arXiv subjects

Alireza Haqi

Publications and source records attributed to Alireza Haqi.

7 recordsLinked to original sources

arXiv2026

On the Gap of Finite Posets

Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.

math.CO↗

arXiv2026

Parallel Sampling from the Ising $p$-Spin Model

We study the parallel complexity of sampling from the high-temperature Ising mixed $p$-spin Gibbs measure, a canonical instance of a mean-field spin glass on the hypercube $\{\pm 1\}^n$. We propose two different algorithms for this problem, corresponding to two different regimes of accuracy. Our first algorithm is a parallel implementation of a Markov chain known as block dynamics, combined with an approximate rejection sampling step that uses an Ising model in a novel way as a proposal distribution to approximate the quadratic interaction terms of the $p$-spin Hamiltonian. For any $\varepsilon > 0$, this algorithm runs in $n^{\tfrac{1}{3}}\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time with $\operatorname{poly}(n, \log(\tfrac{1}{\varepsilon}))$ work, and outputs a sample whose law is $\varepsilon$-close to the $p$-spin measure in total variation distance. Our second algorithm uses Picard iterations to parallelize the Algorithmic Stochastic Localization (ASL) process of El Alaoui, Montanari, and Sellke (2025), and for any $\varepsilon > \varepsilon_n$, takes $\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time and $\operatorname{poly}(\tfrac{n}{\varepsilon})$ work to produce a sample that is $\varepsilon$-close to the $p$-spin measure in the normalized 2-Wasserstein metric. Here, $\varepsilon_n > 0$ is a threshold that goes to $0$ as $n \to \infty$. Our result constitutes a doubly exponential improvement in the $\varepsilon$ dependence of the runtime and an exponential improvement in the $\varepsilon$ dependence of the total work when compared to na\"ive ASL, whose runtime scales as $\exp(\operatorname{poly}(\tfrac{1}{\varepsilon}))$.

cs.DS↗

arXiv2026

On Rounding on the Hypersimplex

We study correlated rounding on the hypersimplex, the base polytope of the uniform matroid. For each point $x$ in the hypersimplex, the goal is to sample a $k$-subset $A(x)$ with marginals $x$, while coupling the samples for all choices of $x$ so that nearby inputs produce nearby sets. We give conditional constant-stretch results for the natural maximum-entropy sequential scheme, based on a conjectured spectral property of the covariance matrix of the maximum-entropy distribution over $k$-subsets; this conjecture has been extensively tested numerically. Under this property, the scheme samples the maximum-entropy $k$-subset distribution with prescribed marginals using a common random ordering and common uniform thresholds. For every $x,y\in[0,1]^n$ with $\sum_i x_i=\sum_i y_i=k$, it satisfies \[ \mathbb{E}\!\left[|A(x)\triangle A(y)|\right] \le 6\|x-y\|_1 . \] Thus, conditional on the spectral hypothesis, the previous $O(\log k)$ bound for hypersimplex correlated rounding would improve to a constant and the open question raised by Naor, Raju, Shetty, Srinivasan, Valieva, and Wajc would have a positive answer. By adding dummy coordinates, the same conditional result gives stretch at most $12$ for the at-most-$k$ polytope.

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arXiv2026

On Thin Perfect Matchings up to Polylogarithmic Factors

We resolve the thin matching problem proposed by Anari, Charikar and Ramakrishnan [ACR23] up to polylogarithmic factors. Given a fractional perfect matching $x$, we say a perfect matching $M$ is $\alpha$-thin w.r.t. $x$ if for any cut $(S,\overline{S})$, we have $$ |M \cap E(S,\overline{S})| \leq \alpha\cdot x(S,\overline{S}).$$ [ACR23] conjectured that for any fractional perfect matching $x$, there exists a perfect matching $M$ which is $O(1)$-thin w.r.t. $x$. First, we show that if $M$ is restricted to be in the support of $x$, then $\alpha \geq \Omega(n)$ and we complement this by designing an efficient algorithm that outputs an $O(n\log n)$-thin perfect matching where $n$ is the number of vertices. Then, we relax this constraint and show that for any fractional perfect matching $x$, there is a perfect matching $M$ (which is not necessarily in the support of $x$) such that $M$ is $\text{polylog}(n)$-thin w.r.t. $x$. All results work for both bipartite and non-bipartite graphs. We also discuss applications to the metric distortion problem.

