arXiv subjects
Publications and source records attributed to Vishesh Jain.
For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.
For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \geq 4$, then \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{2n/g}\operatorname{Bethe}(A).\] The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.
Sampling graph colorings via local Markov chains is a central problem in approximate counting and Markov chain Monte Carlo (MCMC). We address the problem of sampling a random $k$-coloring of a graph with maximum degree $Δ$. The simplest algorithmic approach is to establish rapid mixing of the single-site update chain known as the Metropolis Glauber dynamics, which at each step chooses a random vertex $v$ and proposes a random color $c$, recoloring $v$ to $c$ if the resulting coloring remains proper. It is a long-standing open problem to prove that the Glauber dynamics has polynomial mixing time on all graphs whenever $k\geqΔ+2$. We prove that for every $δ>0$ and all $Δ\geq Δ_0(δ)$, if $k\ge (1+δ)Δ$ then the Glauber dynamics has optimal mixing time of $O_δ(|V| \log |V|)$ on any graph of girth $\geq 7$ and maximum degree $Δ$. Our approach builds on a non-Markovian coupling introduced by Hayes and Vigoda (2003) for the large-degree regime $Δ=Ω(\log n)$ and girth $11$, in which updates at time $t$ may depend on and modify proposed updates at future times. A complete analysis of this framework requires resolving substantial technical obstacles that remain in the original argument, and extending it to the constant-degree regime introduces further difficulties, since non-Markovian updates may fail with constant probability. We overcome these obstacles by developing and analyzing a refined local non-Markovian coupling, and by establishing new local-uniformity results for the Metropolis dynamics, extending prior results for the heat-bath chain due to Hayes (2013). Together, these ingredients provide a complete analysis of the non-Markovian coupling framework in the large-degree regime, while simultaneously strengthening it substantially to obtain optimal mixing all the way down to the constant-degree setting.
The Johnson--Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O(\varepsilon^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\varepsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameters $n,d, \varepsilon$ is \[ Θ\left(\min\left\{d,n-1,\frac{\log(2+\varepsilon^2n)}{\varepsilon^2}\right\}\right). \] We resolve this conjecture in the affirmative. In fact, we prove the stronger statement that the upper bound is attained by a linear map. The matching lower bound, due to Larsen--Nelson and Alon--Klartag, holds even for nonlinear embeddings.
We consider the problem of graphically local metric embedding, i.e. embedding points from an arbitrary finite metric space into a target metric space while preserving, up to a small distortion, only a subset of the pairwise distances specified by a bounded degree graph $G$. We provide a general reduction showing that, in many cases, this is no easier than embedding the points while approximately preserving all pairwise distances. As an illustration of our general reduction, we show that there exists a Euclidean metric space $X$ on $n$ points along with a graph $G = (X,E)$ of maximum degree $3$ such that any embedding of $X$ into $\ell_2^m$ which only preserves distances specified by $E$ up to a relative error of $(1+\varepsilon)$ must satisfy $m = Ω(\varepsilon^{-2}\log n)$. Our lower bound matches the upper bound on the dimension coming from the Johnson-Lindenstrauss lemma for approximately preserving all pairwise distances; previously, such a lower bound was known only for the class of noncontracting embeddings [Schechtman-Shraibman, Discrete & Computational Geometry, 2009]. Moreover, the condition that the maximum degree of the graph is $3$ is best possible: for graphs $G$ of maximum degree $2$ (or more generally, treewidth at most $2$), any metric space embeds $G$-isometrically into any two-dimensional normed space.
We prove cutoff in total variation distance for the discrete-time Kac walk on $S^{n-1}$ started from a coordinate vector. The cutoff occurs at $ C_{\mathrm{BRW}}n\log n$, where $C_{\mathrm{BRW}} \approx 3.8916$ is an explicit constant determined by the speed of the leftmost particle in a branching random walk. In particular, the cutoff location is not at the conjectured time $ 2n\log n$.
