An Efficient, Lean 4 Verified Necessary and Sufficient Condition for Membership in Pedigree Polytopes

Abstract

The set \(P_{n}\) consists of the characteristic vectors of pedigrees, which are combinatorial encodings in one-to-one correspondence with Hamiltonian cycles. For \(n \geq 4\) and \(τ_{n} = \binom{n}{3}-1\), the Membership Problem for the Pedigree Polytope (M3P) asks whether a given vector \(X \in \mathbb{Q}^{τ_{n}}\) belongs to \(\mathrm{conv}(P_n)\). 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 $\mathrm{conv}(P_n)$ is characterised by the feasibility, at every stage $k=4,\dots,n-1$, of a recursively constructed layered network, together with, when needed, a multicommodity flow problem $\mathrm{MCF}(k-1)$ 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.