arXiv subjects

Sanyam Agarwal

Publications and source records attributed to Sanyam Agarwal.

6 recordsLinked to original sources

arXiv2026

Deterministic NC Quadratic Root Counting in Characteristic Two

Counting satisfying assignments of Boolean formulas is a basic problem in theoretical computer science, with $\#3\text{-}\SAT$ as the standard $\SharpP$-complete problem. More generally, counting the solutions of a system of polynomial equations over $\F_2$ is $\SharpP$-complete. Here we focus on the more structured problem of counting the solutions of a single polynomial equation. For polynomial equations over finite fields, Ehrenfeucht and Karpinski \cite{computationalcomplexityofxorandcountingproblems1990} showed a sharp difference between degrees two and three: quadratic root counting is solvable in polynomial time, while the degree-three problem is $\SharpP$-complete. Their quadratic algorithm is sequential. For fixed finite fields, Ishai et al.~\cite{ishai2012randomizing} later gave deterministic parallel algorithms in odd characteristic and randomized parallel algorithms in characteristic two. We give a deterministic $\NC$ algorithm for exactly counting the solutions of a quadratic polynomial equation over every fixed finite field of characteristic two. Our algorithm separates the radical and uses the absolute trace to realize the bit distinguishing the two nondegenerate finite-field types as the Arf invariant \cite{arf1941untersuchungen} of a quadratic form over $\F_2$. It then recovers that invariant from an integral matrix using Browder's determinant criterion \cite{browder2006complete}. This replaces the randomized canonical-form step in the algorithm of Ishai et al.

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arXiv2026

On the Principal Minor Expansion and Complexity of the Symmetrized Determinant

Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over all possible multiplication orders in the Leibniz formula for the determinant. He used the symmetrized determinant to design algorithms estimating the permanent of a matrix. To this end, he showed that there is a $O(n^{r+3})$ algorithm computing $\sdet$, where $r$ is the dimension of the algebra, and is therefore polynomial-time computable for fixed $r$. In this work, we study the algebraic properties and complexity of $\sdet$. While most of the properties of the ordinary determinant don't generalize to $\sdet$ defined on non-commutative algebras, we show that the principal minor expansion of the $\sdet$ is analogous to the ordinary determinant. Second, we prove that there exists a polynomial-sized algebra such that computing the symmetrized determinant is $\sharpP$-hard. Third, we show that the associated polynomial family is $\VNP$-complete over a suitable polynomial-dimensional algebra in the non-commutative setting. Further, when seen as a family of polynomials over the matrix algebra, it is also $\VNP$-complete in the commutative setting. This places the symmetrized determinant among the natural complete families arising from algebraic computation.

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arXiv2025

The Limits of Tractable Marginalization

Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can be commonly expressed by polynomial size arithmetic circuits computing multilinear polynomials. This raises the question, can all functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $\textsf{FP}\neq\#\textsf{P}$ (an assumption implied by $\textsf{P} \neq \textsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small circuits for that function's multilinear representation.

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arXiv2024

Probabilistic Generating Circuits -- Demystified

Zhang et al. (ICML 2021, PLMR 139, pp. 12447-1245) introduced probabilistic generating circuits (PGCs) as a probabilistic model to unify probabilistic circuits (PCs) and determinantal point processes (DPPs). At a first glance, PGCs store a distribution in a very different way, they compute the probability generating polynomial instead of the probability mass function and it seems that this is the main reason why PGCs are more powerful than PCs or DPPs. However, PGCs also allow for negative weights, whereas classical PCs assume that all weights are nonnegative. One of the main insights of our paper is that the negative weights are responsible for the power of PGCs and not the different representation. PGCs are PCs in disguise, in particular, we show how to transform any PGC into a PC with negative weights with only polynomial blowup. PGCs were defined by Zhang et al. only for binary random variables. As our second main result, we show that there is a good reason for this: we prove that PGCs for categorial variables with larger image size do not support tractable marginalization unless NP = P. On the other hand, we show that we can model categorial variables with larger image size as PC with negative weights computing set-multilinear polynomials. These allow for tractable marginalization. In this sense, PCs with negative weights strictly subsume PGCs.

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arXiv2017

AppTechMiner: Mining Applications and Techniques from Scientific Articles

This paper presents AppTechMiner, a rule-based information extraction framework that automatically constructs a knowledge base of all application areas and problem solving techniques. Techniques include tools, methods, datasets or evaluation metrics. We also categorize individual research articles based on their application areas and the techniques proposed/improved in the article. Our system achieves high average precision (~82%) and recall (~84%) in knowledge base creation. It also performs well in application and technique assignment to an individual article (average accuracy ~66%). In the end, we further present two use cases presenting a trivial information retrieval system and an extensive temporal analysis of the usage of techniques and application areas. At present, we demonstrate the framework for the domain of computational linguistics but this can be easily generalized to any other field of research.

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arXiv2016

Which techniques does your application use?: An information extraction framework for scientific articles

Every field of research consists of multiple application areas with various techniques routinely used to solve problems in these wide range of application areas. With the exponential growth in research volumes, it has become difficult to keep track of the ever-growing number of application areas as well as the corresponding problem solving techniques. In this paper, we consider the computational linguistics domain and present a novel information extraction system that automatically constructs a pool of all application areas in this domain and appropriately links them with corresponding problem solving techniques. Further, we categorize individual research articles based on their application area and the techniques proposed/used in the article. k-gram based discounting method along with handwritten rules and bootstrapped pattern learning is employed to extract application areas. Subsequently, a language modeling approach is proposed to characterize each article based on its application area. Similarly, regular expressions and high-scoring noun phrases are used for the extraction of the problem solving techniques. We propose a greedy approach to characterize each article based on the techniques. Towards the end, we present a table representing the most frequent techniques adopted for a particular application area. Finally, we propose three use cases presenting an extensive temporal analysis of the usage of techniques and application areas.

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