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Shayan Oveis Gharan

Publications and source records attributed to Shayan Oveis Gharan.

At least 19 recordsLinked to original sources
arXiv2026

Rank-1-perturbed trickledown theorems: Mixing time of Glauber dynamics for the Sherrington-Kirkpatrick model up to $β\leq \frac{1}{2}+\varepsilon$

We introduce a new family of trickledown theorems, a.k.a., local to global technique to bound the spectral gap of the Glauber dynamics for multi-state spin systems. In this technique instead of upper-bounding the influence matrix of a link of co-dimension 2 by $λI$ (where $λ$ is the second eigenvalue of the link), we upper-bound the influence matrix after a carefully chosen rank-1 shift. The rank-1 shift allows for a significantly smaller upper-bound but it comes at the cost of bounding the average loss due to rank-1 perturbations. As an application we use this method to show that the natural Glauber dynamics mixes in polynomial time to generate samples from the Sherrington-Kirkpatrick model for $β\leq \tfrac{1}{2}+\varepsilon$, for an absolute constant $\varepsilon>0$. At the heart of the proof we manage to bound the loss due to rank-1 perturbations by averaging over all links of co-dimension 2.

cs.DS↗
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 $α$-thin w.r.t. $x$ if for any cut $(S,\overline{S})$, we have $$ |M \cap E(S,\overline{S})| \leq α\cdot x(S,\overline{S}).$$ [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 $α\geq Ω(n)$ 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

High-Dimensional Expanders, the Sparsest Cut Problem, and Steurer's Conjecture

In 2010, Steurer conjectured that any family of $n$ unit-norm vectors $v_1,\dots,v_n$ with polynomially small average correlation $\mathbb{E}_{i,j}|\langle v_i,v_j\rangle|\leq n^{-ε}$ contains linear-sized constant-separated sets. We refute this conjecture in a strong sense using the machinery of sparse high-dimensional expanders: such vector families do not even have linear-sized $\frac{1}{\log^{1/4-o(1)}(n)}$-separated sets. Consequently, we show that there are families of vertex expanders on $n$ vertices for which the (average) $L_2$-mixing time to the uniform distribution of any reweighted simple random walk is at least $\log^{5/4-o(1)} n$.

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arXiv2026

Sampling from the Hardcore Model on Random Regular Bipartite Graphs above the Uniqueness Threshold

We design an efficient sampling algorithm to generate samples from the hardcore model on random regular bipartite graphs as long as $λ\lesssim \frac{1}{\sqrtΔ}$, where $Δ$ is the degree. Combined with recent work of Jenssen, Keevash and Perkins this implies an FPRAS for the partition function of the hardcore model on random regular bipartite graphs at any fugacity. Our algorithm is shown by analyzing two new Markov chains that work in complementary regimes. Our proof then proceeds by showing the corresponding simplicial complexes are top-link spectral expanders and appealing to the trickle-down theorem to prove fast mixing.

cs.DS↗
arXiv2026

Trickle-down Theorems via C-Lorentzian Polynomials II: Pairwise Spectral Influence and Improved Dobrushin's Condition

Let $μ$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$-partite simplicial complex with distribution $μ$ on maximal faces. For any pair of vertices $u,v\in V$, define the pairwise spectral influence $\mathcal{I}_{u,v}$ as follows. Let $σ$ be a choice of spins $s_w\in S_w$ for every $w\in V\setminus\{u,v\}$, and construct a matrix in $\mathbb{R}^{(S_u\cup S_v)\times (S_u\cup S_v)}$ where for any $s_u\in S_u, s_v\in S_v$, the $(us_u,vs_v)$-entry is the probability that $s_v$ is the spin of $v$ conditioned on $s_u$ being the spin of $u$ and on $σ$. Then $\mathcal{I}_{u,v}$ is the maximal second eigenvalue of this matrix, over all choices of spins for all $w\in V\setminus\{u,v\}$. Equivalently, $\mathcal{I}_{u,v}$ is the maximum local spectral expansion of links of codimension $2$ that include a spin for every $w \in V \setminus \{u,v\}$. We show that if the largest eigenvalue of the pairwise spectral influence matrix with entries $\mathcal{I}_{u,v}$ is bounded away from 1, i.e. $λ_{\max}(\mathcal{I})\leq 1-ε$ (and $X$ is connected), then the Glauber dynamics mixes rapidly and generate samples from $μ$. This improves/generalizes the classical Dobrushin's influence matrix as the $\mathcal{I}_{u,v}$ lower-bounds the classical influence of $u\to v$. As an application, we prove that the Glauber dynamics mixes rapidly up to (approximately) the phase transition for the multi-state hardcore model--a widely studied model in telecommunication networks and statistical physics (generalizing the hardcore model) introduced by Mazel and Suhov. As a by-product of our results, we also prove improved/almost optimal trickle-down theorems for partite simplicial complexes. Our proof builds on the trickle-down theorems via $\mathcal{C}$-Lorentzian polynomials machinery recently developed by the authors and Lindberg.

