arXiv subjects

Iman Gholami

Publications and source records attributed to Iman Gholami.

7 recordsLinked to original sources

arXiv2026

Beyond the Library: An Agentic Framework for Autoformalizing Research Mathematics

While Large Language Models (LLMs) have demonstrated exceptional capabilities in mathematical reasoning, they frequently produce subtle errors that evade human detection. Formal mathematical languages like Lean 4 offer mechanical proof checking, strongly motivating the need for autoformalization: the automatic translation of natural language mathematics into verifiable code. Recent trends indicate that general-purpose LLMs, heavily optimized for standard programming, now outperform smaller models explicitly fine-tuned for Lean. Leveraging this shift, we introduce *Theo*, an agentic autoformalization framework powered by general coding LLMs. At the core of our system is an orchestrator that manages a multi-agent pipeline tailored for research-level mathematics. Because cutting-edge research frequently relies on concepts outside the scope of existing libraries like Mathlib, our system dynamically extends necessary type definitions and validates them via a novel Auxiliary Lemma technique before formalizing the primary theorems. We applied our approach to PutnamBench, producing machine-checked Lean proofs for a random sample of 32 problems. Furthermore, we evaluate our system on seven research papers---five from the ACM Symposium on Theory of Computing (STOC) and two recent OpenAI manuscripts---spanning combinatorics, communication complexity, mechanism design, learning theory, number theory, discrete geometry, and graph theory. We successfully formalize their main theorems and proofs and validate the generated formalizations with human experts; notably, two developments require no axioms beyond Lean's kernel. All of our formalizations are available at https://beyondthelibrary.github.io/formal_arxiv/.

cs.AI↗

arXiv2026

Quiet Planting for $k$-SAT, Multiple Solutions of Arbitrary Geometry

Recent work on "quiet planting" in combinatorial optimization aims to generate instances with a hidden solution that is hard to recover, typically by making the planted distribution statistically indistinguishable from uniform for specific algorithms, such as statistical queries. A prominent example is planted $k$-SAT, where $O(n^{k/2})$ clauses can be planted while maintaining indistinguishability from uniform instances, evidenced by prior hardness results which also align with findings in SAT refutation. Despite extensive research and practical use in benchmarking SAT solvers, the challenge of quietly planting multiple solutions while preserving hardness has remained an open problem. This work initiates the study of quiet planting with an arbitrary number of solutions, proposing the first method to construct quiet planting distributions for $k$-SAT formulas that accommodate more than one solution. We provide statistical query lower bounds for distinguishing these planted instances from uniform ones, and our method allows for planting solutions with arbitrary geometric relationships, including varying Hamming distances. A key innovation facilitating multiple solutions is the ability to incorporate arbitrary correlations between variable selection in clauses and their negation patterns, departing from prior approaches. We also investigate the worst-case complexity of SAT by showing the difficulty in distinguishing satisfiable instances with numerous solutions from unsatisfiable ones, addressing an open problem of Hsieh, Mohanty, and Xu (CCC'22). Technically, we generalize $(r-1)$-wise uniformness in clause distributions, proving hardness if marginal negation distributions are $(r-1)$-wise uniform. We also reveal a connection to binary linear codes, showing a $[k, t, r]$ code can guide planting up to $2^t - 1$ solutions on $k$ variables.

cs.CC↗

arXiv2025

Prize-Collecting Forest with Submodular Penalties: Improved Approximation

Constrained forest problems form a class of graph problems where specific connectivity requirements for certain cuts within the graph must be satisfied by selecting the minimum-cost set of edges. The prize-collecting version of these problems introduces flexibility by allowing penalties to be paid to ignore some connectivity requirements. Goemans and Williamson introduced a general technique and developed a 2-approximation algorithm for constrained forest problems. Further, Sharma, Swamy, and Williamson extended this work by developing a 2.54-approximation algorithm for the prize-collecting version of these problems. Motivated by the generality of their framework, which includes problems such as Steiner trees, Steiner forests, and their variants, we pursued further exploration. We present a significant improvement by achieving a 2-approximation algorithm for this general model, matching the approximation factor of the constrained forest problems.

cs.DS↗

arXiv2025

Breaking a Long-Standing Barrier: 2-$\varepsilon$ Approximation for Steiner Forest

