arXiv subjects

Dhara Thakkar

Publications and source records attributed to Dhara Thakkar.

11 recordsLinked to original sources

arXiv2026

Algorithms for Finite Group Epimorphism Testing

The Group Epimorphism Problem (GpEpi) asks, given two finite groups $G_1$ and $G_2$, whether there exists a surjective group homomorphism, or epimorphism, from $G_1$ to $G_2$. When the input groups are given by their multiplication (Cayley) tables, the problem admits a quasipolynomial-time algorithm in general, but little is known about its complexity for structured classes of finite groups. In this paper, we study the computational complexity of GpEpi for several well-studied classes of finite groups. Our main results are polynomial-time epimorphism tests for several classes of groups for which polynomial-time isomorphism testing was previously known: Groups with Abelian normal Hall subgroups with cyclic complement; Groups with (product of) elementary Abelian normal Hall subgroup with elementary Abelian complement; and Groups with some constraints on their Abelian chief factors.

cs.DS↗

arXiv2026

The Geometry of Dissipative Complexity: A Levi-Type Decomposition Theorem for Markovian Quantum Dynamics via Lie Wedges and Invariant Cones

Lie algebras describe how control Hamiltonians combine in closed quantum systems. In open Markovian systems, however, dissipation introduces irreversible directions that the Lie algebra alone does not retain. A dynamical Lie wedge preserves this information as a convex cone of locally admissible generators. The classical Levi decomposition separates a finite-dimensional Lie algebra into semisimple and solvable parts. In this work, we prove a corresponding decomposition and reconstruction theorem for dynamical Lie wedges. The theorem decomposes the wedge into semisimple and solvable data, records how these components are coupled, and provides a converse reconstruction of the wedge. The framework yields four structural types of the generated algebra. We further prove that the total dissipation strength of every admissible generator depends only on its coordinate in the solvable radical. In particular, if the generated Lie algebra is semisimple, every admissible generator is Hamiltonian. Together, these results extend Lie-algebraic structural methods to irreversible Markovian control and clarify how dissipation is encoded in the generator geometry.

quant-ph↗

arXiv2026

QEDBENCH: Quantifying the Alignment Gap in Automated Evaluation of University-Level Mathematical Proofs

As Large Language Models (LLMs) saturate elementary benchmarks, the research frontier has shifted from generation to the reliability of automated evaluation. We demonstrate that standard "LLM-as-a-Judge" protocols suffer from a systematic Alignment Gap when applied to upper-undergraduate to early graduate level mathematics. To quantify this, we introduce QEDBench, the first large-scale dual-rubric alignment benchmark to systematically measure alignment with human experts on university-level math proofs by contrasting course-specific rubrics against expert common knowledge criteria. By deploying a dual-evaluation matrix (7 judges x 5 solvers) against 1,000+ hours of human evaluation, we reveal that certain frontier evaluators like Claude Opus 4.5, DeepSeek-V3, Qwen 2.5 Max, and Llama 4 Maverick exhibit significant positive bias (up to +0.18, +0.20, +0.30, +0.36 mean score inflation, respectively). Furthermore, we uncover a critical reasoning gap in the discrete domain: while Gemini 3.0 Pro achieves state-of-the-art performance (0.91 average human evaluation score), other reasoning models like GPT-5 Pro and Claude Sonnet 4.5 see their performance significantly degrade in discrete domains. Specifically, their average human evaluation scores drop to 0.72 and 0.63 in Discrete Math, and to 0.74 and 0.50 in Graph Theory. In addition to these research results, we also release QEDBench as a public benchmark for evaluating and improving AI judges. Our benchmark is publicly published at https://github.com/qqliu/Yale-QEDBench.

cs.LG↗

arXiv2025

An Arithmetic Characterization of 2-Generated Numbers

A group $G$ is said to be $k$-generated if it has a generating set with $k$ elements. A positive integer $n$ is called a \emph{2-generated number} if every group of order $n$ is 2-generated. In this article, we establish an arithmetic characterization of 2-generated numbers expressed in terms of the prime factorization of $n$.

math.GR↗

arXiv2025

Group Order is in QCMA

In this work, we show that verifying the order of a finite group given as a black-box is in the complexity class QCMA. This solves an open problem asked by Watrous in 2000 in his seminal paper on quantum proofs and directly implies that the Group Non-Membership problem is also in the class QCMA, which further proves a conjecture proposed by Aaronson and Kuperberg in 2006. Our techniques also give improved quantum upper bounds on the complexity of many other group-theoretical problems, such as group isomorphism in black-box groups.

quant-ph↗

arXiv2025

Complexity of Constructing Minimal Faithful Permutation Representations for Fitting-free Groups

In this paper, we investigate the complexity of computing minimal faithful permutation representations for groups without abelian normal subgroups (a.k.a. Fitting-free groups). When our groups are given as quotients of permutation groups, we exhibit a polynomial-time algorithm for constructing such representations. Furthermore, in the setting of permutation groups, we obtain an $\textsf{NC}$ procedure for computing the minimal faithful permutation degree, and a randomized $\textsf{NC}$ ($\textsf{RNC}$) algorithm for computing a minimal faithful permutation representation. This improves upon the work of Das and Thakkar (STOC 2024, SIAM J. Comput. 2026), who established a Las Vegas polynomial-time algorithm for computing the minimal faithful permutation degree for this class in the setting of permutation groups.

