The Parameterized Complexity of Finding Point Sets with Hereditary Properties

We consider problems where the input is a set of points in the plane and an integer $k$, and the task is to find a subset $S$ of the input points of size $k$ such that $S$ satisfies some property. We focus on properties that depend only on the order type of the points and are monotone under point removals. We show that not all such problems are fixed-parameter tractable parameterized by $k$, by exhibiting a property defined by three forbidden patterns for which finding a $k$-point subset with the property is $\mathrm{W}[1]$-complete and (assuming the exponential time hypothesis) cannot be solved in time $n^{o(k/\log k)}$. However, we show that problems of this type are fixed-parameter tractable for all properties that include all collinear point sets, properties that exclude at least one convex polygon, and properties defined by a single forbidden pattern.