An Efficient, Lean 4 Verified Necessary and Sufficient Condition for Membership in Pedigree Polytopes
Abstract
The set Pn consists of the characteristic vectors of pedigrees, which are combinatorial encodings in one-to-one correspondence with Hamiltonian cycles. For n ≥ 4 and τn = binomn3−1, the Membership Problem for the Pedigree Polytope (M3P) asks whether a given vector X ∈ ℚτn belongs to conv(Pn). This paper gives a complete account, with the full proof statements and machine-verified Lean~4 certificates, of the necessary and sufficient (\NS) condition established in Chapter~5 of \emph{Pedigree Polytopes: New Insights on Computational Complexity of Combinatorial Optimisation Problems} (Arthanari, 2023): membership of X in conv(Pn) is characterised by the feasibility, at every stage k=4,…,n−1, of a recursively constructed layered network, together with, when needed, a multicommodity flow problem MCF(k−1) achieving its maximum possible flow zmax. We present the construction, the \NS{} condition itself, and the proofs of this characterisation, together with the resulting polynomial-time algorithm for M3P. Both directions of the \NS{} condition --- sufficiency and necessity (the multicommodity-flow feasibility argument of the book's Chapter~5) --- are machine-verified in Lean~4/Mathlib4 with zero unresolved \texttt{sorry}s in either dependency chain. The \NS{} characterisation therefore follows directly from the two machine-checked implications. The complete Lean~4 project is publicly available. A companion introductory article (arXiv:2507.09069) presents the same material for a broader readership, without the Lean~4 verification; the present paper adds full statements, proof sketches, and the complete machine-checked certificates.
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BibTeX
@misc{arthanari2026efficient,
title = {{An Efficient, Lean 4 Verified Necessary and Sufficient Condition for Membership in Pedigree Polytopes}},
author = {T. S. Arthanari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15914},
eprint = {2610.15914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15914},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - An Efficient, Lean 4 Verified Necessary and Sufficient Condition for Membership in Pedigree Polytopes
AU - T. S. Arthanari
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15914
UR - https://arxiv.org/abs/2610.15914
AB - The set P_n consists of the characteristic vectors of pedigrees, which are combinatorial encodings in one-to-one correspondence with Hamiltonian cycles. For n ≥ 4 and τ_n = binomn3−1, the Membership Problem for the Pedigree Polytope (M3P) asks whether a given vector X ∈ ℚ^τ_n belongs to conv(P_n). This paper gives a complete account, with the full proof statements and machine-verified Lean~4 certificates, of the necessary and sufficient (\NS) condition established in Chapter~5 of \emph{Pedigree Polytopes: New Insights on Computational Complexity of Combinatorial Optimisation Problems} (Arthanari, 2023): membership of X in conv(P_n) is characterised by the feasibility, at every stage k=4,…,n−1, of a recursively constructed layered network, together with, when needed, a multicommodity flow problem MCF(k−1) achieving its maximum possible flow z_max. We present the construction, the \NS{} condition itself, and the proofs of this characterisation, together with the resulting polynomial-time algorithm for M3P. Both directions of the \NS{} condition --- sufficiency and necessity (the multicommodity-flow feasibility argument of the book's Chapter~5) --- are machine-verified in Lean~4/Mathlib4 with zero unresolved \texttt{sorry}s in either dependency chain. The \NS{} characterisation therefore follows directly from the two machine-checked implications. The complete Lean~4 project is publicly available. A companion introductory article (arXiv:2507.09069) presents the same material for a broader readership, without the Lean~4 verification; the present paper adds full statements, proof sketches, and the complete machine-checked certificates.
ER -