math.CO↗
arXiv2025

Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences

Let $X$ be a $d$-partite $d$-dimensional simplicial complex with parts $T_1,\dots,T_d$ and let $μ$ be a distribution on the facets of $X$. Informally, we say $(X,μ)$ is a path complex if for any $i<j<k$ and $F \in T_i,G \in T_j, K\in T_k$, we have $\mathbb{P}_μ[F,K | G]=\mathbb{P}_μ[F|G]\cdot\mathbb{P}_μ[K|G].$ We develop a new machinery with $\mathcal{C}$-Lorentzian polynomials to show that if all links of $X$ of co-dimension 2 have spectral expansion at most $1/2$, then $X$ is a $1/2$-local spectral expander. We then prove that one can derive fast-mixing results and log-concavity statements for top-link spectral expanders. We use our machinery to prove fast mixing results for sampling maximal flags of flats of distributive lattices (a.k.a. linear extensions of posets) subject to external fields, and to sample maximal flags of flats of "typical" modular lattices. We also use it to re-prove the Heron-Rota-Welsh conjecture and to prove a conjecture of Chan and Pak which gives a generalization of Stanley's log-concavity theorem. Lastly, we use it to prove near optimal trickle-down theorems for "sparse complexes" such as constructions by Lubotzky-Samuels-Vishne, Kaufman-Oppenheim, and O'Donnell-Pratt.

math.CO↗
arXiv2025

Unweighted One-Sided Code Sparsifiers and Thin Subgraphs

For a linear code $\mathcal{C} \subseteq \mathbb{F}_2^n$ and $α\in [0,1]$, call a set $S \subseteq [n]$ an (unweighted) one-sided $α$-sparsifier of $\mathcal{C}$ if for all $c \in \mathcal{C}$, $\mathrm{wt}(c_S)\geq α\cdot \mathrm{wt}(c)$, where $c_S$ is the projection of $c$ onto the coordinates in $S$ and $\mathrm{wt}(c)$ is the Hamming weight of $c$. \\ We show that every $k$-dimensional linear code $\mathcal{C}\subseteq \mathbb{F}_2^n$ has at least $2^{n - k}$ many unweighted one-sided $1/2$-sparsifiers and hence one of size at most $n/2 + O(\sqrt{n k})$. As an application, letting $\mathcal{C} \subseteq \mathbb{F}_2^E$ denote the cut-space of a graph $G=(V, E)$, we show a lower bound of $2^{\lvert E \rvert- (\lvert V \rvert - 1)}$ on the number of $1/2$-thin subgraphs of $G$ and the existence of a $1/2$-thin subgraph with at least $\lvert E \rvert /2-O(\sqrt{\lvert E \rvert \cdot \lvert V \rvert})$ edges. In contrast to previous results on thin subgraphs, our proofs are purely "combinatorial".

math.CO↗
arXiv2024

On approximability of the Permanent of PSD matrices

We study the complexity of approximating the permanent of a positive semidefinite matrix $A\in \mathbb{C}^{n\times n}$. 1. We design a new approximation algorithm for $\mathrm{per}(A)$ with approximation ratio $e^{(0.9999 + γ)n}$, exponentially improving upon the current best bound of $e^{(1+γ-o(1))n}$ [AGOS17,YP22]. Here, $γ\approx 0.577$ is Euler's constant. 2. We prove that it is NP-hard to approximate $\mathrm{per}(A)$ within a factor $e^{(γ-ε)n}$ for any $ε>0$. This is the first exponential hardness of approximation for this problem. Along the way, we prove optimal hardness of approximation results for the $\|\cdot\|_{2\to q}$ ``norm'' problem of a matrix for all $-1 < q < 2$.

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arXiv2023

An Improved Trickle-Down Theorem for Partite Complexes

We prove a strengthening of the trickle down theorem for partite complexes. Given a $(d+1)$-partite $d$-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are $\frac{1-δ}{d}$-(one-sided) spectral expanders, then the link of any face of co-dimension $k$ is an $O(\frac{1-δ}{kδ})$-(one-sided) spectral expander, for all $3\leq k\leq d+1$. For an application, using our theorem as a black-box, we show that links of faces of co-dimension $k$ in recent constructions of bounded degree high dimensional expanders have spectral expansion at most $O(1/k)$ fraction of the spectral expansion of the links of the worst faces of co-dimension $2$.

cs.DM↗
arXiv2023

On Optimization and Counting of Non-Broken Bases of Matroids

Given a matroid $M=(E,{\cal I})$, and a total ordering over the elements $E$, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in ${\cal I}$ with no broken circuit. The set of NBC independent sets of any matroid $M$ define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.

