... ... @@ -160,7 +160,11 @@ statement with a condition on $R$ might be helpful: \end{figure} \FloatBarrier The interesting score change is the following quadratic one between the pixels $u$ and $v$: between the pixels $u$ and $v$. Here $p_c$ denotes the amount of crop $c$ pixel $p$ contains before the swap. $p_c'$ is defined analogously for after the swap. \begin{align*} &\sum_{c_i \in \Cps} \sum_{c_j \in \Cps} R(c_i, c_j) \cdot (u'_i \cdot v'_{j} - u_i \cdot v_{j})\\ &= \sum_{c_i \in \{c_\alpha, c_\beta \}} ... ... @@ -222,9 +226,25 @@ statement with a condition on $R$ might be helpful: &+ \lambda \cdot (R(c_\beta, c_\alpha) - R(c_\alpha, c_\alpha)) \cdot v_\alpha\\ &+ \lambda \cdot (R(c_\beta, c_\beta) - R(c_\alpha, c_\beta)) \cdot v_\beta \end{align*} Which, if every crop likes all other crops more than itself, can be made positive with the appropriate $\lambda$. Then the quadratic term in the formula above will always be positive, and we can focus only on all of the linear constraints. Of special note here is the quadratic term. If every crop likes all other crops at least as much as it likes itself, then the quadratic term is positive. The quadratic term being positive means we can always swap in at least one direction without the score getting worse. The optimal swap direction is determined by all of the linear functions in $\lambda$ we put to the side throughout this proof. Therefore a relationship function $R$ fulfilling $R(c_i, c_j) \geq R(c_i, c_i)$ ensures there is always a good way to swap crops among neighboring pixels. Being allowed to swap even amongst neighboring pixels allows us to disregard the pattern structure we had for the standard swapping method and we can by the same reasoning conclude that this new method has after termination at most $\binom{C}{2}$ still fractional pixels. \end{proof} \ No newline at end of file