arXiv subjects

Zhiguo Fu

Publications and source records attributed to Zhiguo Fu.

15 recordsLinked to original sources

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Improving Role Consistency in Multi-Agent Collaboration via Quantitative Role Clarity

In large language model (LLM)-driven multi-agent systems, disobey role specification (failure to adhere to the defined responsibilities and constraints of an assigned role, potentially leading to an agent behaving like another) is a major failure mode \cite{DBLP:journals/corr/abs-2503-13657}. To address this issue, in the present paper, we propose a quantitative role clarity to improve role consistency. Firstly, we construct a role assignment matrix $S(\phi)=[s_{ij}(\phi)]$, where $s_{ij}(\phi)$ is the semantic similarity between the $i$-th agent's behavior trajectory and the $j$-th agent's role description. Then we define role clarity matrix $M(\phi)$ as $\text{softmax}(S(\phi))-I$, where $\text{softmax}(S(\phi))$ is a row-wise softmax of $S(\phi)$ and $I$ is the identity matrix. The Frobenius norm of $M(\phi)$ quantifies the alignment between agents' role descriptions and their behaviors trajectory. Moreover, we employ the role clarity matrix as a regularizer during lightweight fine-tuning to improve role consistency, thereby improving end-to-end task performance. Experiments on the ChatDev multi-agent system show that our method substantially improves role consistency and task performance: with Qwen and Llama, the role overstepping rate decreases from $46.4\%$ to $8.4\%$ and from $43.4\%$ to $0.2\%$, respectively, and the role clarity score increases from $0.5328$ to $0.9097$ and from $0.5007$ to $0.8530$, respectively, the task success rate increases from $0.6769$ to $0.6909$ and from $0.6174$ to $0.6763$, respectively.

cs.AI↗

arXiv2023

Approximability of the Four-Vertex Model

We study the approximability of the four-vertex model, a special case of the six-vertex model.We prove that, despite being NP-hard to approximate in the worst case, the four-vertex model admits a fully polynomial randomized approximation scheme (FPRAS) when the input satisfies certain linear equation system over GF(2).The FPRAS is given by a Markov chain known as the worm process, whose state space and rapid mixing rely on the solution of the linear equation system. This is the first attempt to design an FPRAS for the six-vertex model with unwindable constraint functions.Additionally, we explore the applications of this technique on planar graphs, providing efficient sampling algorithms.

cs.CC↗

arXiv2022

Beyond Windability: An FPRAS for The Six-Vertex Model

The six-vertex model is an important model in statistical physics and has deep connections with counting problems. There have been some fully polynomial randomized approximation schemes (FPRAS) for the six-vertex model [30, 10], which all require that the constraint functions are windable. In the present paper, we give an FPRAS for the six-vertex model with an unwindable constraint function by Markov Chain Monte Carlo method (MCMC). Different from [10], we use the Glauber dynamics to design the Markov Chain depending on a circuit decomposition of the underlying graph. Moreover, we prove the rapid mixing of the Markov Chain by coupling, instead of canonical paths in [10].

math.PR↗

arXiv2022

The Implicit Regularization of Momentum Gradient Descent with Early Stopping

The study on the implicit regularization induced by gradient-based optimization is a longstanding pursuit. In the present paper, we characterize the implicit regularization of momentum gradient descent (MGD) with early stopping by comparing with the explicit $\ell_2$-regularization (ridge). In details, we study MGD in the continuous-time view, so-called momentum gradient flow (MGF), and show that its tendency is closer to ridge than the gradient descent (GD) [Ali et al., 2019] for least squares regression. Moreover, we prove that, under the calibration $t=\sqrt{2/\lambda}$, where $t$ is the time parameter in MGF and $\lambda$ is the tuning parameter in ridge regression, the risk of MGF is no more than 1.54 times that of ridge. In particular, the relative Bayes risk of MGF to ridge is between 1 and 1.035 under the optimal tuning. The numerical experiments support our theoretical results strongly.

cs.LG↗

arXiv2020

From Holant to Quantum Entanglement and Back

Holant problems are intimately connected with quantum theory as tensor networks. We first use techniques from Holant theory to derive new and improved results for quantum entanglement theory. We discover two particular entangled states $|{\Psi_6}\rangle$ of 6 qubits and $|{\Psi_8}\rangle$ of 8 qubits respectively, that have extraordinary and unique closure properties in terms of the Bell property. Then we use entanglement properties of constraint functions to derive a new complexity dichotomy for all real-valued Holant problems containing an odd-arity signature. The signatures need not be symmetric, and no auxiliary signatures are assumed.