The Steiner Forest problem, also known as the Generalized Steiner Tree problem, is a fundamental optimization problem on edge-weighted graphs where, given a set of vertex pairs, the goal is to select a minimum-cost subgraph such that each pair is connected. This problem generalizes the Steiner Tree problem, first introduced in 1811, for which the best known approximation factor is 1.39 [Byrka, Grandoni, Rothvo{\ss}, and Sanit\`a, 2010] (Best Paper award, STOC 2010). The celebrated work of [Agrawal, Klein, and Ravi, 1989] (30-Year Test-of-Time award, STOC 2023), along with refinements by [Goemans and Williamson, 1992] (SICOMP'95), established a 2-approximation for Steiner Forest over 35 years ago. Jain's (FOCS'98) pioneering iterative rounding techniques later extended these results to higher connectivity settings. Despite the long-standing importance of this problem, breaking the approximation factor of 2 has remained a major challenge, raising suspicions that achieving a better factor -- similar to Vertex Cover -- might indeed be hard. Notably, fundamental works, including those by Gupta and Kumar (STOC'15) and Gro{\ss} et al. (ITCS'18), introduced 96- and 69-approximation algorithms, possibly with the hope of paving the way for a breakthrough in achieving a constant-factor approximation below 2 for the Steiner Forest problem. In this paper, we break the approximation barrier of 2 by designing a novel deterministic algorithm that achieves a $2 - 10^{-11}$ approximation for this fundamental problem. As a key component of our approach, we also introduce a novel dual-based local search algorithm for the Steiner Tree problem with an approximation guarantee of $1.943$, which is of independent interest.

cs.DS↗

arXiv2024

Bi-Criteria Metric Distortion

Selecting representatives based on voters' preferences is a fundamental problem in social choice theory. While cardinal utility functions offer a detailed representation of preferences, ordinal rankings are often the only available information due to their simplicity and practical constraints. The metric distortion framework addresses this issue by modeling voters and candidates as points in a metric space, with distortion quantifying the efficiency loss from relying solely on ordinal rankings. Existing works define the cost of a voter with respect to a candidate as their distance and set the overall cost as either the sum (utilitarian) or maximum (egalitarian) of these costs across all voters. They show that deterministic algorithms achieve a best-possible distortion of 3 for any metric when considering a single candidate. This paper explores whether one can obtain a better approximation compared to an optimal candidate by relying on a committee of $k$ candidates ($k \ge 1$), where the cost of a voter is defined as its distance to the closest candidate in the committee. We answer this affirmatively in the case of line metrics, demonstrating that with $O(1)$ candidates, it is possible to achieve optimal cost. Our results extend to both utilitarian and egalitarian objectives, providing new upper bounds for the problem. We complement our results with lower bounds for both the line and 2-D Euclidean metrics.

cs.GT↗

arXiv2024

Prize-Collecting Steiner Tree: A 1.79 Approximation

Prize-Collecting Steiner Tree (PCST) is a generalization of the Steiner Tree problem, a fundamental problem in computer science. In the classic Steiner Tree problem, we aim to connect a set of vertices known as terminals using the minimum-weight tree in a given weighted graph. In this generalized version, each vertex has a penalty, and there is flexibility to decide whether to connect each vertex or pay its associated penalty, making the problem more realistic and practical. Both the Steiner Tree problem and its Prize-Collecting version had long-standing $2$-approximation algorithms, matching the integrality gap of the natural LP formulations for both. This barrier for both problems has been surpassed, with algorithms achieving approximation factors below $2$. While research on the Steiner Tree problem has led to a series of reductions in the approximation ratio below $2$, culminating in a $\ln(4)+\epsilon$ approximation by Byrka, Grandoni, Rothvo{\ss}, and Sanit\`a, the Prize-Collecting version has not seen improvements in the past 15 years since the work of Archer, Bateni, Hajiaghayi, and Karloff, which reduced the approximation factor for this problem from $2$ to $1.9672$. Interestingly, even the Prize-Collecting TSP approximation, which was first improved below $2$ in the same paper, has seen several advancements since then. In this paper, we reduce the approximation factor for the PCST problem substantially to 1.7994 via a novel iterative approach.

cs.DS↗

arXiv2023

2-Approximation for Prize-Collecting Steiner Forest

Approximation algorithms for the prize-collecting Steiner forest problem (PCSF) have been a subject of research for over three decades, starting with the seminal works of Agrawal, Klein, and Ravi and Goemans and Williamson on Steiner forest and prize-collecting problems. In this paper, we propose and analyze a natural deterministic algorithm for PCSF that achieves a $2$-approximate solution in polynomial time. This represents a significant improvement compared to the previously best known algorithm with a $2.54$-approximation factor developed by Hajiaghayi and Jain in 2006. Furthermore, K{\"{o}}nemann, Olver, Pashkovich, Ravi, Swamy, and Vygen have established an integrality gap of at least $9/4$ for the natural LP relaxation for PCSF. However, we surpass this gap through the utilization of a combinatorial algorithm and a novel analysis technique. Since $2$ is the best known approximation guarantee for Steiner forest problem, which is a special case of PCSF, our result matches this factor and closes the gap between the Steiner forest problem and its generalized version, PCSF.

cs.DS↗