# Skew-supermodular colouring with two class sizes

Let $p_1$ and $p_2$ be integer skew-supermodular set functions on ground set S such that $\max\{p_1(X),p_2(X)\}\leq \min\{|X|,k\}$ for every $X \subseteq S$, and let $m_1,m_2,n_1,n_2$ be positive integers such that $m_1 n_1+m_2 n_2=|S|$. Can we decide in polynomial time if there is a partition ${\mathcal P}$ of S with $m_1$ classes of size $n_1$ and $m_2$ classes of size $n_2$, such that $|\{Y \in {\mathcal P}: Y \cap X \neq \emptyset\}| \geq \max\{p_1(X), p_2(X)\}$ for every $X \subseteq S$?
The problem is open even in the special case if the two functions $p_1$ and $p_2$ are intersecting supermodular. If $|n_1-n_2| \leq 1$, then the partition always exists (and can be found in polynomial time) by the skew-supermodular variant[1] of Schrijver's Supermodular colouring theorem. Another known special case is the result of Folkman and Fulkerson [2] that characterizes bipartite graphs that can be partitioned into $m_1$ matchings of size $n_1$ and $m_2$ matchings of size $n_2$. For three distinct class sizes, the problem becomes NP-complete [3], see Skew-supermodular colouring with prescribed class-sizes.