Simultaneous Embedding of Two Paths on the Grid
🤖 AI Summary
This study addresses the problem of simultaneously embedding an x-monotone and a y-monotone path on the integer grid without self-intersections, with the objective of minimizing either the length of the longest edge or the perimeter of the bounding region. By leveraging techniques from computational geometry and combinatorial optimization, and exploiting the interplay between grid constraints and path monotonicity, the authors prove that minimizing the longest edge length is NP-hard. Furthermore, they present the first polynomial-time algorithm for minimizing the embedding perimeter in a specific class of monotone paths, achieving a time complexity of \(O(n^{3/2})\). This work advances the understanding of constrained graph embeddings and provides an efficient solution for a practically relevant variant of the problem.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
simultaneous embedding
paths
grid
NP-hard
monotone
Innovation
Methods, ideas, or system contributions that make the work stand out.
simultaneous embedding
grid embedding
NP-hardness
monotone paths
perimeter minimization
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