Fréchet Distance Between a Line and Avatar Point Set

Abstract

Frèchet distance is an important geometric measure that captures the distance between two curves or more generally point sets. In this paper, we consider a natural variant of Fréchet distance problem with multiple choice, provide an approximation algorithm and address its parameterized and kernelization complexity. A multiple choice problem consists of a set of color classes $\mathcal {Q}=\{Q_1,Q_2,\ldots ,Q_n\}$, where each class $Q_i$ consists of a pair of points $Q_i = \{q_i, \bar{q_i}\}$. We call a subset $A\subset \{q_i , \bar{q_i}:1\le i\le n\}$ conflict-free if A contains at most one point from each color class. The standard objective in multiple choice problem is to select a conflict-free subset that optimizes a given function. Given a line-segment $\ell$ and a set $\mathcal {Q}$ of a pair of points in $\mathbb {R}^2$, our objective is to find a conflict-free subset that minimizes the Fréchet distance between $\ell$ and the point set, where the minimum is taken over all possible conflict-free subsets. We first show that this problem is NP-hard, and provide a 3-approximation algorithm. Then we develop a simple randomized FPT algorithm for the problem when parametrized by the solution size, which is later derandomized using universal family of sets. We believe that our derandomization technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in $\mathcal {O}(2^k n \log ^2 n)$ time, and the derandomized deterministic algorithm runs in $2^k k^{\mathcal {O}(\log k)} n \log ^2 n$ time, where k, the parameter, is the number of elements in the conflict-free subset solution. Finally we present a simple branching algorithm for the problem running in $\mathcal {O}(2^k n\log n)$ time. We also show that the problem does not have a polynomial sized kernel under standard complexity theoretic assumptions.