Mixed Linear Layouts: Complexity, Heuristics, and Experiments

TL;DR

This paper investigates mixed linear graph layouts combining stack and queue pages, proves NP-completeness for recognition problems, and introduces a heuristic that improves conflict minimization in such layouts based on experimental results.

Contribution

It provides the first complexity results for recognizing mixed layouts and proposes a new heuristic that outperforms previous methods in conflict reduction.

Findings

NP-completeness of recognition problems for certain mixed layouts

The new heuristic reduces conflicts more effectively in experiments

Heuristic improves over previous approaches for linear layouts with s=q=1

Abstract

A $k$-page linear graph layout of a graph $G = (V,E)$ draws all vertices along a line $\ell$ and each edge in one of $k$ disjoint halfplanes called pages, which are bounded by $\ell$. We consider two types of pages. In a stack page no two edges should cross and in a queue page no edge should be nested by another edge. A crossing (nesting) in a stack (queue) page is called a conflict. The algorithmic problem is twofold and requires to compute (i) a vertex ordering and (ii) a page assignment of the edges such that the resulting layout is either conflict-free or conflict-minimal. While linear layouts with only stack or only queue pages are well-studied, mixed $s$-stack $q$-queue layouts for $s,q \ge 1$ have received less attention. We show NP-completeness results on the recognition problem of certain mixed linear layouts and present a new heuristic for minimizing conflicts. In a computational experiment for the case $s, q = 1$ we show that the new heuristic is an improvement over previous heuristics for linear layouts.