arXiv subjects

Steven Heilman

Publications and source records attributed to Steven Heilman.

At least 19 recordsLinked to original sources

arXiv2026

Sharp Hardness for MAX-3-CUT and Quantum MAX-CUT

Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $\alpha_3+\epsilon$ for every $\epsilon>0$, where $\alpha_3\approx.83600811464$ is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in $[-1/2,2/5]$ and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of $\alpha_{\rm BOV}+\epsilon$ for every $\epsilon>0$, where $\alpha_{\rm BOV}\approx 0.9563372685$ is the approximation ratio of the Bri\"et-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their $S^{k-1}$-valued Borell inequality for correlations in $[-.5843,.5843]$ for all $k\geq3$.

cs.CC↗

arXiv2026

On the Convergence of Adam, Revisited

We show that projected Adam for online optimization with arbitrary moment decay parameters $\beta_1,\beta_2\in[0,1)$ can have average regret bounded away from zero. A similar result of Reddi-Kale-Kumar from 2018 required $\beta_1<\sqrt{\beta_2}$. Similar to their result, we use a three-periodic sequence of linear functions on $[-1,1]$ with slopes $c,-1,-1$, though we use $c$ slightly larger than $2$. This nonzero average regret result extends to Adam variants such as AdamW, RMSProp, NAdam, Adan, AdaMax, Muon, and to an i.i.d. variant of the three-periodic sequence of slopes for Adam.

cs.LG↗

arXiv2026

An Upper Bound on Grothendieck's Constant

We show that Grothendieck's real constant $K_G$ can be upper bounded by projecting vectors onto a random plane through the origin and thresholding a degree five Hermite polynomial. This resolves a conjecture of Braverman-Makarychev-Makarychev-Naor from 2011, who required an extra randomization step in their rounding scheme and proved $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-500}$. As a corollary of our result, we prove the bound $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-217}$ by thresholding degree three Hermite polynomials in the plane. We finally give a rigorous computer-assisted proof that $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-5}$ using interval arithmetic and degree three Hermite polynomial thresholding.

math.FA↗

arXiv2026

Trees and Graphs with Non Log-concave Dominating Set Sequence via AI Tools

We give new examples of graphs and trees with dominating set sequences that are not log-concave. These examples were generated by PatternBoost, a transformer-based reinforcement learning software developed by Charton-Ellenberg-Wagner-Williamson. We also show: for any positive integer $m$, there exists a tree whose dominating set sequence is not log-concave for at least $m$ indices by modifying a similar construction of Bautista-Ramos for the independent set sequence. We show that a large class of caterpillar graphs has log-concave dominating set sequences. A continuous analogue of the sequence is also log-concave for all graphs.

math.CO↗

arXiv2026

Independent Sets and Continued Fractions

Linek's 1989 problem asks whether the numbers of independent sets of trees avoid infinitely many positive integers. We show that the set of natural numbers realized as the number of independent sets of a tree has a lower growth exponent of $0.1966$. We further prove that the set of positive integers representable by connected planar graphs has asymptotic density one. Lastly, we establish a phase transition: the number of independent sets of graphs with fewer than $d|V|$ edges for any $d<1$ is contained in a set of density zero, whereas, following Shkredov's recent breakthrough on Zaremba's conjecture in continued fraction theory, there exists a constant $D$ such that the number of independent sets of graphs with at most $D|V|$ edges covers all positive integers.

math.CO↗

arXiv2026

A Lower Bound for Grothendieck's Constant

We show that Grothendieck's real constant $K_{G}$ satisfies $K_G\geq c+10^{-26}$, improving on the lower bound of $c=1.676956674215576\ldots$ of Davie and Reeds from 1984 and 1991, respectively.

math.FA↗

arXiv2025

A Fourier approach to Levine's hat puzzle

We consider Lionel Levine's notorious hat puzzle with two players. Each player has a stack of hats on their head, and each hat is chosen independently to be either black or white. After observing only the other player's hats, players simultaneously choose one of their own hats. The players win if both chosen hats are black. In this note, we observe an upper bound on the probability of success, using Chang's lemma, a result in Boolean harmonic analysis.

