From Finite Vector Field Data to Combinatorial Dynamical Systems in the Sense of Forman

The paper defines the ILP and cost matrix and claims that any global minimizer induces a combinatorial dynamical system (Theorem 1.1/5.1) . Its proof argues (i) the constraint Σ_i a_{ik} + Σ_j a_{kj} − a_{kk} = 1 enforces that each simplex participates exactly once, yielding Dom V ∪ Im V = K and Dom V ∩ Im V = Crit V, and (ii) any non-admissible off-diagonal a_{ij}=1 can be replaced by a_{ii}=a_{jj}=1 at strictly lower cost because C_{ij} = max(2α+1,3) > 2α = C_{ii}+C_{jj} . The model’s solution gives the same two steps, but more cleanly via row/column sums (r_k, c_k) to establish partial injectivity and the Dom/Im identities, and then the local replacement argument for non-admissible pairs. Minor wording imprecision in the paper (“admissible matching” vs. “matching feasibility”) aside, both arguments establish the same result with essentially the same proof skeleton.