arXiv subjects

Prasad Tetali

Publications and source records attributed to Prasad Tetali.

At least 19 recordsLinked to original sources

arXiv2026

The Problem Is the Problem: Towards Scalable Mathematical Discovery

AI systems are increasingly capable of contributing to mathematical research. In research practice, frontier-model reasoning is a limited resource, and expert mathematical review is even more sharply constrained. Allocating these scarce resources well is therefore central to making AI-assisted mathematical discovery efficient. In most current AI-for-math workflows, human effort is concentrated at the beginning and end, in selecting suitable research problems and later reviewing the resulting artifacts. These two stages are becoming bottlenecks for research-level mathematics. We address them by proposing a new human-AI discovery paradigm. The human input is no longer a single problem selected in advance, but a research direction in which the experts have interest and expertise. The system then searches a broad literature corpus for candidate problems in that direction. Inspired by search and recommender systems, we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade that automates the search for suitable problems and focuses human attention on artifacts that have passed several stages of filtering. In a combinatorics pilot, the pipeline starts from 5,245 combinatorics papers, recovers 6,453 candidate conjectures or open problems, and filters them to 4,717 apparently well-posed and still-open conjectures. Subsequent reasoning and automated triage stages surface 598 potential resolutions and select 77 items for author-team review. Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies--Jenssen--Perkins--Roberts, Erd\H{o}s--Straus, Ikenmeyer--Pak--Panova, and Lund--Saraf--Wolf. These results demonstrate the effectiveness of this new mode of human-AI collaboration for mathematical discovery.

cs.AI↗

arXiv2026

Faster Mixing for Triangulations via Transport Flows

We prove an $\widetilde O(n^2)$ bound for the relaxation time and the log-Sobolev time (inverse log-Sobolev constant) of the classical triangulation flip chain on a convex $(n+2)$-gon, implying a mixing time of $\widetilde O(n^2)$. The previous state of the art for the mixing time of this chain, due to Eppstein and Frishberg, was $\widetilde O(n^3)$, while the best known lower bound on the mixing time, due to Molloy, Reed, and Steiger, is $Ω(n^{3/2})$. Our relaxation time bound makes significant progress towards Aldous' conjectured bound of $Θ(n^{3/2})$ for the relaxation time. We improve upon the analysis of Eppstein and Frishberg by further developing the framework of transport flows introduced in the work of Chen et al. In this light, our results can be seen as a more efficient way of using combinatorial decompositions to obtain functional inequalities for Markov chains. We hope our ideas will find other applications in the future.

math.CO↗

arXiv2026

ImProver: Agent-Based Automated Proof Optimization

Large language models (LLMs) have been used to generate formal proofs of mathematical theorems in proofs assistants such as Lean. However, we often want to optimize a formal proof with respect to various criteria, depending on its downstream use. For example, we may want a proof to adhere to a certain style, or to be readable, concise, or modularly structured. Having suitably optimized proofs is also important for learning tasks, especially since human-written proofs may not optimal for that purpose. To this end, we study a new problem of automated proof optimization: rewriting a proof so that it is correct and optimizes for an arbitrary criterion, such as length or readability. As a first method for automated proof optimization, we present ImProver, a large-language-model agent that rewrites proofs to optimize arbitrary user-defined metrics in Lean. We find that naively applying LLMs to proof optimization falls short, and we incorporate various improvements into ImProver, such as the use of symbolic Lean context in a novel Chain-of-States technique, as well as error-correction and retrieval. We test ImProver on rewriting real-world undergraduate, competition, and research-level mathematics theorems, finding that ImProver is capable of rewriting proofs so that they are substantially shorter, more modular, and more readable.

cs.AI↗

arXiv2023

Hardness and Approximation of Submodular Minimum Linear Ordering Problems

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $f(\cdot)$ due to an ordering $σ$ of the items (say $[n]$), i.e., $\min_σ \sum_{i\in [n]} f(E_{i,σ})$, where $E_{i,σ}$ is the set of items mapped by $σ$ to indices $[i]$. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata, Tetali, and Tripathi [ITT2012], using Lovász extension of submodular functions. We show a $(2-\frac{1+\ell_{f}}{1+|E|})$-approximation for monotone submodular MLOP where $\ell_{f}=\frac{f(E)}{\max_{x\in E}f(\{x\})}$ satisfies $1 \leq \ell_f \leq |E|$. Our theory provides new approximation bounds for special cases of the problem, in particular a $(2-\frac{1+r(E)}{1+|E|})$-approximation for the matroid MLOP, where $f = r$ is the rank function of a matroid. We further show that minimum latency vertex cover (MLVC) is $\frac{4}{3}$-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.

