Eulerian orientations and Hadamard codes: A novel connection via counting

TL;DR

This paper uncovers a new link between Eulerian orientations and Hadamard codes by analyzing counting problems with specific local constraints, revealing polynomial-time solutions for certain classes and hardness when combining them.

Contribution

It introduces two special classes of local constraints for counting Eulerian orientations and establishes a novel connection to Hadamard codes, along with complexity results.

Findings

Polynomial-time algorithms for specific classes of Eulerian orientation counting problems.

Characterization of base classes of constraints by Hadamard codes.

ifficulties in combining classes, leading to -hardness results.

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.