 Solution

For fair division, let $J=\{1,\dots ,m\}$ and let the additive valuation of agent $i$ be $v_i(S)={\sum }_{j\in S}{v}_{ij}$. Use one variable $X_i$ per agent, with constraint domain $\mathcal{P}(J)$, the power set of the items. Empty bundles are permitted in this original formulation.

Require pairwise disjoint sets:Each such binary constraint involves exactly $X_i,X_{\ell }$. Require complete allocation by the single all-variable constraintFinally impose the envy-free allocation binary constraintseach on $X_i,X_{\ell }$. The two directions for a pair use different agents' additive valuations and must both hold. These conditions give precisely an ordered set partition of the items that is an envy-free allocation.