cs.DS↗

arXiv2023

Note on the number of antichains in generalizations of the Boolean lattice

We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t =O\Bigl(\frac{n}{\log n}\Bigr)$, the number $α([t]^n)$ of antichains of the poset $[t]^n$ is at most \[\exp_2\Bigl(1+O\Bigl(\Bigl(\frac{t\log^3 n}{n}\Bigr)^{1/2}\Bigr)\Bigr)N(t,n)\,,\] where $N(t,n)$ is the size of a largest level of $[t]^n$. This, in particular, says that if $t \ll n/\log^3 n$ as $n \rightarrow \infty$, then $\logα([t]^n)=(1+o(1))N(t,n)$, giving a (partially) positive answer to a question of Moshkovitz and Shapira for $t, n$ in this range. Particularly for $t=3$, we prove a better upper bound: \[\logα([3]^n)\le(1+4\log 3/n)N(3,n),\] which is the best known upper bound on the number of antichains of $[3]^n$.

math.CO↗

arXiv2022

On the zeroes of hypergraph independence polynomials

We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $Δ$ have a zero-free disk almost as large as the optimal disk for graphs of maximum degree $Δ$ established by Shearer (of radius $\sim 1/(e Δ)$). Up to logarithmic factors in $Δ$ this is optimal, even for hypergraphs with all edge-sizes strictly greater than $2$. We conjecture that for $k\ge 3$, $k$-uniform linear hypergraphs have a much larger zero-free disk of radius $Ω(Δ^{- \frac{1}{k-1}} )$. We establish this in the case of linear hypertrees.

math.CO↗

arXiv2022

Efficient sampling and counting algorithms for the Potts model on $\mathbb Z^d$ at all temperatures

For $d \ge 2$ and all $q\geq q_{0}(d)$ we give an efficient algorithm to approximately sample from the $q$-state ferromagnetic Potts and random cluster models on finite tori $(\mathbb Z / n \mathbb Z )^d$ for any inverse temperature $β\geq 0$. This shows that the physical phase transition of the Potts model presents no algorithmic barrier to efficient sampling, and stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen--Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models at all temperatures. Our algorithms are based on representing the random cluster model as a contour model using Pirogov--Sinai theory, and then computing an accurate approximation of the logarithm of the partition function by inductively truncating the resulting cluster expansion. The main innovation of our approach is an algorithmic treatment of unstable ground states, which is essential for our algorithms to apply to all inverse temperatures $β$. By treating unstable ground states our work gives a general template for converting probabilistic applications of Pirogov-Sinai theory to efficient algorithms.

math.PR↗

arXiv2022

Multi Purpose Routing: New Perspectives and Approximation Algorithms

The cost due to delay in services may be intrinsically different for various applications of vehicle routing such as medical emergencies, logistical operations, and ride-sharing. We study a fundamental generalization of the Traveling Salesman Problem, namely $L_p$ TSP, where the objective is to minimize an aggregated measure of the delay in services, quantified by the Minkowski $p$-norm of the delay vector. We present efficient combinatorial and Linear Programming algorithms for approximating $L_p$ TSP on general metrics. We provide several approximation algorithms for the $L_p$ TSP problem, including $4.27$ & $10.92$-approximation algorithms for single & multi vehicle $L_2$ TSP, called the Traveling Firefighter Problem. Among other contributions, we provide an $8$-approximation and a $1.78$ inapproximability for All-Norm TSP problem, addressing scenarios where one does not know the ideal cost function, or is seeking simultaneous approximation with respect to any cost function.

