arXiv subjects

Yogesh Dahiya

Publications and source records attributed to Yogesh Dahiya.

6 recordsLinked to original sources

arXiv2026

Quantum-Classical Equivalence for AND-Functions

A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form $$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), $$ when the outer function $f$ is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function $f$, the bounded-error quantum and classical deterministic communication complexities of the function $f \circ \mathrm{AND}_2$ are polynomially related, up to polylogarithmic factors in $n$. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of $f$. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.

cs.CC↗

arXiv2025

Self-Reinforced Preferential Attachment

We consider a preferential attachment random graph with self-reinforcement. Each time a new vertex comes in, it attaches itself to an old vertex with a probability that is proportional to the sum of the degrees of that old vertex at all prior times. The resulting growing graph is a random tree whose vertices have degrees that grow polynomially fast in time. We compute the growth exponent, show that it is strictly larger than the growth exponent in the absence of self-reinforcement, and develop insight into how the self-reinforcement affects the growth. Proofs are based on a stochastic approximation scheme.

math.PR↗

arXiv2025

Exact versus Approximate Representations of Boolean Functions in the De Morgan Basis

A seminal result of Nisan and Szegedy (STOC, 1992) shows that for any total Boolean function, the degree of the real polynomial that computes the function, and the minimal degree of a real polynomial that point-wise approximates the function, are at most polynomially separated. Extending this result from degree to other complexity measures like sparsity of the polynomial representation, or total weight of the coefficients, remains poorly understood. In this work, we consider this problem in the De Morgan basis, and prove an analogous result for the sparsity of the polynomials at a logarithmic scale. Our result further implies that the exact $\ell_1$ norm and its approximate variant are also similarly related to each other at a log scale. This is in contrast to the Fourier basis, where the analog of our results are known to be false. Our proof is based on a novel random restriction method. Unlike most existing random restriction methods used in complexity theory, our random restriction process is adaptive and is based on how various complexity measures simplify during the restriction process.

cs.CC↗

arXiv2024

Urns with Multiple Drawings and Graph-Based Interaction

Consider a finite undirected graph and place an urn with balls of two colours at each vertex. At every discrete time step, for each urn, a fixed number of balls are drawn from that same urn with probability $p$, and from a randomly chosen neighbour of that urn with probability $1-p$. Based on what is drawn, the urns then reinforce themselves or their neighbours. For every ball of a given colour in the sample, in case of Pólya-type reinforcement, a constant multiple of balls of that colour is added while in case of Friedman-type reinforcement, balls of the other colour are reinforced. These different choices for reinforcement give rise to multiple models. In this paper, we study the convergence of the fraction of balls of either colour across urns for all of these models. We show that in most cases the urns synchronize, that is, the fraction of balls of either colour in each urn converges to the same limit almost surely. A different kind of asymptotic behaviour is observed on bipartite graphs. We also prove similar results for the case of finite directed graphs.

math.PR↗

arXiv2022

On (Simple) Decision Tree Rank

In the decision tree computation model for Boolean functions, the depth corresponds to query complexity, and size corresponds to storage space. The depth measure is the most well-studied one, and is known to be polynomially related to several non-computational complexity measures of functions such as certificate complexity. The size measure is also studied, but to a lesser extent. Another decision tree measure that has received very little attention is the minimal rank of the decision tree, first introduced by Ehrenfeucht and Haussler in 1989. This measure is closely related to the logarithm of the size, but is not polynomially related to depth, and hence it can reveal additional information about the complexity of a function. It is characterised by the value of a Prover-Delayer game first proposed by Pudlák and Impagliazzo in the context of tree-like resolution proofs. In this paper we study this measure further. We obtain an upper bound on depth in terms of rank and Fourier sparsity. We obtain upper and lower bounds on rank in terms of (variants of) certificate complexity. We also obtain upper and lower bounds on the rank for composed functions in terms of the depth of the outer function and the rank of the inner function. This allow us to easily recover known asympotical lower bounds on logarithm of the size for Iterated AND-OR and Iterated 3-bit Majority. We compute the rank exactly for several natural functions and use them to show that all the bounds we have obtained are tight. We also show that rank in the simple decision tree model can be used to bound query complexity, or depth, in the more general conjunctive decision tree model. Finally, we improve upon the known size lower bound for the Tribes function and conclude that in the size-rank relationship for decision trees, obtained by Ehrenfeucht and Haussler, the upper bound for Tribes is asymptotically tight.

cs.CC↗

arXiv2015

Want Answers? A Reddit Inspired Study on How to Pose Questions

Questions form an integral part of our everyday communication, both offline and online. Getting responses to our questions from others is fundamental to satisfying our information need and in extending our knowledge boundaries. A question may be represented using various factors such as social, syntactic, semantic, etc. We hypothesize that these factors contribute with varying degrees towards getting responses from others for a given question. We perform a thorough empirical study to measure effects of these factors using a novel question and answer dataset from the website Reddit.com. To the best of our knowledge, this is the first such analysis of its kind on this important topic. We also use a sparse nonnegative matrix factorization technique to automatically induce interpretable semantic factors from the question dataset. We also document various patterns on response prediction we observe during our analysis in the data. For instance, we found that preference-probing questions are scantily answered. Our method is robust to capture such latent response factors. We hope to make our code and datasets publicly available upon publication of the paper.

cs.CL↗