A survey of graph layout problems

Abstract

Graph layout problems are a particular class of combinatorial optimization problems whose goal is to find a linear layout of an input graph in such way that a certain objective cost is optimized. This survey considers their motivation, complexity, approximation properties, upper and lower bounds, heuristics and probabilistic analysis on random graphs. The result is a complete view of the current state of the art with respect to layout problems from an algorithmic point of view.

Optimal Linear Arrangement of Interval Graphs

“…We have given simple approximation algorithms for those classes. There are several other linear layout problems, like CUTWIDTH, whose complexity is not resolved for the class of interval graphs [4].…”

“…The problem is solvable in polynomial time for trees [6,3,18], and for some other restricted graph classes such as grids or hypercubes [4]. There is an approximation algorithm for general graphs with performance ratio O(log n) [17].…”

“…Furthermore, Sprague has shown how to implement Kleitman and Vohra's algorithm to answer the decision problem in O(n log n) time, and thus produce a minimum bandwidth layout in O(n log 2 n) time [19]. We refer the reader to [4] for a survey of known results on the OLA, bandwidth and other related layout problems.…”

A Refined Evaluation Function for the MinLA Problem

“…MinLA is known to be NP-hard for general graphs [4], though there exist polynomial cases such as trees, rooted trees, hypercubes, meshes, outerplanar graphs, and others (see [2] for a detailed survey). To tackle the MinLA problem, a number of heuristic algorithms have been developed.…”

“…His aim was to design error-correcting codes with minimal average absolute errors on certain classes of graphs. MinLA arises also in other research fields like biological applications, graph drawing, VLSI layout and software diagram layout [2,11].…”