Theorem 7, p. 1434 — the VRHS production cut (32) along a chain c₁ ≺ ⋯ ≺ c_n is valid for PCPSP-F, and for PCPSP-P under (33)

Consider an instance of the PCPSP-C satisfying the standing assumptions. Fix a destination $d\in\mathcal D$, a period $t\in\mathcal T$ and clusters $c_1,\dots,c_n\in\mathcal C$ ($n\ge 1$) such that

1. $c_1\prec c_2\prec\dots\prec c_n$,
2. $q(rcl(c_1)\setminus rcl(c_n))<U^d_t$,
3. $q(rcl(c_1))>U^d_t$.

Let $\Delta_k=rcl(c_k)\setminus rcl(c_{k+1})$ and $\delta_k=q(\Delta_k)$ for $k=1,\dots,n-1$, and $\Delta_n=rcl(c_n)$, $\delta_n=U^d_t-q(rcl(c_1)\setminus rcl(c_n))$. Let $0\le\alpha_b\le 1$ for $b\in c_n$, and consider, with $w_{c,t}=\sum_{t'=1}^{t}x_{c,t'}$, the inequality

Then (32) holds at every $(x,y)$ feasible for the PCPSP-F that satisfies the destination capacity rows (9). If moreover

then (32) holds at every $(x,y)$ feasible for the PCPSP-P that satisfies (9).

The VRHS cuts couple the precedence structure along a chain with the capacity of one destination: they tighten the linear relaxation of the PCPSP-C at points that send heavy successors of a partially extracted cluster to a capacitated destination.

Formalization Note "Valid for PCPSP-F (PCPSP-P)" is read as: valid at every point feasible for (1)–(7) with integrality (10) (resp. (11)) whose side constraints $Gy\le g$ include the destination capacity rows (9) (p. 1427: "constraints (5) typically include conditions of the form (8), (9)"); (9) is an explicit hypothesis. Condition 1 is stated for consecutive pairs; the chain is c : ℕ → C read at $1,\dots,n$. Periods are Fin T (index $t$ is period $t+1$); $y$ is taken at the single period $t$ and $w$ is cumulative.