math.PR↗

arXiv2023

Sphere Valued Noise Stability and Quantum MAX-CUT Hardness

We prove a vector-valued inequality for the Gaussian noise stability (i.e. we prove a vector-valued Borell inequality) for Euclidean functions taking values in the two-dimensional sphere, for all correlation parameters at most $1/10$ in absolute value. This inequality was conjectured (for all correlation parameters at most $1$ in absolute value) by Hwang, Neeman, Parekh, Thompson and Wright. Such an inequality is needed to prove sharp computational hardness of the product state Quantum MAX-CUT problem, assuming the Unique Games Conjecture. In fact, assuming the Unique Games Conjecture, we show that the product state of Quantum MAX-CUT is NP-hard to approximate within a multiplicative factor of $.9859$. In contrast, a polynomial time algorithm is known with approximation factor $.956\ldots$.

quant-ph↗

arXiv2023

Three Candidate Plurality is Stablest for Correlations at most 1/10

We prove the three candidate Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell from 2005 for correlations $\rho$ satisfying $-1/43<\rho<1/10$: the Plurality function is the most noise stable three candidate election method with small influences, when the corrupted votes have correlation $-1/43<\rho<1/10$ with the original votes. The previous best result of this type only achieved positive correlations at most $10^{-10^{10}}$. Our result follows by solving the three set Standard Simplex Conjecture of Isaksson-Mossel from 2011 for all correlations $-1/43<\rho<1/10$. The Gaussian Double Bubble Theorem corresponds to the case $\rho\to1^{-}$, so in some sense, our result is a generalization of the Gaussian Double Bubble Theorem. Our result is also notable since it is the first result for any $\rho<0$, which is the only relevant case for computational hardness of MAX-3-CUT. In fact, assuming the Unique Games Conjecture, we show that MAX-3-CUT is NP-hard to approximate within a multiplicative factor of $.98937$, which improves on the known (unconditional) NP-hardness of approximation within a factor of $1-(1/102)$, proven in 1997. As an additional corollary, we conclude that three candidate Borda Count is stablest for all $-1/43<\rho<1/10$.

math.PR↗

arXiv2022

Noise Stability of Ranked Choice Voting

We conjecture that Borda count is the ranked choice voting method that best preserves the outcome of an election with randomly corrupted votes, among all fair voting methods with small influences satisfying the Condorcet Loser Criterion. This conjecture is an adaptation of the Plurality is Stablest Conjecture to the setting of ranked choice voting. Since the plurality function does not satisfy the Condorcet Loser Criterion, our new conjecture is not directly related to the Plurality is Stablest Conjecture. Nevertheless, we show that the Plurality is Stablest Conjecture implies our new Borda count is Stablest conjecture. We therefore deduce that Borda count is stablest for elections with three candidates when the corrupted votes are nearly uncorrelated with the original votes. We also adapt a dimension reduction argument to this setting, showing that the optimal ranked choice voting method is "low-dimensional." The Condorcet Loser Criterion asserts that a candidate must lose an election if each other candidate is preferred in head-to-head comparisons. Lastly, we discuss a variant of our conjecture with the Condorcet Winner Criterion as a constraint instead of the Condorcet Loser Criterion. In this case, we have no guess for the most stable ranked choice voting method.

cs.GT↗

arXiv2022

Hyperstable Sets with Voting and Algorithmic Hardness Applications

The noise stability of a Euclidean set $A$ with correlation $\rho$ is the probability that $(X,Y)\in A\times A$, where $X,Y$ are standard Gaussian random vectors with correlation $\rho\in(0,1)$. It is well-known that a Euclidean set of fixed Gaussian volume that maximizes noise stability must be a half space. For a partition of Euclidean space into $m>2$ parts each of Gaussian measure $1/m$, it is still unknown what sets maximize the sum of their noise stabilities. In this work, we classify partitions maximizing noise stability that are also critical points for the derivative of noise stability with respect to $\rho$. We call a partition satisfying these conditions hyperstable. Uner the assumption that a maximizing partition is hyperstable, we prove: * a (conditional) version of the Plurality is Stablest Conjecture for $3$ or $4$ candidates. * a (conditional) sharp Unique Games Hardness result for MAX-m-CUT for $m=3$ or $4$ * a (conditional) version of the Propeller Conjecture of Khot and Naor for $4$ sets. We also show that a symmetric set that is hyperstable must be star-shaped. For partitions of Euclidean space into $m>2$ parts of fixed (but perhaps unequal) Gaussian measure, the hyperstable property can only be satisfied when all of the parts have Gaussian measure $1/m$. So, as our main contribution, we have identified a possible strategy for proving the full Plurality is Stablest Conjecture and the full sharp hardness for MAX-m-CUT: to prove both statements, it suffices to show that sets maximizing noise stability are hyperstable. This last point is crucial since any proof of the Plurality is Stablest Conjecture must use a property that is special to partitions of sets into equal measures, since the conjecture is false in the unequal measure case.

