Eulerian orientations and Hadamard codes: A novel connection via counting
Shuai Shao, Zhuxiao Tang
TL;DR
The authors study the counting problem for Eulerian orientations with vertex-local constraints and reveal a deep link to Hadamard codes. They introduce tractable classes formed by affine signatures together with either $\delta_1$-affine or $\delta_0$-affine kernels, and prove a chain-reaction algorithm yields polynomial-time solvability for $\#EO(\mathscr{A}\cup\mathscr{D}_1)$ and $\#EO(\mathscr{A}\cup\mathscr{D}_0)$. A precise characterization shows non-trivial $\delta_1$-affine (and $\delta_0$-affine) kernels correspond to $m$-multiples of basic kernels, which in turn are exactly the wings of a butterfly and balanced 1-Hadamard codes. The paper also proves $\#P$-hardness when both tractable classes are present, highlighting a sharp boundary between tractable and hard regimes and linking counting problems to coding-theory structures in a novel way. The results advance the Holant/#EO dichotomy program in ARS-free settings and open avenues for applying Hadamard-code insights to combinatorial counting problems.
Abstract
We discover a novel connection between two classical mathematical notions, Eulerian orientations and Hadamard codes by studying the counting problem of Eulerian orientations (\#EO) with local constraint functions imposed on vertices. We present two special classes of constraint functions and a chain reaction algorithm, and show that the \#EO problem defined by each class alone is polynomial-time solvable by the algorithm. These tractable classes of functions are defined inductively, and quite remarkably the base level of these classes is characterized perfectly by the well-known Hadamard code. Thus, we establish a novel connection between counting Eulerian orientations and coding theory. We also prove a \#P-hardness result for the \#EO problem when constraint functions from the two tractable classes appear together.