arXiv subjects

Ijay Narang

Publications and source records attributed to Ijay Narang.

7 recordsLinked to original sources

arXiv2026

Structural Corrections to the Bethe Approximation of the Permanent

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

cs.DS↗

arXiv2026

The Hard-Core Model on Bipartite Spectral Expanders: Counting and Sampling at All Fugacities

We study approximate counting and sampling algorithms for the hard-core model on $\Delta$-regular bipartite graphs under a spectral expansion condition. Let $M_G$ be the biadjacency matrix of $G$. For every fixed $\xi\in(0,1)$, we give an FPRAS for the hard-core partition function and an efficient approximate sampler whenever \[ \lambda\leq \frac{1-\xi}{\sigma_2(M_G)}. \] The main idea is to introduce a family of quadratic tilts in the left-right occupation imbalance and show that each tilted measure can be sampled efficiently using Glauber dynamics. A discrete Gaussian identity expresses the original hard-core model as an exact positive mixture of these tilted measures; truncation and simulated annealing then yield efficient counting and sampling algorithms. For the complementary high-fugacity regime, we refine the polymer-model approach and show that the required phase-dominance and cluster expansion conditions follow from the singular-spectrum bound alone. Combining the two regimes, we obtain efficient approximate counting and sampling at every fugacity $\lambda>0$ whenever \[ \sigma_2(M_G)\leq c\left(\frac{\Delta^2}{\log(\mathrm e\Delta)}\right)^{1/3} \] for an absolute constant $c>0$. In particular, this recovers all-fugacity algorithms for random $\Delta$-regular bipartite graphs for all sufficiently large $\Delta$, while providing an efficiently verifiable certificate of their success on a given instance.

cs.DS↗

arXiv2026

Computational Thresholds for Balanced and Fixed-Slice Independent Sets in Bipartite Graphs

Motivated by recent work of Kocurek, Oveis Gharan, and Tjowasi, which gives an efficient sampling algorithm for the hard-core model on random regular bipartite graphs by decomposing into fixed-size slices, we study the worst-case tractability of approximate counting and sampling of fixed-size slices for bipartite independent set problems. Let $G=(L\sqcup R,E)$ be a bipartite graph with $|L|=|R|=n$ and maximum degree $\Delta$. The fixed-slice problem asks to sample uniformly from independent sets satisfying $|I\cap L|=\alpha_L n$ and $|I\cap R|=\alpha_R n$. We show that if the overall density $\alpha$ lies in the interval $(\frac{1}{\Delta}, \tfrac{1}{2})$, and the densities on the two sides are more balanced than the typical phase densities of a random $\Delta$-regular bipartite graph, then there is no FPRAS or efficient sampling scheme unless $\mathbf{NP}=\mathbf{RP}$. We then study a related fugacity model in which the densities are not fixed, but the independent set is required to be balanced between the two sides of the bipartition. For $\lambda>0$, the balanced hard-core model is the ordinary hard-core model with fugacity $\lambda$, conditioned on the event $|I\cap L|=|I\cap R|$. We prove that this model has the same computational threshold as the hard-core model on general bounded-degree graphs. That is, for every fixed $\Delta\ge 3$, if $\lambda<\lambda_c(\Delta)$, then the balanced partition function admits an FPTAS and the balanced hard-core distribution admits an efficient sampling scheme. Conversely, if $\lambda>\lambda_c(\Delta)$, then no FPRAS or efficient sampler exists on this graph class unless $\mathbf{NP}=\mathbf{RP}$.

