arXiv subjects

Aditya Potukuchi

Publications and source records attributed to Aditya Potukuchi.

At least 19 recordsLinked to original sources

arXiv2026

Non-existence probabilities and lower tails in the critical regime via Belief Propagation

We compute the logarithmic asymptotics of the non-existence probability (and more generally the lower-tail probability) for a wide variety of combinatorial problems for a range of parameters in the `critical regime' between the regime amenable to hypergraph container methods and that amenable to Janson's inequality. Examples include lower tails and non-existence probabilities for subgraphs of random graphs and for $k$-term arithmetic progressions in random sets of integers. Our methods apply in the general framework of estimating the probability that a $p$-random subset of vertices in a $k$-uniform hypergraph induces significantly fewer hyperedges than expected. We show that under some simple structural conditions on the hypergraph and an upper bound on $p$ determined by a phase transition in the hard-core model on the infinite $k$-uniform, $\Delta$-regular, linear hypertree, this probability can be accurately approximated by the Bethe free energy evaluated at the unique fixed point of a Belief Propagation operator on the hypergraph.

math.CO↗

arXiv2024

Lower tails for triangles inside the critical window

We study the probability that the random graph $G(n,p)$ is triangle-free. When $p =o(n^{-1/2})$ or $p = \omega(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and via regularity or hypergraph container methods in the latter case. We prove for the first time an asymptotic formula for the logarithm of this probability when $p = c n^{-1/2}$ for $c$ a sufficiently small constant. More generally, we study lower-tail large deviations for triangles in random graphs: the probability that $G(n,p)$ has at most $\eta$ times its expected number of triangles, when $p = c n^{-1/2}$ for $c$ and $\eta \in [0,1)$ constant. Our results apply for all $c$ if $\eta \ge .4993$ and for $c$ small enough otherwise. For $\eta$ small (including the case of triangle-freeness), we prove that a phase transition occurs as $c$ varies, in the sense of a non-analyticity of the rate function, while for $\eta \ge .4993$ we prove that no phase transition occurs. On the other hand for the random graph $G(n,m)$, with $m = b n^{3/2}$, we show that a phase transition occurs in the lower-tail problem for triangles as $b$ varies for \emph{every} $\eta \in [0,1)$. Our method involves ingredients from algorithms and statistical physics including the cluster expansion and concentration inequalities for contractive Markov chains.

math.PR↗

arXiv2024

Sampling and counting triangle-free graphs near the critical density

We study the following combinatorial counting and sampling problems: can we efficiently sample from the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ conditioned on triangle-freeness? Can we efficiently approximate the probability that $G(n,p)$ is triangle-free? These are prototypical instances of forbidden substructure problems ubiquitous in combinatorics. The algorithmic questions are instances of approximate counting and sampling for a hypergraph hard-core model. Estimating the probability that $G(n,p)$ has no triangles is a fundamental question in probabilistic combinatorics and one that has led to the development of many important tools in the field. Through the work of several authors, the asymptotics of the logarithm of this probability are known if $p =o( n^{-1/2})$ or if $p =\omega( n^{-1/2})$. The regime $p = \Theta(n^{-1/2})$ is more mysterious, as this range witnesses a dramatic change in the the typical structural properties of $G(n,p)$ conditioned on triangle-freeness. As we show, this change in structure has a profound impact on the performance of sampling algorithms. We give two different efficient sampling algorithms for triangle-free graphs (and complementary algorithms to approximate the triangle-freeness large deviation probability), one that is efficient when $p < c/\sqrt{n}$ and one that is efficient when $p > C/\sqrt{n}$ for constants $c, C>0$. The latter algorithm involves a new approach for dealing with large defects in the setting of sampling from low-temperature spin models.

cs.DS↗

arXiv2024

Error-Correcting Graph Codes

In this paper, we construct Error-Correcting Graph Codes. An error-correcting graph code of distance $\delta$ is a family $C$ of graphs on a common vertex set of size $n$, such that if we start with any graph in $C$, we would have to modify the neighborhoods of at least $\delta n$ vertices in order to obtain some other graph in $C$. This is a natural graph generalization of the standard Hamming distance error-correcting codes for binary strings. Yohananov and Yaakobi were the first to construct codes in this metric, constructing good codes for $\delta < 1/2$, and optimal codes for a large-alphabet analogue. We extend their work by showing 1. Combinatorial results determining the optimal rate vs. distance trade-off nonconstructively. 2. Graph code analogues of Reed-Solomon codes and code concatenation, leading to positive distance codes for all rates and positive rate codes for all distances. 3. Graph code analogues of dual-BCH codes, yielding large codes with distance $\delta = 1-o(1)$. This gives an explicit ''graph code of Ramsey graphs''. Several recent works, starting with the paper of Alon, Gujgiczer, K\"orner, Milojevi\'c, and Simonyi, have studied more general graph codes; where the symmetric difference between any two graphs in the code is required to have some desired property. Error-correcting graph codes are a particularly interesting instantiation of this concept.