Consider the following online filling game. An array of length $n$ is initially empty. At each time step one observes an independent sample from $\mathrm{Unif}[0,1]$ and must either discard it or place it irrevocably into an empty position of the array, while preserving the constraint that the occupied entries are non-decreasing from left to right. Among all possible strategies, what is the optimal expected time required to fill the array? Let $v_n$ denote this optimal expected completion time. Our main result determines $v_n$ up to lower-order terms: \[ v_n=\left(\frac12+o(1)\right)n\log n. \] More precisely, no strategy, even if randomized and adaptive, can have expected completion time below $\left(\frac12-o(1)\right)n\log n$, while we provide an explicit deterministic strategy whose expected completion time is at most $\left(\frac12+o(1)\right)n\log n$. For comparison, the natural coupon-collector strategy, which partitions $[0,1]$ into $n$ equal intervals and reserves one array position for each interval, has expected completion time $(1+o(1))n\log n$. We also consider a with-replacement version of the game, in which previously placed entries may be overwritten. For this variant, we give a deterministic strategy with expected completion time $O(n\sqrt{\log n})$, thereby establishing a separation between the two models.
In 1984, Frankl and Füredi asked for the maximum density of an $n$-vertex $r$-graph in which every $(r+1)$-set of vertices spans $0$ or $2$ edges. They gave a construction with asymptotic density $2^{1-r}$. We significantly improve this bound by constructing such $r$-graphs with density $Ω(r^{-3})$, thereby improving the dependence on $r$ from exponential to polynomial. We also obtain lower bounds for the more general problem in which every $(r+1)$-set spans an even number of edges from $\{0,2,\ldots,2k\}$.
Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state $|ψ\rangle$. Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform $|ψ\rangle$ into another state $|ϕ\rangle$. Famously, Nielsen's theorem [Phys. Rev. Lett., 1999] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the local spectrum of $|ϕ\rangle$ should majorise the local spectrum of $|ψ\rangle$). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states $|ψ\rangle, |ϕ\rangle$ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states $|ψ\rangle, |ϕ\rangle$ are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication. Via Nielsen's theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices from the so-called trace-normalised complex Wishart-Laguerre ensemble. Concretely, let $X$ and $Y$ be independent $n \times m$ random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen's conjecture says that the probability that the spectrum of $X X^\dagger / \operatorname{tr}(X X^\dagger)$ majorises the spectrum of $Y Y^\dagger / \operatorname{tr}(Y Y^\dagger)$ tends to zero as both $n$ and $m$ grow large. We prove this conjecture, and we also confirm some related predictions of Cunden, Facchi, Florio and Gramegna [J. Phys. A., 2020; Phys. Rev. A., 2021].
For a graph $G$, let $τ(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $τ(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for all sufficiently large $n$. This is optimal up to the choice of the constant $c$ and resolves a conjecture of Chan-Kontorovich-Pak regarding a problem of Sedláček from the late 1960s.
Quantile shares, introduced by Babichenko, Feldman, Holzman, and Narayan [STOC 2024], offer an ordinal, self-maximizing, and interpretable benchmark for fair division of indivisible goods, but their universal feasibility is known only conditional on the rainbow Erdős matching conjecture (EMC). Specifically, Babichenko et al. showed that assuming the rainbow EMC in the near-perfect matching regime, the $(1/2e)$-quantile share is universally feasible. In contrast, a simple argument shows that the $q$-quantile share can be infeasible for any $q > 1/e$. We introduce a one-parameter refinement of quantile shares, the $c$-thinned quantile share, obtained by thinning the inclusion probability in the random benchmark bundle by a factor of $c$ for a fixed constant $c\in(0,1]$. Our main result is that there exists a universal constant $c >0$ for which the $c$-thinned $e^{-c}$-quantile share is unconditionally universally feasible; this is best possible in the sense that for any $c \in (0,1]$, the $c$-thinned $q$-quantile share can be infeasible for any $q > e^{-c}$. Prior to this work, the only nontrivial share known to be universally feasible was Feige's residual maximin share. The thinning viewpoint also lets us remove the factor-two loss in the conditional result for the original quantile share: assuming the rainbow EMC, the $(1/e)$-quantile share is universally feasible.
Entropic independence is a structural property of measures that underlies modern proofs of functional inequalities, notably (modified) log-Sobolev inequalities, via ``annealing'' or local-to-global schemes. Existing sufficient criteria for entropic independence typically require spectral independence and/or uniform bounds on marginals under \emph{all} pinnings, which can fail in natural canonical-ensemble models even when strong mixing properties are expected. We introduce \emph{sparse localization}: a restricted localization framework, in the spirit of Chen--Eldan, in which one assumes $\ell_2$-independence only for a sparse family of pinnings (those fixing at most $cn$ coordinates for any $c > 0$), yet still deduces quadratic entropic stability and entropic independence with an explicit multiplicative loss of order $c^{-1}$. As an application, we give a rigorous proof of approximate conservation of entropy for the uniform distribution on independent sets of a given size in bounded degree graphs.
