A Maximum Linear Arrangement Problem on Directed Graphs

TL;DR

This paper introduces the MaxDLA problem, a directed graph variant of MinLA, explores its theoretical properties, complexity, and algorithms, and establishes bounds and polynomial solutions for specific graph classes.

Contribution

It defines MaxDLA, relates it to MaxDiCut, proves NP-hardness for planar digraphs, and provides polynomial algorithms for trees and special graph orientations.

Findings

MaxDLA is related to MaxDiCut with specific bounds.

MaxDiCut is NP-Hard for planar digraphs.

Polynomial algorithms exist for trees and certain orientations.

Abstract

We propose a new arrangement problem on directed graphs, Maximum Directed Linear Arrangement (MaxDLA). This is a directed variant of a similar problem for undirected graphs, in which however one seeks maximum and not minimum; this problem known as the Minimum Linear Arrangement Problem (MinLA) has been much studied in the literature. We establish a number of theorems illustrating the behavior and complexity of MaxDLA. First, we relate MaxDLA to Maximum Directed Cut (MaxDiCut) by proving that every simple digraph $D$ on $n$ vertices satisfies $\frac{n}{2}$$maxDiCut(D) \leq MaxDLA(D) \leq (n-1)MaxDiCut(D)$. Next, we prove that MaxDiCut is NP-Hard for planar digraphs (even with the added restriction of maximum degree 15); it follows from the above bounds that MaxDLA is also NP-Hard for planar digraphs. In contrast, Hadlock (1975) and Dorfman and Orlova (1972) showed that the undirected Maximum Cut problem is solvable in polynomial time on planar graphs. On the positive side, we present a polynomial-time algorithm for solving MaxDLA on orientations of trees with degree bounded by a constant, which translates to a polynomial-time algorithm for solving MinLA on the complements of those trees. This pairs with results by Goldberg and Klipker (1976), Shiloach (1979) and Chung (1984) solving MinLA in polynomial time on trees. Finally, analogues of Harper's famous isoperimetric inequality for the hypercube, in the setting of MaxDLA, are shown for tournaments, orientations of graphs with degree at most two, and transitive acyclic digraphs.