cs.IT↗

arXiv2024

Unbalanced Random Matching Markets with Partial Preferences

Properties of stable matchings in the popular random-matching-market model have been studied for over 50 years. In a random matching market, each agent has complete preferences drawn uniformly and independently at random. Wilson (1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching markets, the proposers are matched to their $\ln n$th choice on average. In this paper, we consider markets where agents have partial (truncated) preferences, that is, the proposers only rank their top $d$ partners. Despite the long history of the problem, the following fundamental question remained unanswered: \emph{what is the smallest value of $d$ that results in a perfect stable matching with high probability?} In this paper, we answer this question exactly -- we prove that a degree of $\ln^2 n$ is necessary and sufficient. That is, we show that if $d < (1-\epsilon) \ln^2 n$ then no stable matching is perfect and if $d > (1+ \epsilon) \ln^2 n$, then every stable matching is perfect with high probability. This settles a recent conjecture by Kanoria, Min and Qian (2021). We generalize this threshold for unbalanced markets: we consider a matching market with $n$ agents on the shorter side and $n(\alpha+1)$ agents on the longer side. We show that for markets with $\alpha =o(1)$, the sharp threshold characterizing the existence of perfect stable matching occurs when $d$ is $\ln n \cdot \ln \left(\frac{1 + \alpha}{\alpha + (1/n(\alpha+1))} \right)$. Finally, we extend the line of work studying the effect of imbalance on the expected rank of the proposers (termed the ``stark effect of competition''). We establish the regime in unbalanced markets that forces this stark effect to take shape in markets with partial preferences.

cs.GT↗

arXiv2023

On the evolution of structure in triangle-free graphs

We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite. Erd\H{o}s, Kleitman, and Rothschild showed that almost every triangle-free graph is bipartite. Osthus, Pr\"omel, and Taraz later showed that for $m \ge (1+\epsilon)\frac{\sqrt{3}}{4}n^{3/2}\sqrt{\log n}$, almost every triangle-free graph on $n$ vertices and $m$ edges is bipartite. Here we give a precise characterization of the distribution of edges within each part of the max cut of a uniformly chosen triangle-free graph $G$ on $n$ vertices and $m$ edges, for a larger range of densities with $m=\Theta(n^{3/2} \sqrt{\log n})$. Using this characterization, we describe the evolution of the structure of typical triangle-free graphs as the density changes. We show that as the number of edges decreases below $\frac{\sqrt{3}}{4} n^{3/2}\sqrt{\log n}$, the following structural changes occur in $G$: -Isolated edges, then trees, then more complex subgraphs emerge as `defect edges', edges within parts of a max cut of $G$. The distribution of defect edges is first that of independent Erd\H{o}s-R\'{e}nyi random graphs, then that of independent exponential random graphs, conditioned on a small maximum degree and no triangles. -There is a sharp threshold for $3$-colorability at $m \sim \frac{\sqrt{2}}{4} n^{3/2}\sqrt{\log n}$ and a sharp threshold between $4$-colorability and unbounded chromatic number at $m\sim\frac{1}{4}n^{3/2}\sqrt{\log n}$. -Giant components emerge in the defect edges at $m\sim\frac{1}{4} n^{3/2}\sqrt{\log n}$. We use these results to prove asymptotic formulas for the number of triangle-free graphs at these densities. We likewise prove analogous results for the random graph $G(n,p)$ conditioned on triangle-freeness.

math.CO↗

arXiv2023

Approximately counting independent sets in dense bipartite graphs via subspace enumeration

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph -- in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to ``high-temperature'' problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. The proof exploits the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e. bounded threshold rank), which via subspace enumeration lets us enumerate small cuts efficiently.

cs.DS↗

arXiv2023

Blocking sets, minimal codes and trifferent codes

We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-$2$ subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least $3$. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set. Our main motivation for these new bounds is their application to trifferent codes, which are sets of ternary codes of length $n$ with the property that for any three distinct codewords there is a coordinate where they all have distinct values. Over the finite field $\mathbb{F}_3$, we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length $n$ has size at most $3^{n/4.55}$, improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length $n$ and size at least $\frac{1}{3}\left( 9/5 \right)^{n/4}$, thus (asymptotically) matching the best lower bound on trifferent codes. We also give explicit constructions of affine blocking sets with respect to codimension-$2$ subspaces that are a constant factor bigger than the best known lower bound. By restricting to~$\mathbb{F}_3$, we obtain linear trifferent codes of size at least $3^{23n/312}$, improving the current best explicit construction that has size $3^{n/112}$.

math.CO↗

arXiv2022

Algorithms for the ferromagnetic Potts model on expanders

We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger's algorithm to count cuts that may be of independent interest. It is \#BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of \#BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high temperature proof follows more standard polymer model analysis, our result holds in the largest known range of parameters $d$ and $q$.

