Simultaneous Embedding of Two Paths on the Grid

TL;DR

This paper investigates the complexity of embedding two paths on a grid without crossings, proving NP-hardness for minimizing the longest edge and providing an efficient method for perimeter minimization under monotonicity constraints.

Contribution

It establishes NP-hardness for the longest edge minimization and offers a polynomial-time algorithm for perimeter minimization with monotone paths.

Findings

Minimizing the longest edge is NP-hard.

Perimeter can be minimized in O(n^{3/2}) time for monotone paths.

Embedding without crossings is computationally challenging.

Abstract

We study the problem of simultaneous geometric embedding of two paths without self-intersections on an integer grid. We show that minimizing the length of the longest edge of such an embedding is NP-hard. We also show that we can minimize in $O(n^{3/2})$ time the perimeter of an integer grid containing such an embedding if one path is $x$-monotone and the other is $y$-monotone.