math.PR↗

arXiv2022

Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

Let $\Omega$ be a measurable Euclidean set in $\mathbb{R}^{n}$ that is symmetric, i.e. $\Omega=-\Omega$, such that $\Omega\times\mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume. We conclude that either $\Omega$ or $\Omega^{c}$ is convex. Moreover, except for the case $H(x)=\langle x,N(x)\rangle+\lambda$ with $H\geq0$ and $\lambda<0$, we show there exist a radius $r>0$ and an integer $0\leq k\leq n-1$ such that after applying a rotation, the boundary of $\Omega$ must satisfy $\partial\Omega= rS^{k}\times\mathbb{R}^{n-k-1}$, with $\sqrt{n-1}\leq r\leq\sqrt{n+1}$ when $k\geq1$. Here $S^{k}$ denotes the unit sphere of $\mathbb{R}^{k+1}$ centered at the origin, and $n\geq1$ is an integer. One might say this result nearly resolves the symmetric Gaussian conjecture of Barthe from 2001.

math.PR↗

arXiv2022

Dimension-Free Noninteractive Simulation from Gaussian Sources

Let $X$ and $Y$ be two real-valued random variables. Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be independent identically distributed copies of $(X,Y)$. Suppose there are two players A and B. Player A has access to $X_{1},X_{2},\ldots$ and player B has access to $Y_{1},Y_{2},\ldots$. Without communication, what joint probability distributions can players A and B jointly simulate? That is, if $k,m$ are fixed positive integers, what probability distributions on $\{1,\ldots,m\}^{2}$ are equal to the distribution of $(f(X_{1},\ldots,X_{k}),\,g(Y_{1},\ldots,Y_{k}))$ for some $f,g\colon\mathbb{R}^{k}\to\{1,\ldots,m\}$? When $X$ and $Y$ are standard Gaussians with fixed correlation $\rho\in(-1,1)$, we show that the set of probability distributions that can be noninteractively simulated from $k$ Gaussian samples is the same for any $k\geq m^{2}$. Previously, it was not even known if this number of samples $m^{2}$ would be finite or not, except when $m\leq 2$. Consequently, a straightforward brute-force search deciding whether or not a probability distribution on $\{1,\ldots,m\}^{2}$ is within distance $0<\epsilon<|\rho|$ of being noninteractively simulated from $k$ correlated Gaussian samples has run time bounded by $(5/\epsilon)^{m(\log(\epsilon/2) / \log|\rho|)^{m^{2}}}$, improving a bound of Ghazi, Kamath and Raghavendra. A nonlinear central limit theorem (i.e. invariance principle) of Mossel then generalizes this result to decide whether or not a probability distribution on $\{1,\ldots,m\}^{2}$ is within distance $0<\epsilon<|\rho|$ of being noninteractively simulated from $k$ samples of a given finite discrete distribution $(X,Y)$ in run time that does not depend on $k$, with constants that again improve a bound of Ghazi, Kamath and Raghavendra.

math.PR↗

arXiv2021

A Variational Proof of Robust Gaussian Noise Stability

Using the calculus of variations, we prove that a Euclidean set of fixed Gaussian measure that nearly maximizes Gaussian noise stability is close to a half space. The main result proves a modification of a conjecture of Eldan from 2013: a robust Borell inequality that removes a logarithmic dependence on the distance of the set to a half space. For sets of Gaussian measure $1/2$, we prove Eldan's 2013 conjecture. The noise stability of a Euclidean set $A$ with correlation $\rho$ is the probability that $(X,Y)\in A\times A$, where $X,Y$ are standard Gaussian random vectors with correlation $\rho\in(-1,1)$. Barchiesi, Brancolini and Julin proved that a Euclidean set of fixed Gaussian measure that nearly minimizes Gaussian surface area is close to a half space, using a variational "penalty function" method. Our proof adapts their method to the more general setting of noise stability. We also show that half spaces are the only sets that are stable (in the sense of second variation) for noise stability, generalizing a result of McGonagle and Ross for Gaussian surface area.