cs.DS↗

arXiv2022

Determinant Maximization via Matroid Intersection Algorithms

Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics \cite{pukelsheim2006optimal}, convex geometry \cite{Khachiyan1996}, fair allocations\linebreak \cite{anari2016nash}, combinatorics \cite{AnariGV18}, spectral graph theory \cite{nikolov2019proportional}, network design, and random processes \cite{kulesza2012determinantal}. In an instance of a determinant maximization problem, we are given a collection of vectors $U=\{v_1,\ldots, v_n\} \subset \RR^d$, and a goal is to pick a subset $S\subseteq U$ of given vectors to maximize the determinant of the matrix $\sum_{i\in S} v_i v_i^\top $. Often, the set $S$ of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint $\left(|S|\leq k\right)$ or matroid constraint ($S$ is a basis of a matroid defined on the vectors). In this paper, we give a polynomial-time deterministic algorithm that returns a $r^{O(r)}$-approximation for any matroid of rank $r\leq d$. This improves previous results that give $e^{O(r^2)}$-approximation algorithms relying on $e^{O(r)}$-approximate \emph{estimation} algorithms \cite{NikolovS16,anari2017generalization,AnariGV18,madan2020maximizing} for any $r\leq d$. All previous results use convex relaxations and their relationship to stable polynomials and strongly log-concave polynomials. In contrast, our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improve any solution by finding an \emph{alternating negative cycle} in the \emph{exchange graph} defined by the matroids. While the $\det(.)$ function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm.

cs.DS↗

arXiv2022

$λ_\infty$ & Maximum Variance Embedding: Measuring and Optimizing Connectivity of A Graph Metric

Bobkov, Houdré, and the last author [2000] introduced a Poincaré-type functional parameter, $λ_\infty$, of a graph and related it to connectivity of the graph via Cheeger-type inequalities. A work by the second author, Raghavendra, and Vempala [2013] related the complexity of $λ_\infty$ to the so-called small-set expansion (SSE) problem and further set forth the desiderata for NP-hardness of this optimization problem. We confirm the conjecture that computing $λ_\infty$ is NP-hard for weighted trees. Beyond measuring connectivity in many applications we want to optimize it. This, via convex duality, leads to a problem in machine learning known as the Maximum Variance Embedding (MVE). The output is a function from vertices to a low dim Euclidean space, subject to bounds on Euclidean distances between neighbors. The objective is to maximize output variance. Special cases of MVE into $n$ and $1$ dims lead to absolute algebraic connectivity [1990] and spread constant [1998], that measure connectivity of the graph and its Cartesian $n$-power, respectively. MVE has other applications in measuring diffusion speed and robustness of networks, clustering, and dimension reduction. We show that computing MVE in tree-width dims is NP-hard, while only one additional dim beyond width of a given tree-decomposition makes the problem in P. We show that MVE of a tree in 2 dims defines a non-convex yet benign optimization landscape, i.e., local=global optima. We further develop a linear time combinatorial algorithm for this case. Finally, we denote approximate Maximum Variance Embedding is tractable in significantly lower dims. For trees and general graphs, for which Maximum Variance Embedding cannot be solved in less than $2$ and $Ω(n)$ dims, we provide $1+\varepsilon$ approximation algorithms for embedding into $1$ and $O(\log n /\varepsilon^2)$ dims, respectively.

cs.CC↗

arXiv2021

On the Bipartiteness Constant and Expansion of Cayley Graphs

Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = λ_1(A)\geq \dots \ge λ_n \ge -1$. It is a classical fact that $λ_n = -1$ if and only if $G$ is bipartite. Our main result provides a quantitative separation of $λ_n$ from $-1$ in the case of Cayley graphs, in terms of their expansion. Denoting $h_{out}$ by the (outer boundary) vertex expansion of $G$, we show that if $G$ is a non-bipartite Cayley graph (constructed using a group and a symmetric generating set of size $d$) then $λ_n \ge -1 + ch_{out}^2/d^2\,,$ for $c$ an absolute constant. We exhibit graphs for which this result is tight up to a factor depending on $d$. This improves upon a recent result by Biswas and Saha who showed $λ_n \ge -1 + h_{out}^4/(2^9d^8)\,.$ We also note that such a result could not be true for general non-bipartite graphs.

