Past exam of the computer science course of the University of Oxford / 2023 / Combinatorial optimisation / 1 / b

The intersection–union definition of a submodular set function isThe diminishing returns characterization of submodularity says that its marginal increments decrease as the conditioning set grows:To obtain diminishing returns from submodularity, apply the intersection–union inequality to $A\cup \{x\}$ and $B$. Their intersection is $A$ and their union is $B\cup \{x\}$, giving exactly the displayed comparison.

Conversely, assume diminishing returns, and enumerate $A\setminus B=\{x_1,\dots ,x_m\}$. Put $C_j=(A\cap B)\cup \{x_1,\dots ,x_j\}$ and $D_j=B\cup \{x_1,\dots ,x_j\}$. Since $C_{j-1}\subseteq D_{j-1}$ and $x_j\notin D_{j-1}$, summing the marginal increment comparisons gives the telescoping sumRearrangement proves submodularity. When $A\setminus B$ is empty, both sums are zero, so the proof includes that case.