(i) Consider a network with node set $N$ and set of directed arcs $A$ equipped with functions $d^{+}: A \rightarrow \mathbb{Z}$ and $d^{-}: A \rightarrow \mathbb{Z}$ with $d^{-} \leqslant d^{+}$. Given $u: N \rightarrow \mathbb{R}$ we define the differential $\Delta u: A \rightarrow \mathbb{R}$ by $\Delta u(j)=u\left(i^{\prime}\right)-u(i)$ for $j=\left(i, i^{\prime}\right) \in A$. We say that $\Delta u$ is a feasible differential if

$$d^{-}(j) \leqslant \Delta u(j) \leqslant d^{+}(j) \text { for all } j \in A$$

Write down a necessary and sufficient condition on $d^{+}, d^{-}$for the existence of a feasible differential and prove its necessity.

Assuming Minty's Lemma, describe an algorithm to construct a feasible differential and outline how this algorithm establishes the sufficiency of the condition you have given.

(ii) Let $E \subseteq S \times T$, where $S, T$ are finite sets. A matching in $E$ is a subset $M \subseteq E$ such that, for all $s, s^{\prime} \in S$ and $t, t^{\prime} \in T$,

$$\begin{array}{lll} (s, t),\left(s^{\prime}, t\right) \in M & \text { implies } & s=s^{\prime} \\ (s, t),\left(s, t^{\prime}\right) \in M & \text { implies } & t=t^{\prime} \end{array}$$

A matching $M$ is maximal if for any other matching $M^{\prime}$ with $M \subseteq M^{\prime}$ we must have $M=M^{\prime}$. Formulate the problem of finding a maximal matching in $E$ in terms of an optimal distribution problem on a suitably defined network, and hence in terms of a standard linear optimization problem.

[You may assume that the optimal distribution subject to integer constraints is integervalued.]