We study the adjacent-transposition chain on the symmetric group $\mathfrak{S}_n$ with a regular parameter vector $\vec{p} = (p_{i,j})_{i\neq j}$. Fill's spectral gap conjecture, recently resolved in the affirmative by Greaves-Zhu, states that among all regular parameter vectors, the spectral gap of the transition matrix is minimized by the uniform vector $p_{i,j}= 1/2$ for all $i\neq j$. We prove the stronger statement that among all regular parameter vectors, the spectral gap is minimized if and only if $\vec{p}$ has a neutral label, i.e., there exists $c \in [n]$ such that $p_{c,i} = 1/2$ for all $i\neq c$. Moreover, in this case, we show that the multiplicity of the second largest eigenvalue is equal to the number of neutral labels, unless the number of neutral labels is $n-2$ or $n$, in which case the multiplicity is $n-1$. This confirms a conjecture of Fill.
Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be exactly 1 (by taking $G$ to be empty or complete), but for all other values of $\ell$, one might suspect that this fraction is always significantly smaller than 1. In this paper we prove an essentially optimal result along these lines: if $\ell$ is not 0 or $\binom{k}{r}$, then this fraction is at most $(1/e) + \varepsilon$, assuming $k$ is sufficiently large in terms of $r$ and $\varepsilon>0$, and $G$ is sufficiently large in terms of $k$. Previously, this was only known for a very limited range of values of $r,k,\ell$ (due to Kwan-Sudakov-Tran, Fox-Sauermann, and Martinsson-Mousset-Noever-Trujić). Our result answers a question of Alon-Hefetz-Krivelevich-Tyomkyn, who suggested this as a hypergraph generalisation of their "edge-statistics conjecture". We also prove a much stronger bound when $\ell$ is far from 0 and $\binom{k}{r}$.
The homogeneous Kuramoto model on a graph $G = (V,E)$ is a network of $|V|$ identical oscillators, one at each vertex, where every oscillator is coupled bidirectionally (with unit strength) to its neighbors in the graph. A graph $G$ is said to be globally synchronizing if, for almost every initial condition, the homogeneous Kuramoto model converges to the all-in-phase synchronous state. Confirming a conjecture of Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend, we show that with high probability, the random graph process becomes globally synchronizing as soon as it is connected. This is best possible, since connectivity is a necessary condition for global synchronization.
Consider shuffling a deck of $n$ cards, labeled $1$ through $n$, as follows: at each time step, pick one card uniformly with your right hand and another card, independently and uniformly with your left hand; then swap the cards. How long does it take until the deck is close to random? Diaconis and Shahshahani showed that this process undergoes cutoff in total variation distance at time $t = \lfloor n\log{n}/2 \rfloor$. Confirming a conjecture of N.~Berestycki, we prove the definitive ``hitting time'' version of this result: let $τ$ denote the first time at which all cards have been touched. The total variation distance between the stopped distribution at $τ$ and the uniform distribution on permutations is $o_n(1)$; this is best possible, since at time $τ-1$, the total variation distance is at least $(1+o_n(1))e^{-1}$.
For a finite graph $F$ and a value $p \in [0,1]$, let $I(F,p)$ denote the largest $y$ for which there is a sequence of graphs of edge density approaching $p$ so that the induced $F$-density of the sequence approaches $y$. We show that for all $F$ on at least three vertices and all $p \in (0,1)$, the binomial random graph $G(n,p)$ has induced $F$-density strictly less than $I(F,p).$ This provides a negative answer to a problem posed by Liu, Mubayi and Reiher. Our approach is in the limiting setting of graphons, and we in fact show a stronger result: the binomial random graph is never a \emph{local} maximum in the space of graphons of edge density $p$. This is done by finding a sequence of balanced perturbations of arbitrarily small norm that increase the $F$-density.
Let $G = (V,E)$ be a graph on $n$ vertices and let $m^*(G)$ denote the size of a maximum matching in $G$. We show that for any $δ> 0$ and for any $1 \leq k \leq (1-δ)m^*(G)$, the down-up walk on matchings of size $k$ in $G$ mixes in time polynomial in $n$. Previously, polynomial mixing was not known even for graphs with maximum degree $Δ$, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time $O_{Δ,δ}(n\log{n})$. In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting.