Fréchet distance between a line and avatar point set

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 Q = {Q1, Q2, . . ., Qn}, where each class Qi consists of a pair of points Qi = {qi, q̄i}. We call a subset A ⊂ {qi, q̄i : 1 ≤ i ≤ 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 ℓ and set Q of a pair of points in double-struck R2, our objective is to find a conflict free subset that minimizes the Fréchet distance between ℓ 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 which is later derandomized using universal family of sets. We believe that this technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in O(2kn log2 n) time, and the derandomized deterministic algorithm runs in O(2kkO(log k)n log2 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 O(2kn2 log n) time. We also show that the problem is unlikely to have a polynomial sized kernel under standard complexity theoretic assumption.