On the Parameterized Complexity of Red-Blue Points Separation

TL;DR

This paper investigates the computational complexity of separating red and blue points with lines, showing that the general problem is unlikely to be fixed-parameter tractable, but axis-parallel variants are solvable efficiently under certain conditions.

Contribution

It proves the parameterized hardness of the general separation problem and provides fixed-parameter algorithms for the axis-parallel case.

Findings

General problem unlikely to be solved faster than $n^{O(k)}$ time under ETH.

Axis-parallel line separation is fixed-parameter tractable in the size of the blue set.

Algorithm for axis-parallel separation runs in $O^*(9^{|B|})$ time.

Abstract

We study the following geometric separation problem: Given a set $R$ of red points and a set $B$ of blue points in the plane, find a minimum-size set of lines that separate $R$ from $B$. We show that, in its full generality, parameterized by the number of lines $k$ in the solution, the problem is unlikely to be solvable significantly faster than the brute-force $n^{O(k)}$-time algorithm, where $n$ is the total number of points. Indeed, we show that an algorithm running in time $f(k)n^{o(k/ \log k)}$, for any computable function $f$, would disprove ETH. Our reduction crucially relies on selecting lines from a set with a large number of different slopes (i.e., this number is not a function of $k$). Conjecturing that the problem variant where the lines are required to be axis-parallel is FPT in the number of lines, we show the following preliminary result. Separating $R$ from $B$ with a minimum-size set of axis-parallel lines is FPT in the size of either set, and can be solved in time $O^*(9^{|B|})$ (assuming that $B$ is the smallest set).