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.
@InProceedings{shao_et_al:LIPIcs.ITCS.2025.86, author = {Shao, Shuai and Tang, Zhuxiao}, title = {{Eulerian Orientations and Hadamard Codes: A Novel Connection via Counting}}, booktitle = {16th Innovations in Theoretical Computer Science Conference (ITCS 2025)}, pages = {86:1--86:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-361-4}, ISSN = {1868-8969}, year = {2025}, volume = {325}, editor = {Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.86}, URN = {urn:nbn:de:0030-drops-227146}, doi = {10.4230/LIPIcs.ITCS.2025.86}, annote = {Keywords: Eulerian orientations, Hadamard codes, Counting problems, Tractable classes} }
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