§2, p. 96 — at λ_j = c_2j the variables that are zero in the (PR) solution can be set to zero in (PR_λ)

Let $m\ge2$, let $r_{ij}\ge0$, let $j\mapsto i_j$ be a cheapest-agent selection, and take the multipliers $\lambda_j=c_{2j}$, the second smallest of $c_{1j},\dots,c_{mj}$. Let $x$ be feasible for (PR$_\lambda$) (binary, with $\sum_j r_{ij}x_{ij}\le b_i$ for every agent $i$). Define $x'$ by keeping $x_{ij}$ when $i=i_j$ and setting $x'_{ij}=0$ when $i\ne i_j$, i.e. when $x_{ij}$ is zero in the (PR) solution. Then $x'$ is feasible for (PR$_\lambda$) and

So, at these multipliers, (PR$_\lambda$) may be solved over points supported on the (PR) assignment pattern. This is the third observation in the paper's verification of its principal result.

Formalization Note $r_{ij}\ge0$ is implicit in the paper ("the resource required by agent $i$ to do task $j$") and is needed: it is what keeps $x'$ within the budgets. Agents and tasks are 0-based.