Consider an oriented complete graph, i.e. a tournament $G=(V,E)$
Ex:
Let $A(i\in V)=\{j\in V|(j,i)\in V\}\cup\{i\}$. Is there an efficient way of knowing if it is possible to find a set of half planes $h_i\subset \mathbb{R}^2, i\in V$ s.t. $k\notin A(i)\rightarrow \bigcap_{j\in A(i)}h_j\not\subset h_k$
Ex:
context: shows up in the study of causality in 2+1 spacetime.
Take as your half-planes $$ h_i = \left\{(x,y) \in \mathbb R^2: \frac{x}{i} + \frac{y}{n+1-i} \ge 1\right\}. $$ For $n=5$, for example, we get:
The idea here is that each half-plane $h_i$ has a segment where its boundary is the boundary of the $n$-fold intersection $\bigcap_{j=1}^n h_j$. Let $P_i$ be a point slightly outside $h_i$ near the midpoint of that segment; then $P_i \notin h_i$ but $P_i \in h_j$ for all $j \ne i$. For a concrete example, take $P_i = (\frac{i^2-1}{n+1}, \frac{(n+1-i)^2-1}{n+1})$, though other points will also work.
Whatever the tournament is, and whatever the sets $A(i)$ are, if $k \notin A(i)$, then $$\bigcap_{j\in A(i)}h_j\not\subset h_k$$ because $P_k$ is in the intersection on the left-hand side but not in $h_k$.