arXiv subjects

Deepak Ponvel Chermakani

Publications and source records attributed to Deepak Ponvel Chermakani.

11 recordsLinked to original sources

arXiv2020

Lower bounds for the maximum number of runners that cause loneliness, and its application to Isolation

We consider (n+1) runners with given constant unique integer speeds running along the circumference of a circle whose circumferential length is one, and all runners starting from the same point. We define and give lower bounds to a first problem PMAX of finding, for every runner r, the maximum number of runners that can be simultaneously separated from runner r by a distance of atleast d. For d=1/(2^(floor(lg(n)))), a lower bound for PMAX is ( n - ((n-1)/floor(lg(n))) ), which makes the fraction of simultaneously separated runners tend to 1 as n tends to infinity. Next, we define and give upper bounds to a second problem ISOLATE of finding, for every runner r, the minimum number of steps needed to isolate r, assuming that the runners that can be simultaneously separated from r by atleast d, are removed at each step. For d=1/(2^(floor(lg(n)))), an upper bound for ISOLATE is ( lg(n - 1)/lg(floor(lg(n))) ).

cs.CG↗

arXiv2015

Optimal Aggregation of Blocks into Subproblems in Linear-Programs with Block-Diagonal-Structure

Wall-clock-time is minimized for a solution to a linear-program with block-diagonal-structure, by decomposing the linear-program into as many small-sized subproblems as possible, each block resulting in a separate subproblem, when the number of available parallel-processing-units is at least equal to the number of blocks. This is not necessarily the case when the parallel processing capability is limited, causing multiple subproblems to be serially solved on the same processing-unit. In such a situation, it might be better to aggregate blocks into larger sized subproblems. The optimal aggregation strategy depends on the computing-platform used, and minimizes the average-case running time for the set of subproblems. We show that optimal aggregation is NP-hard when blocks are of unequal size, and that optimal aggregation can be achieved within polynomial-time when blocks are of equal size.

cs.DM↗

arXiv2013

Efficiently determining Convergence in Polynomial Recurrence Sequences

We derive the necessary and sufficient condition, for a given Polynomial Recurrence Sequence to converge to a given target rational K. By converge, we mean that the Nth term of the sequence, is equal to K, as N tends to positive infinity. The basic idea of our approach is to construct a univariate polynomial equation in x, whose coefficients correspond to the terms of the Sequence. The approach then obtains the condition by analyzing five cases that cover all possible real values of x. The condition can be evaluated within time that is a polynomial function of the size of the description of the Polynomial Recurrence Sequence, hence convergence or non-convergence can be efficiently determined.

cs.DM↗

arXiv2012

NP-Hardness of optimizing the sum of Rational Linear Functions over an Asymptotic-Linear-Program

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a real-variable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NP-Complete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.

cs.CC↗

arXiv2012

NP-Completeness of deciding the feasibility of Linear Equations over binary-variables with coefficients and constants that are 0, 1, or -1

We convert, within polynomial-time and sequential processing, NP-Complete Problems into a problem of deciding feasibility of a given system S of linear equations with constants and coefficients of binary-variables that are 0, 1, or -1. S is feasible, if and only if, the NP-Complete problem has a feasible solution. We show separate polynomial-time conversions to S, from the SUBSET-SUM and 3-SAT problems, both of which are NP-Complete. The number of equations and variables in S is bounded by a polynomial function of the size of the NP-Complete problem, showing that deciding the feasibility of S is strongly-NP-Complete. We also show how to apply the approach used for the SUBSET-SUM problem to decide the feasibility of Integer Linear Programs, as it involves reducing the coefficient-magnitudes of variables to the logarithm of their initial values, though the number of variables and equations are increased.

cs.CC↗

arXiv2012

Efficiently expressing feasibility problems in Linear Systems, as feasibility problems in Asymptotic-Linear-Programs

We present a polynomial-time algorithm that obtains a set of Asymptotic Linear Programs (ALPs) from a given linear system S, such that one of these ALPs admits a feasible solution if and only if S admits a feasible solution. We also show how to use the same algorithm to determine whether or not S admits a non-trivial solution for any desired subset of its variables. S is allowed to consist of linear constraints over real variables with integer coefficients, where each constraint has either a lesser-than-or-equal-to, or a lesser-than, or a not-equal-to relational operator. Each constraint of the obtained ALPs has a lesser-than-or-equal-to relational operator, and the coefficients of its variables vary linearly with respect to the time parameter that tends to positive infinity.

cs.CC↗

arXiv2012

A Non-Triviality Certificate for Scalars and its application to Linear Systems

We present an approach of taking a linear weighted Average of N given scalars, such that this Average is zero, if and only if, all N scalars are zero. The weights for the scalars in this Average vary asymptotically with respect to a large positive real. We use this approach with a previous result on Asymptotic Linear Programming, to develop an O(M^4) Algorithm that decides whether or not a system of M Linear Inequalities is feasible, and, whether or not any desired subset of the variables in this system, is permitted to have a non-trivial solution.

cs.CC↗

arXiv2012

Expressing Reachability in Linear Recurrences, as Infinite Determinants and Rational Polynomial Equations

We present two tools, which could be useful in determining whether or not a non-Homogenous Linear Recurrence can reach a desired rational. First, we derive the determinant that is equal to the ith term in a non-Homogenous Linear Recurrence. We use this to derive the infinite determinant that is zero, if and only if, the desired rational can be reached by some term in the recurrence. Second, we derive an infinite summation of rational Polynomials, such that this summation can be equal to 1, if and only if, the desired rational can be reached by some term in the recurrence.

cs.DM↗

arXiv2010

Repeating Patterns in Linear Programs that express NP-Complete Problems

One of my recent papers transforms an NP-Complete problem into the question of whether or not a feasible real solution exists to some Linear Program. The unique feature of this Linear Program is that though there is no explicit bound on the minimum required number of linear inequalities, which is most probably exponential to the size of the NP-Complete problem, the Linear Program can still be described efficiently. The reason for this efficient description is that coefficients keep repeating in some pattern, even as the number of inequalities is conveniently assumed to tend to Infinity. I discuss why this convenient assumption does not change the feasibility result of the Linear Program. I conclude with two Conjectures, which might help to make an efficient decision on the feasibility of this Linear Program.

cs.CC↗

arXiv2010

A new Rational Generating Function for the Frobenius Coin Problem

An important question arising from the Frobenius Coin Problem is to decide whether or not a given monetary sum S can be obtained from N coin denominations. We develop a new Generating Function G(x), where the coefficient of x^i is equal to the number of ways in which coins from the given denominations can be arranged as a stack whose total monetary worth is i. We show that the Recurrence Relation for obtaining G(x), is linear, enabling G(x) to be expressed as a rational function, that is, G(x) = P(x)/Q(x), where both P(x) and Q(x) are Polynomials whose degrees are bounded by the largest coin denomination.

cs.DM↗

arXiv2008

Another approach to decide on real root existence for univariate Polynomials, and a multivariate extension for 3-SAT

We present six Theorems on the univariate real Polynomial, using which we develop a new algorithm for deciding the existence of atleast one real root for univariate integer Polynomials. Our algorithm outputs that no positive real root exists, if and only if, the given Polynomial is a factor of a real Polynomial with positive coefficients. Next, we define a transformation that transforms any instance of 3-SAT into a multivariate real Polynomial with positive coefficients, if and only if, the instance is not satisfiable.

math.NA↗