cs.DS↗

arXiv2021

On the Geometry of Stable Steiner Tree Instances

In this note we consider the Steiner tree problem under Bilu-Linial stability. We give strong geometric structural properties that need to be satisfied by stable instances. We then make use of, and strengthen, these geometric properties to show that $1.562$-stable instances of Euclidean Steiner trees are polynomial-time solvable. We also provide a connection between certain approximation algorithms and Bilu-Linial stability for Steiner trees.

cs.DS↗

arXiv2021

Enumerating independent sets in Abelian Cayley graphs

We show that any connected Cayley graph $\Gamma$ on an Abelian group of order $2n$ and degree $\tilde{\Omega}(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $\Gamma$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Pl\"{u}nnecke-Rusza-Petridis inequality from additive combinatorics.

math.CO↗

arXiv2021

Approximately counting independent sets in bipartite graphs via graph containers

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to $d$-regular, bipartite graphs satisfying a weak expansion condition: when $d$ is constant, and the graph is a bipartite $\Omega( \log^2 d/d)$-expander, we obtain an FPTAS for the number of independent sets. Previously such a result for $d>5$ was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a $d$-regular, bipartite $\alpha$-expander, with $\alpha>0$ fixed, we give an FPTAS for the hard-core model partition function at fugacity $\lambda=\Omega(\log d / d^{1/4})$. Finally we present an algorithm that applies to all $d$-regular, bipartite graphs, runs in time $\exp\left( O\left( n \cdot \frac{ \log^3 d }{d } \right) \right)$, and outputs a $(1 + o(1))$-approximation to the number of independent sets.

cs.DS↗

arXiv2021

Independent sets of a given size and structure in the hypercube

We determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $\beta \in [0,1]$ as $d \to \infty$, extending a result of Galvin for $\beta \in [1-1/\sqrt{2},1]$. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard core model at any fixed fugacity $\lambda>0$. In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.

math.CO↗

arXiv2020

On the list recoverability of randomly punctured codes

We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes.

math.CO↗

arXiv2019

On the $\text{AC}^0[\oplus]$ complexity of Andreev's Problem

Andreev's Problem states the following: Given an integer $d$ and a subset of $S \subseteq \mathbb{F}_q \times \mathbb{F}_q$, is there a polynomial $y = p(x)$ of degree at most $d$ such that for every $a \in \mathbb{F}_q$, $(a,p(a)) \in S$? We show an $\text{AC}^0[\oplus]$ lower bound for this problem. This problem appears to be similar to the list recovery problem for degree $d$-Reed-Solomon codes over $\mathbb{F}_q$ which states the following: Given subsets $A_1,\ldots,A_q$ of $\mathbb{F}_q$, output all (if any) the Reed-Solomon codewords contained in $A_1\times \cdots \times A_q$. For our purpose, we study this problem when $A_1, \ldots, A_q$ are random subsets of a given size, which may be of independent interest.

cs.CC↗

arXiv2019

A spectral bound on hypergraph discrepancy

Let $\mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m \times n$ incidence matrix of $\mathcal{H}$ and let us denote $\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|$. We show that the discrepancy of $\mathcal{H}$ is $O(\sqrt{t} + \lambda)$. As a corollary, this gives us that for every $t$, the discrepancy of a random $t$-regular hypergraph with $n$ vertices and $m \geq n$ edges is almost surely $O(\sqrt{t})$ as $n$ grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.

math.CO↗

arXiv2019

Simplified inpproximability of hypergraph coloring via t-agreeing families

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khachatrian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs of quasi NP-hardness of the following problems: $\bullet$ coloring a $3$ colorable $4$-uniform hypergraph with $(\log n)^\delta$ many colors $\bullet$ coloring a $3$ colorable $3$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors $\bullet$ coloring a $2$ colorable $6$-uniform hypergraph with $(\log n)^\delta$ many colors $\bullet$ coloring a $2$ colorable $4$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors where $n$ is the number of vertices of the hypergraph and $\delta>0$ is a universal constant.

cs.CC↗

arXiv2019

Improved efficiency for covering codes matching the sphere-covering bound

A covering code is a subset $\mathcal{C} \subseteq \{0,1\}^n$ with the property that any $z \in \{0,1\}^n$ is close to some $c \in \mathcal{C}$ in Hamming distance. For every $\epsilon,\delta>0$, we show a construction of a family of codes with relative covering radius $\delta + \epsilon$ and rate $1 - \mathrm{H}(\delta) $ with block length at most $\exp(O((1/\epsilon) \log (1/\epsilon)))$ for every $\epsilon > 0$. This improves upon a folklore construction which only guaranteed codes of block length $\exp(1/\epsilon^2)$. The main idea behind this proof is to find a distribution on codes with relatively small support such that most of these codes have good covering properties.

cs.IT↗