arXiv subjects

Chandra Kanta Mohapatra

Publications and source records attributed to Chandra Kanta Mohapatra.

3 recordsLinked to original sources

arXiv2026

Optimal Union Probability Interval Is NP-Hard

A problem dating back to Boole [Laws of Thought, Walton & Maberly,1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343--359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif.\ Intell.\ 28 (1986) 71--87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J.Complexity 4 (1988) 1--11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found.\ Trends Mach.\ Learn. 1 (2008) 1--305]. This paper (i) describes the pertinent geometry of Boole's problem via coordinate projections of an elementary polytope arising essentially from Hailperin's linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math.\ Programming 50 (1991) 395--414] and Boros et al. [Math.\ Oper.\ Res. 39 (2014) 1311--1329 and 51 (2026) 134--148].

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arXiv2022

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. And while \emph{nearly linear time} algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck \cite{BM74}, fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans \cite{Umans08} and Kedlaya \& Umans \cite{Kedlaya11} gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables $n$ is at most $d^{o(1)}$ where the degree of the input polynomial in every variable is less than $d$. They also stated the question of designing fast algorithms for the large variable case (i.e. $n \notin d^{o(1)}$) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field $\F_{q}$ of characteristic $p$ which evaluates an $n$-variate polynomial of degree less than $d$ in each variable on $N$ inputs in time $$\left((N + d^n)^{1 + o(1)}\text{poly}(\log q, d, p, n)\right)$$ provided that $p$ is at most $d^{o(1)}$, and $q$ is at most $(\exp(\exp(\exp(\cdots (\exp(d)))))$, where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. $n \notin d^{o(1)}$), this is the first {nearly linear} time algorithm for this problem over any (large enough) field.Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the applications to data structure upper bounds for polynomial evaluation and to an upper bound on the rigidity of Vandermonde matrices.

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arXiv2019

Schur Polynomials do not have small formulas if the Determinant doesn't!

Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Sta99], in Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84, Sta99]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [HRTS00, OW15] and Geometric complexity theory [IP17]. However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless \emph{every} polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class $\mathrm{VBP}$ is equal to the complexity class $\mathrm{VF}$, there exist Schur polynomials which do not have polynomial size algebraic formulas. As a consequence of our proof, we also show that computing the determinant of certain \emph{generalized} Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not \emph{multilinear}. A key ingredient of our proof is the study of composition of \emph{well behaved} algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest.

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