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arXiv2026

Fast Spanning Tree Sampling in Broadcast Congested Clique

We present the first polylogarithmic-round algorithm for sampling a random spanning tree in the (Broadcast) Congested Clique model. For any constant $c > 0$, our algorithm outputs a sample from a distribution whose total variation distance from the uniform spanning tree distribution is at most $O(n^{-c})$ in at most $c \cdot \log^{O(1)}(n)$ rounds. The exponent hidden in $\log^{O(1)}(n)$ is an absolute constant independent of $c$ and $n$. This is an exponential improvement over the previous best algorithm of Pemmaraju, Roy, and Sobel (PODC 2025) for the Congested Clique model.

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arXiv2025

Parallel Sampling via Autospeculation

We present parallel algorithms to accelerate sampling via counting in two settings: any-order autoregressive models and denoising diffusion models. An any-order autoregressive model accesses a target distribution $\mu$ on $[q]^n$ through an oracle that provides conditional marginals, while a denoising diffusion model accesses a target distribution $\mu$ on $\mathbb{R}^n$ through an oracle that provides conditional means under Gaussian noise. Standard sequential sampling algorithms require $\widetilde{O}(n)$ time to produce a sample from $\mu$ in either setting. We show that, by issuing oracle calls in parallel, the expected sampling time can be reduced to $\widetilde{O}(n^{1/2})$. This improves the previous $\widetilde{O}(n^{2/3})$ bound for any-order autoregressive models and yields the first parallel speedup for diffusion models in the high-accuracy regime, under the relatively mild assumption that the support of $\mu$ is bounded. We introduce a novel technique to obtain our results: speculative rejection sampling. This technique leverages an auxiliary ``speculative'' distribution~$\nu$ that approximates~$\mu$ to accelerate sampling. Our technique is inspired by the well-studied ``speculative decoding'' techniques popular in large language models, but differs in key ways. Firstly, we use ``autospeculation,'' namely we build the speculation $\nu$ out of the same oracle that defines~$\mu$. In contrast, speculative decoding typically requires a separate, faster, but potentially less accurate ``draft'' model $\nu$. Secondly, the key differentiating factor in our technique is that we make and accept speculations at a ``sequence'' level rather than at the level of single (or a few) steps. This last fact is key to unlocking our parallel runtime of $\widetilde{O}(n^{1/2})$.

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arXiv2022

Improving Order with Queues

Given a sequence of $n$ numbers and $k$ parallel First-in-First-Out (FIFO) queues, how close can one bring the sequence to sorted order? It is known that $k$ queues suffice to sort the sequence if the Longest Decreasing Subsequence (LDS) of the input sequence is at most $k$. But, what if the number of queues is too small for sorting completely? - We give a simple algorithm, based on Patience Sort, that reduces the LDS by $k - 1$. We also show, that the algorithm is optimal, i.e., for any $L > 0$ there exists a sequence of LDS $L$ such that the LDS cannot be reduced below $L - k + 1$ with $k$ queues. - Merging two sorted queues is at the core of Merge Sort. In contrast, two sequences of LDS two cannot always be merged into a sequence of LDS two. We characterize when it is possible and give an algorithm to decide whether it is possible. Merging into a sequence of LDS three is always possible. - A down-step in a sequence is an item immediately followed by a smaller item. We give an optimal algorithm for reducing the number of down-steps. The algorithm is online. Our research was inspired by an application in car manufacturing.

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