cs.DS↗

arXiv2026

Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials

We introduce a robust OR polynomial framework for composing positive semidefinite certificates across OR constraints. We demonstrate the power of this method in two applications. The first is on acute-free families. A set $\mathcal F=\{(x_i^{(1)},\ldots,x_i^{(r)})\}_{i=1}^M \subseteq (S^{n-1})^r$ is $r$-way acute-free if, for every $i\neq j$, there is a coordinate $t\in[r]$ such that $\langle x_i^{(t)},x_j^{(t)}\rangle\leq 0$. We write $M_r(n)$ for the maximum size of such a set, and $M_r^{\pm}(n)$ for the hypercube restriction. On the hypercube, $r$-way acute-free sets are independent sets for some strong power graph $G_n^{\boxtimes r}$. The Lov\'asz theta number $\vartheta(G_n)$ is multiplicative but exponentially loose, whereas the Schrijver number $\vartheta'(G_n)$ gives the correct order, but is not multiplicative. We bypass this obstruction by proving a general quasi-tensorization result for the Schrijver number. That is, for every collection of graphs $G_1,\ldots, G_r$ satisfying $\vartheta'(G_i)\geq 2$, there is an absolute constant $C$ such that $\vartheta'(G_1\boxtimes \cdots \boxtimes G_r) \leq \prod_{i=1}^r \vartheta'(G_i)^{C\log r \log \vartheta'(G_i)}$. Applying this result gives that $M_r^{\pm}(n) \le M_r(n) \le (2n)^{C_0 r\log r\log(2n)}$ for some absolute constant $C_0$. The second application is on multicolor Ramsey numbers. The $r$-color Ramsey number $R_r(k)$ is the minimum $n$ such that every $r$-coloring of the edges of the complete graph on $n$ vertices contains a monochromatic copy of $K_k$. In a breakthrough result, Balister et al. [arXiv:2410.17197] showed that $R_r(k)\le \exp(-\Omega(k/r^{12}))r^{rk}$ via a geometric lemma. By improving the $r$ dependency in their geometric lemma via the OR polynomial framework, we prove that $R_r(k)\le \exp(-\Omega(k/(r^9(\log r)^6)))r^{rk}$.

math.CO↗

arXiv2026

Optimal detection of planted stars via a random energy model

We study the problem of detecting a planted star in the Erd{\H{o}}s--R{\'e}nyi random graph $G(n,m)$, formulated as a hypothesis test. We determine the scaling window for critical detection in $m$ in terms of the star size, and characterize the asymptotic total variation distance between the null and alternative hypotheses in this window. In the course of the proofs we show a condensation phase transition in the likelihood ratio that closely resembles that of the random energy model from spin glass theory.

math.ST↗

arXiv2026

Constant-Stepsize Stochastic Approximation: Finite-Time Convergence, Gaussian Approximation, and Tail Bounds

Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency, yet the distribution of the iterates is typically intractable. Classical asymptotics results give $X_k^{(\alpha)} \approx X^{(\alpha)} \approx x^\star+\sqrt{\alpha}Y$, where $X^{(\alpha)}$ is the steady state and $Y$ is an appropriate Gaussian limit, by progressively taking the time $k\uparrow\infty$ and stepsize $\alpha\downarrow0$. Such limit results, however, do not quantify finite-time, finite-stepsize errors. We develop an explicit pre-limit characterization for SA with i.i.d.\ and Markovian noise. We establish existence and uniqueness of the stationary law, a geometric Wasserstein convergence to stationarity, and almost-sure and $L^3$ convergence of the steady state to the root $x^\star$, identifying the scale $\sqrt{\alpha}$ as first-order fluctuation. At this scale, we derive a higher-order quantitative Gaussian approximation with a Wasserstein error, using Stein's method and Poisson equation techniques. We further obtain non-uniform Berry--Esseen-type tail bounds, incorporating both steady-state approximation and finite-time convergence errors. We instantiate the theory for strongly convex smooth SGD, linear SA, and nonlinear contractive SA. Beyond strong convexity, for general convex SGD, we identify a Gibbs limiting law and prove a pre-limit Wasserstein approximation error under stability and Stein-equation hypothesis, which are validated numerically.

cs.LG↗

arXiv2025

Sharp Inner Product Correlations for Hypercube Bijections

We resolve a conjecture of Rob Morris concerning bijections on the hypercube. Specifically, we show that for any bijection $f : \{-1,1\}^n \to \{-1,1\}^n$, \[ \Pr_{x,y \in \{-1,1\}^n}\big[ \langle x,y \rangle \ge 0 \;\text{and}\; \langle f(x),f(y) \rangle \ge 0 \big] \;\;\ge\; \tfrac{1}{4} - O(1/\sqrt{n}), \] implying the same lower bound for the joint event under any two bijections. Our proof proceeds by applying the spectral decomposition of the Hamming association scheme, which allows us to reformulate the problem as a linear program over the Birkhoff polytope. This makes it possible to isolate the contribution of the nontrivial spectrum, which we show is asymptotically negligible, leaving the dominant contribution arising from the principal eigenvalue.

math.CO↗