cs.CC↗

arXiv2019

Beyond #CSP: A Dichotomy for Counting Weighted Eulerian Orientations with ARS

We define and explore a notion of unique prime factorization for constraint functions, and use this as a new tool to prove a complexity classification for counting weighted Eulerian orientation problems with arrow reversal symmetry (ARS). We prove that all such problems are either polynomial-time computable or #P-hard. We show that the class of weighted Eulerian orientation problems subsumes all weighted counting constraint satisfaction problems (#CSP) on Boolean variables. More significantly, we establish a novel connection between #CSP and counting weighted Eulerian orientation problems that is global in nature. This connection is based on a structural determination of all half-weighted affine linear subspaces over $\mathbb{Z}_2$, which is proved using M\"obius inversion.

cs.CC↗

arXiv2017

On Blockwise Symmetric Matchgate Signatures and Higher Domain \#CSP

For any $n\geq 3$ and $ q\geq 3$, we prove that the {\sc Equality} function $(=_n)$ on $n$ variables over a domain of size $q$ cannot be realized by matchgates under holographic transformations. This is a consequence of our theorem on the structure of blockwise symmetric matchgate signatures. %due to the rank of the matrix form of the blockwise symmetric standard signatures, %where $(=_n)$ is an equality signature on domain $\{0, 1, \cdots, q-1\}$. This has the implication that the standard holographic algorithms based on matchgates, a methodology known to be universal for \#CSP over the Boolean domain, cannot produce P-time algorithms for planar \#CSP over any higher domain $q\geq 3$.

cs.CC↗

arXiv2017

New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification

We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in ${\mathbb C}$ for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar graphs. The classification has an explicit criterion. The new P-time cases in (2) provably cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local connection to #CSP, defined in terms of a "loop space". This is the first substantive advance toward a planar Holant classification with not necessarily symmetric constraints. We introduce M\"obius transformation on ${\mathbb C}$ as a powerful new tool in hardness proofs for counting problems.

cs.CC↗

arXiv2017

Complexity Classification of the Eight-Vertex Model

We prove a complexity dichotomy theorem for the eight-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or \#P-hard. The dichotomy criterion is explicit. For tractability, we find some new classes of problems computable in polynomial time. For \#P-hardness, we employ M\"{o}bius transformations to prove the success of interpolations.

cs.CC↗

arXiv2017

Complexity Classification Of The Six-Vertex Model

We prove a complexity dichotomy theorem for the six-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit.

cs.CC↗

arXiv2016

Holographic Algorithm with Matchgates Is Universal for Planar $\#$CSP Over Boolean Domain

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).

cs.CC↗

arXiv2015

A Holant Dichotomy: Is the FKT Algorithm Universal?

We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a holographic transformation a \emph{universal} strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable in general? This dichotomy is a culmination of previous ones, including those for Spin Systems, Holant, and #CSP. A recurring theme has been that a holographic reduction to FKT is a universal strategy. Surprisingly, for planar Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In previous work, an important tool was a dichotomy for #CSP^d, which denotes #CSP where every variable appears a multiple of d times. However its proof violates planarity. We prove a dichotomy for planar #CSP^2. We apply this planar #CSP^2 dichotomy in the proof of the planar Holant dichotomy. As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is polynomial-time computable when the incidence graph is planar and k >= 5. The same problem is #P-hard when k=3 or k=4, which is also a consequence of our dichotomy. When k=2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor of all hyperedge sizes is at least 5.

cs.CC↗

arXiv2013

A Collapse Theorem for Holographic Algorithms with Matchgates on Domain Size at Most 4

Holographic algorithms with matchgates are a novel approach to design polynomial time computation. It uses Kasteleyn's algorithm for perfect matchings, and more importantly a holographic reduction . The two fundamental parameters of a holographic reduction are the domain size $k$ of the underlying problem, and the basis size $\ell$. A holographic reduction transforms the computation to matchgates by a linear transformation that maps to (a tensor product space of) a linear space of dimension $2^{\ell}$. We prove a sharp basis collapse theorem, that shows that for domain size 3 and 4, all non-trivial holographic reductions have basis size $\ell$ collapse to 1 and 2 respectively. The main proof techniques are Matchgates Identities, and a Group Property of matchgates signatures.

cs.CC↗

arXiv2013

Holographaic Alogorithms on Bases of Rank 2

An essential problem in the design of holographic algorithms is to decide whether the required signatures can be realized by matchgates under a suitable basis transformation (SRP). For holographic algorithms on domain size 2, [1, 2, 4, 5] have built a systematical theory. In this paper, we reduce SRP on domain size k>2 to SRP on domain size 2 for holographic algorithms on bases of rank 2. Furthermore, we generalize the collapse theorem of [3] to domain size k>2.

cs.CC↗