cs.DS↗

arXiv2023

The Frobenius Problem for the Proth Numbers

Let $n$ be a positive integer greater than $2$. We define \textit{the Proth numerical semigroup}, $P_{k}(n)$, generated by $\{k 2^{n+i}+1 \,\mid\, i \in \mathbb{N}\}$, where $k$ is an odd positive number and $k < 2^{n}$. In this paper, we introduce the Frobenius problem for the Proth numerical semigroup $P_{k}(n)$ and give formulas for the embedding dimension of $P_{k}(n)$. We solve the Frobenius problem for $P_{k}(n)$ by giving a closed formula for the Frobenius number. Moreover, we show that $P_{k}(n)$ has an interesting property such as being Wilf.

math.CO↗

arXiv2023

On the Complexity of the Eigenvalue Deletion Problem

For any fixed positive integer $r$ and a given budget $k$, the $r$-\textsc{Eigenvalue Vertex Deletion} ($r$-EVD) problem asks if a graph $G$ admits a subset $S$ of at most $k$ vertices such that the adjacency matrix of $G\setminus S$ has at most $r$ distinct eigenvalues. The edge deletion, edge addition, and edge editing variants are defined analogously. For $r = 1$, $r$-EVD is equivalent to the Vertex Cover problem. For $r = 2$, it turns out that $r$-EVD amounts to removing a subset $S$ of at most $k$ vertices so that $G\setminus S$ is a cluster graph where all connected components have the same size. We show that $r$-EVD is NP-complete even on bipartite graphs with maximum degree four for every fixed $r > 2$, and FPT when parameterized by the solution size and the maximum degree of the graph. We also establish several results for the special case when $r = 2$. For the vertex deletion variant, we show that $2$-EVD is NP-complete even on triangle-free and $3d$-regular graphs for any $d\geq 2$, and also NP-complete on $d$-regular graphs for any $d\geq 8$. The edge deletion, addition, and editing variants are all NP-complete for $r = 2$. The edge deletion problem admits a polynomial time algorithm if the input is a cluster graph, while the edge addition variant is hard even when the input is a cluster graph. We show that the edge addition variant has a quadratic kernel. The edge deletion and vertex deletion variants are FPT when parameterized by the solution size alone. Our main contribution is to develop the complexity landscape for the problem of modifying a graph with the aim of reducing the number of distinct eigenvalues in the spectrum of its adjacency matrix. It turns out that this captures, apart from Vertex Cover, also a natural variation of the problem of modifying to a cluster graph as a special case, which we believe may be of independent interest.

cs.DS↗

arXiv2023

The Minimum Generating Set Problem

Let $G$ be a finite group. In order to determine the smallest cardinality $d(G)$ of a generating set of $G$ and a generating set with this cardinality, one should repeat many times the test whether a subset of $G$ of small cardinality generates $G$. We prove that if a chief series of $G$ is known, then the numbers of these generating tests can be drastically reduced. At most $|G|^{13/5}$ subsets must be tested. This implies that the minimum generating set problem for a finite group $G$ can be solved in polynomial time.

math.GR↗

arXiv2023

Algorithms for the Minimum Generating Set Problem

For a finite group $G$, the size of a minimum generating set of $G$ is denoted by $d(G)$. Given a finite group $G$ and an integer $k$, deciding if $d(G)\leq k$ is known as the minimum generating set (MIN-GEN) problem. A group $G$ of order $n$ has generating set of size $\lceil \log_p n \rceil$ where $p$ is the smallest prime dividing $n=|G|$. This fact is used to design an $n^{\log_p n+O(1)}$-time algorithm for the group isomorphism problem of groups specified by their Cayley tables (attributed to Tarjan by Miller, 1978). The same fact can be used to give an $n^{\log_p n+O(1)}$-time algorithm for the MIN-GEN problem. We show that the MIN-GEN problem can be solved in time $n^{(1/4)\log_p n+O(1)}$ for general groups given by their Cayley tables. This runtime incidentally matches with the runtime of the best known algorithm for the group isomorphism problem. We show that if a group $G$, given by its Cayley table, is the product of simple groups then a minimum generating set of $G$ can be computed in time polynomial in $|G|$. Given groups $G_i$ along with $d(G_i)$ for $i\in [r]$ the problem of computing $d(\Pi_{i\in[r]} G_i)$ is nontrivial. As a consequence of our result for products of simple groups we show that this problem also can be solved in polynomial time for Cayley table representation. For the MIN-GEN problem for permutation groups, to the best of our knowledge, no significantly better algorithm than the brute force algorithm is known. For an input group $G\leq S_n$, the brute force algorithm runs in time $|G|^{O(n)}$ which can be $2^{\Omega(n^2)}$. We show that if $G\leq S_n$ is a primitive permutation group then the MIN-GEN problem can be solved in time quasi-polynomial in $n$. We also design a $\mathrm{DTIME}(2^n)$ algorithm for computing a minimum generating set of permutation groups all of whose non-abelian chief factors have bounded orders.

math.GR↗

arXiv2023

Linear Space Data Structures for Finite Groups with Constant Query-time

A finite group of order $n$ can be represented by its Cayley table. In the word-RAM model the Cayley table of a group of order $n$ can be stored using $O(n^2)$ words and can be used to answer a multiplication query in constant time. It is interesting to ask if we can design a data structure to store a group of order $n$ that uses $o(n^2)$ space but can still answer a multiplication query in constant time. We design a constant query-time data structure that can store any finite group using $O(n)$ words where $n$ is the order of the group. Farzan and Munro (ISSAC 2006) gave an information theoretic lower bound of $\Omega(n)$ on the number of words to store a group of order $n$. Since our data structure achieves this lower bound and answers queries in constant time, it is optimal in both space usage and query-time. A crucial step in the process is essentially to design linear space and constant query-time data structures for nonabelian simple groups. The data structures for nonableian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups (CFSG).

cs.DS↗