Eulerian orientations and Hadamard codes: A novel connection via counting

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.

💡 Research Summary

The paper investigates the counting problem of Eulerian orientations (#EO) under local vertex constraints, and discovers a deep connection between this problem and Hadamard codes. An Eulerian orientation of an undirected graph is a direction assignment to each edge such that every vertex has the same number of incoming and outgoing edges. The authors formalize #EO as follows: each vertex v is equipped with a 0‑1 constraint function (signature) f_v of arity equal to the degree of v. The signature’s support S(f) consists of binary strings that satisfy the local restriction; for an EO‑signature every string in the support has exactly half 1‑bits (i.e., Hamming weight = degree/2). The value of #EO(F) for a set F of signatures is the number of global edge orientations that satisfy all local constraints.

It is known that unrestricted #EO is #P‑complete (Mihail‑Winkler 1996), but certain restrictions make the problem tractable. The authors introduce two new families of signatures, denoted D₁ (δ₁‑affine) and D₀ (δ₀‑affine), which extend the previously studied affine class A. A δ₁‑affine signature is a tensor product δ₁ ⊗ g where δ₁ is the unary signature that forces a variable to be 1, and g is a signature such that pinning any variable of g to 0 yields an affine signature (or another δ₁‑affine signature). The definition of δ₀‑affine is symmetric with 0 and 1 swapped. Neither D₁ nor D₀ contains A, and the tractable classes considered are A ∪ D₁ and A ∪ D₀.

The main algorithmic contribution is a “chain reaction” procedure. A δ₁ (or δ₀) signature acts like a neutron: it initiates a propagation that creates new δ₁ (or δ₀) signatures on neighboring vertices. This propagation continues until no new δ‑signatures appear, at which point all remaining signatures are affine and can be handled by known polynomial‑time methods. The authors prove that #EO(A ∪ D₁) and #EO(A ∪ D₀) are therefore tractable (Theorem 3). However, when both δ₁‑affine and δ₀‑affine signatures are present, the chain reaction collapses (δ₁ and δ₀ annihilate each other), and the problem becomes #P‑hard (Theorem 4).

A crucial structural result (Theorem 6) characterizes the “kernels” of D₁ and D₀, i.e., the signatures that sit at the first level of the inductive definitions. The authors show that a δ₁‑affine kernel’s support is exactly an m‑multiple of a balanced 1‑Hadamard code H₁ᵇ₂ᵏ for some k ≥ 3 and m ≥ 1, or it is a three‑element support not containing δ₀. The balanced Hadamard code is obtained from Sylvester’s construction of Hadamard matrices: rows of H₂ᵏ are mapped from {1, −1} to {1, 0}, the all‑1 row is removed, leaving a binary linear code where every codeword has Hamming weight 2ᵏ − 1 (half the length). The δ₀‑affine kernels are symmetric, corresponding to the balanced 0‑Hadamard code. This establishes a precise algebraic link between the tractable signatures and a classic error‑correcting code.

The hardness proof for the mixed class D₀ ∪ D₁ relies heavily on the kernel characterization. By embedding arbitrary instances of the general #EO problem into graphs that contain both δ₁‑ and δ₀‑signatures, the authors construct a parsimonious reduction, showing that any #P‑hard counting problem can be expressed as #EO(D₀ ∪ D₁).

Beyond the immediate results, the paper situates its contributions within the broader Holant framework, a general counting paradigm that subsumes #CSP and many statistical‑physics models. Previous dichotomies for real‑valued Holant problems depended on an “arrow reversal symmetry” (ARS) that aligns with complex‑valued signatures. By working with 0‑1 signatures without assuming ARS, the authors uncover new tractable families that lie outside all previously known classes, indicating that the full classification of complex‑valued Holant problems is more intricate than anticipated.

The paper also notes concurrent independent work (Meng, Wang, Xia 2024‑2025) that extends the tractable landscape and establishes an FP‑NP/#P dichotomy for #EO problems, further confirming the relevance of the Hadamard‑code connection.

In summary, the authors (1) define two novel constrained signature classes D₁ and D₀, (2) devise a chain‑reaction algorithm that solves #EO for A ∪ D₁ and A ∪ D₀ in polynomial time, (3) prove that the kernels of these classes are exactly balanced Hadamard codes, and (4) demonstrate #P‑hardness when both types of signatures coexist. This work creates the first explicit bridge between counting Eulerian orientations and coding theory, enriches the taxonomy of tractable counting problems, and opens new avenues for both complexity classification and applications of coding theory to combinatorial counting.