Problem
Research questions and friction points this paper is trying to address.
Holant*
complexity dichotomy
domain size 3
complex-valued constraint functions
computational complexity
Group Action and Spin Representation Lead to a Holant* Complexity Dichotomy on Domain Size 3
🤖 AI Summary
This study addresses the open problem of establishing a computational complexity dichotomy for Holant* problems with symmetric constraint functions of arity three over the complex domain. Methodologically, the analysis employs group actions, SL(2,ℂ) spin representations, generalized orthogonal groups, and bilinear forms. By introducing core concepts such as frameworks, shells, and annihilators, this work constructs the first unified analytical framework for high-dimensional complex-valued constraints. The primary contribution lies in establishing a decidable tractability criterion, rigorously proving that instances satisfying it are solvable in polynomial time, while all others are #P-hard. Consequently, this research achieves a complete classification of Holant* complexity in this setting, bridging a critical gap in the study of high-dimensional complex-valued functions, revealing deep underlying algebraic structures, and providing comprehensive foundations for both algorithm design and hardness proofs.
📝 Abstract
We prove a complexity dichotomy theorem for $\mathrm{Holant}^*$ problems over complex-valued symmetric constraint functions $\mathcal{F}$ on domain size $3$. We give a decidable tractability criterion and prove that if $\mathcal{F}$ satisfies the criterion, then $\mathrm{Holant}^*(\mathcal{F})$ is solvable in polynomial time, and otherwise it is #P-hard. This is the first Holant dichotomy for a set of complex-valued constraint functions on higher domains. We show that complex-valued constraint functions have a rich structure not observed in real-valued constraint functions. This structure is only revealed when we analyze them in a bipartite Holant setting with a non-standard bilinear form and provides the backbone to the proof of the dichotomy. We use group actions and the spin representation of $\mathrm{SL}(2, \mathbb{C})$ in $\mathrm{SO}(3, \mathbb{C})$ and the generalized orthogonal group to facilitate this analysis. We also introduce $\textit{frames}$ and $\textit{shells}$. Frames linearize the group action and provide a unified framework for proving #P-hardness when used in conjunction with shells. Furthermore, we characterize the lower dimensional constraint functions by $\textit{annihilators}$, which makes it possible to analyze $\textit{essentially Boolean domain}$ functions in domain size $3$.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Holant* dichotomy
group action
spin representation
frames and shells
complex-valued constraint functions
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