math.CO↗

arXiv2021

The Traveling Firefighter Problem

We introduce the $L_p$ Traveling Salesman Problem ($L_p$-TSP), given by an origin, a set of destinations, and underlying distances. The objective is to schedule a destination visit sequence for a traveler of unit speed to minimize the Minkowski $p$-norm of the resulting vector of visit/service times. For $p = \infty$ the problem becomes a path variant of the TSP, and for $p = 1$ it defines the Traveling Repairman Problem (TRP), both at the center of classical combinatorial optimization. We provide an approximation preserving polynomial-time reduction of $L_p$-TSP to the segmented-TSP Problem [Sitters '14] and further study the case of $p = 2$, which we term the Traveling Firefighter Problem (TFP), when the cost due to a delay in service is quadratic in time. We also study the all-norm-TSP problem [Golovin et al. '08], in which the objective is to find a route that is (approximately) optimal with respect to the minimization of any norm of the visit times, and improve corresponding (in)approximability bounds on metric spaces.

cs.DS↗

arXiv2021

On the Number of Independent Sets in Uniform, Regular, Linear Hypergraphs

We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up to the first order term for all (fixed) $r\ge 3$, with $d$ and $n$ going to infinity. In the case of strong independent sets, for $r=3$, we provide an upper bound that is tight up to the second-order term, improving on a result of Ordentlich-Roth (2004). The tightness in the strong independent set case is established by an explicit construction of a $3$-uniform, $d$-regular, cross-edge free, linear hypergraph on $n$ vertices which could be of interest in other contexts. We leave open the general case(s) with some conjectures. Our proofs use the occupancy method introduced by Davies, Jenssen, Perkins, and Roberts (2017).

math.CO↗

arXiv2020

Toppleable Permutations, Excedances and Acyclic Orientations

Recall that an excedance of a permutation $π$ is any position $i$ such that $π_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$. We also give a formula for the number of AUSOs of complete multipartite graphs. We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.

math.CO↗

arXiv2020

Improved Approximations for Min Sum Vertex Cover and Generalized Min Sum Set Cover

We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges $E$ with arbitrary covering requirements $k_e$, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge $e$ is considered covered by the first time when $k_e$ many of its vertices appear in the ordering. We give a $4.642$ approximation algorithm for GMSSC, coming close to the best possible bound of $4$, already for the classical special case (with all $k_e=1$) of min sum set cover (MSSC) studied by Feige, Lovász and Tetali, and improving upon the previous best known bound of $12.4$ due to Im, Sviridenko and van der Zwaan. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. This also gives an LP-based $4$ approximation for MSSC. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Another well-known special case is the min sum vertex cover (MSVC) problem, in which the input hypergraph is a graph and $k_e = 1$, for every edge. We give a $16/9$ approximation for MSVC, and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best $1.999946$ approximation of Barenholz, Feige and Peleg. (The claimed $1.79$ approximation result of Iwata, Tetali and Tripathi for the MSVC turned out have an unfortunate, seemingly unfixable, mistake in it.) Finally, we revisit MSSC and consider the $\ell_p$ norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm achieves tight, up to NP-hardness, approximation guarantees of $(p+1)^{1+1/p}$, for all $p\ge 1$. For $p=1$, this gives yet another proof of the $4$ approximation for MSSC.

cs.DS↗

arXiv2019

Volume growth, curvature, and Buser-type inequalities in graphs

We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature, and discuss similar results under other types of discrete Ricci curvature. Following recent work in the continuous setting of Riemannian manifolds (by the first author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, the spectral gap of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the second eigenvalue. We also describe a method for proving Buser's inequality in graphs, particularly under a lower bound assumption on curvature.

math.CO↗

arXiv2019

Transport Proofs Of Some Discrete Variants Of The Pr{é}Kopa-leindler Inequality

We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Pr{é}kopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].

math.PR↗

arXiv2018

Finding cliques using few probes

Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an $n$ vertex graph, and need to output a clique. We show that if the input graph is drawn at random from $G_{n,\frac{1}{2}}$ (and hence is likely to have a clique of size roughly $2\log n$), then for every $δ< 2$ and constant $\ell$, there is an $α< 2$ (that may depend on $δ$ and $\ell$) such that no algorithm that makes $n^δ$ probes in $\ell$ rounds is likely (over the choice of the random graph) to output a clique of size larger than $α\log n$.

math.CO↗