math.PR↗

arXiv2020

Three Candidate Plurality is Stablest for Small Correlations

Using the calculus of variations, we prove the following structure theorem for noise stable partitions: a partition of $n$-dimensional Euclidean space into $m$ disjoint sets of fixed Gaussian volumes that maximize their noise stability must be $(m-1)$-dimensional, if $m-1\leq n$. In particular, the maximum noise stability of a partition of $m$ sets in $\mathbb{R}^{n}$ of fixed Gaussian volumes is constant for all $n$ satisfying $n\geq m-1$. From this result, we obtain: (i) A proof of the Plurality is Stablest Conjecture for $3$ candidate elections, for all correlation parameters $\rho$ satisfying $0<\rho<\rho_{0}$, where $\rho_{0}>0$ is a fixed constant (that does not depend on the dimension $n$), when each candidate has an equal chance of winning. (ii) A variational proof of Borell's Inequality (corresponding to the case $m=2$). The structure theorem answers a question of De-Mossel-Neeman and of Ghazi-Kamath-Raghavendra. Item (i) is the first proof of any case of the Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell (2005) for fixed $\rho$, with the case $\rho\to1^{-}$ being solved recently. Item (i) is also the first evidence for the optimality of the Frieze-Jerrum semidefinite program for solving MAX-3-CUT, assuming the Unique Games Conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the Plurality is Stablest Conjecture is known to be false.

math.PR↗

arXiv2020

Tree/Endofunction Bijections and Concentration Inequalities

We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is a fixed positive integer. The method uses a bijection between mappings $f\colon\{1,\ldots,n\}\to\{1,\ldots,n\}$ and doubly rooted trees on $n$ vertices. The main application is a concentration inequality for the number of vertices connected to an independent set in a uniformly random tree, which is then used to prove partial unimodality of its independent set sequence. So, we give probabilistic arguments for inequalities that often use combinatorial arguments.

math.PR↗

arXiv2020

Designing Stable Elections: A Survey

We survey the design of elections that are resilient to attempted interference by third parties. For example, suppose votes have been cast in an election between two candidates, and then each vote is randomly changed with a small probability, independently of the other votes. It is desirable to keep the outcome of the election the same, regardless of the changes to the votes. It is well known that the US electoral college system is about 5 times more likely to have a changed outcome due to vote corruption, when compared to a majority vote. In fact, Mossel, O'Donnell and Oleszkiewicz proved in 2005 that the majority voting method is most stable to this random vote corruption, among voting methods where each person has a small influence on the election. We discuss some recent progress on the analogous result for elections between more than two candidates. In this case, plurality should be most stable to corruption in votes. We also survey results on adversarial election manipulation (where an adversary can select particular votes to change, perhaps in a non-random way), and we briefly discuss ranked choice voting methods (where a vote is a ranked list of candidates).

math.PR↗

arXiv2020

Independent Sets of Random Trees and of Sparse Random Graphs

An independent set of size $k$ in a finite undirected graph $G$ is a set of $k$ vertices of the graph, no two of which are connected by an edge. Let $x_{k}(G)$ be the number of independent sets of size $k$ in the graph $G$ and let $\alpha(G)=\max\{k\geq0\colon x_{k}(G)\neq0\}$. In 1987, Alavi, Malde, Schwenk and Erd\"{o}s asked if the independent set sequence $x_{0}(G),x_{1}(G),\ldots,x_{\alpha(G)}(G)$ of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8\% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk and Erd\"{o}s is ``four-fifths true'', with high probability. We also show unimodality of the independent set sequence of Erd\"{o}s-Renyi random graphs, when the expected degree of a single vertex is large (with (exponentially) high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size $k$ as $k$ varies is of interest in probability, statistical physics, combinatorics, and computer science.

math.PR↗