the algorithm, both characterizations being conditions on the family of Markov chains which the Metropolis algorithm gives rise to when applied to an optimization problem. The first characterization is that the Metropolis algorithm is successful if in every chain, for every set A of states not containing the optimum, the ratio of the ergodic flow out of A to the capacity of A is high. The second characterization is that in every chain the stationary probability of the optimum is high and that the family of chains mixes rapidly. We illustrate the applicability of our results by giving alternative proofs of certain known results.

Modeling Gene Regulatory Network in Fission Yeast Cell Cycle Using Hybrid Petri Nets

The complexity of models of gene regulatory network stems from the fact that such models are required to represent continuous, discrete, as well as stochastic aspects of gene regulation. Hybrid stochastic Petri nets as models can fairly incorporate all these aspects while keeping the model as simple as possible. This paper constructs an hybrid Petri net model of the fission yeast cell cycle regulation mechanism. Based on our simulation of the Petri net on an existing tool, we draw some conclusions about the regulation mechanism under study. We discuss the kinds of biologically significant questions that can be answered using hybrid Petri nets as models for genetic regulatory mechanisms.

Polynomial-Time Isomorphism of 1-L-Complete Sets

Let C be any complexity class closed under log-lin reductions. We show that all sets complete for C under 1-L reductions are polynomial-time isomorphic to each other. We also generalize the result to reductions computed byfinite-crossingmachines. As a corollary, we show that all sets complete for C under two-way DFA reductions are polynomial-time isomorphic to each other.

Metropolis algorithm for solving shortest lattice vector problem (SVP)

In this paper we study the suitability of the Metropolis Algorithm and its generalization for solving the shortest lattice vector problem (SVP). SVP has numerous applications spanning from robotics to computational number theory, viz., polynomial factorization. At the same time, SVP is a notoriously hard problem. Not only it is NP-hard, there is not even any polynomial approximation known for the problem that runs in polynomial time. What one normally uses is the LLL algorithm which, although a polynomial time algorithm, may give solutions which are an exponential factor away from the optimum. In this paper, we have defined an appropriate search space for the problem which we use for implementation of the Metropolis algorithm. We have defined a suitable neighbourhood structure which makes the diameter of the space polynomially bounded, and we ensure that each search point has only polynomially many neighbours. We can use this search space formulation for some other classes of evolutionary algorithms, e.g., for genetic and go-with-the-winner algorithms. We have implemented the Metropolis algorithm and Hastings generalization of Metropolis algorithm for the SVP. Our results are quite encouraging in all instances when compared with LLL algorithm.

On some bandwith restricted versions of the satisfiability problem of propositional CNF formulas

Abstract In the present paper we study the complexity of some restricted versions of the satisfiability problem for propositional CNF formulas. We define these restrictions through their corresponding languages which are identified using the self-reducibility property of satisfiable propositional CNF formulas. The notion of kernel constructibility (similar to self-reducibility) and that of bandwidth are used to define these languages. The results throw some light on the structure of the satisfiability problem. The proof methods illustrate the application of a certain method for reducing Turing machine acceptance problems to decision problems for logics.