cs.CC↗
arXiv2023

Complete Log Concavity of Coverage-Like Functions

We introduce an expressive subclass of non-negative almost submodular set functions, called strongly 2-coverage functions which include coverage and (sums of) matroid rank functions, and prove that the homogenization of the generating polynomial of any such function is completely log-concave, taking a step towards characterizing the coefficients of (homogeneous) completely log-concave polynomials. As a consequence we obtain that the "level sets" of any such function form an ultra-log concave sequence.

math.CO↗
arXiv2022

A (Slightly) Improved Deterministic Approximation Algorithm for Metric TSP

We show that the max entropy algorithm can be derandomized (with respect to a particular objective function) to give a deterministic $3/2-ε$ approximation algorithm for metric TSP for some $ε> 10^{-36}$. To obtain our result, we apply the method of conditional expectation to an objective function constructed in prior work which was used to certify that the expected cost of the algorithm is at most $3/2-ε$ times the cost of an optimal solution to the subtour elimination LP. The proof in this work involves showing that the expected value of this objective function can be computed in polynomial time (at all stages of the algorithm's execution).

cs.DS↗
arXiv2022

A (Slightly) Improved Bound on the Integrality Gap of the Subtour LP for TSP

We show that for some $ε> 10^{-36}$ and any metric TSP instance, the max entropy algorithm returns a solution of expected cost at most $\frac{3}{2}-ε$ times the cost of the optimal solution to the subtour elimination LP. This implies that the integrality gap of the subtour LP is at most $\frac{3}{2}-ε$. This analysis also shows that there is a randomized $\frac{3}{2}-ε$ approximation for the 2-edge-connected multi-subgraph problem, improving upon Christofides' algorithm.

cs.DS↗
arXiv2021

Matroid Partition Property and the Secretary Problem

A matroid $\mathcal{M}$ on a set $E$ of elements has the $α$-partition property, for some $α>0$, if it is possible to (randomly) construct a partition matroid $\mathcal{P}$ on (a subset of) elements of $\mathcal{M}$ such that every independent set of $\mathcal{P}$ is independent in $\mathcal{M}$ and for any weight function $w:E\to\mathbb{R}_{\geq 0}$, the expected value of the optimum of the matroid secretary problem on $\mathcal{P}$ is at least an $α$-fraction of the optimum on $\mathcal{M}$. We show that the complete binary matroid, ${\cal B}_d$ on $\mathbb{F}_2^d$ does not satisfy the $α$-partition property for any constant $α>0$ (independent of $d$). Furthermore, we refute a recent conjecture of Bérczi, Schwarcz, and Yamaguchi by showing the same matroid is $2^d/d$-colorable but cannot be reduced to an $α2^d/d$-colorable partition matroid for any $α$ that is sublinear in $d$.

cs.DS↗
arXiv2021

A Matrix Trickle-Down Theorem on Simplicial Complexes and Applications to Sampling Colorings

We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree $Δ$ whenever the number of colors is at least $q\geq (\frac{10}{3} + ε)Δ$, where $ε>0$ is arbitrary and the maximum degree satisfies $Δ\geq C$ for a constant $C = C(ε)$ depending only on $ε$. For edge-colorings, this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-ε_0 ) Δ$, where $ε_0 \approx 10^{-5}$ is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenheim's influential result, as a new technique to prove that a high dimensional simplical complex is a local spectral expander.

cs.DS↗
arXiv2021

Log-Concave Polynomials IV: Approximate Exchange, Tight Mixing Times, and Near-Optimal Sampling of Forests

We prove tight mixing time bounds for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomials. For a matroid of rank $k$ on a ground set of $n$ elements, or more generally distributions associated with log-concave polynomials of homogeneous degree $k$ on $n$ variables, we show that the down-up random walk, started from an arbitrary point in the support, mixes in time $O(k\log k)$. Our bound has no dependence on $n$ or the starting point, unlike the previous analyses [ALOV19,CGM19], and is tight up to constant factors. The main new ingredient is a property we call approximate exchange, a generalization of well-studied exchange properties for matroids and valuated matroids, which may be of independent interest. In particular, given function $μ: {[n] \choose k} \to \mathbb{R}_{\geq 0},$ our approximate exchange property implies that a simple local search algorithm gives a $k^{O(k)}$-approximation of $\max_{S} μ(S)$ when $μ$ is generated by a log-concave polynomial, and that greedy gives the same approximation ratio when $μ$ is strongly Rayleigh. As an application, we show how to leverage down-up random walks to approximately sample random forests or random spanning trees in a graph with $n$ edges in time $O(n\log^2 n).$ The best known result for sampling random forest was a FPAUS with high polynomial runtime recently found by \cite{ALOV19, CGM19}. For spanning tree, we improve on the almost-linear time algorithm by [Sch18]. Our analysis works on weighted graphs too, and is the first to achieve nearly-linear running time for these problems.

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