A Maximum Linear Arrangement Problem on Directed Graphs

TL;DR

This paper introduces Maximum Directed Linear Arrangement (MaxDLA), a directed analogue of MinLA, and connects it to MaxDiCut to study complexity and structure. It establishes tight bounds linking MaxDLA to MaxDiCut and proves NP-hardness for planar digraphs, while also providing a polynomial-time algorithm for MaxDLA on orientations of trees with bounded degree. The authors further explore isoperimetric-type properties, presenting nested-maximum arrangements for special digraph classes such as tournaments, low-degree orientations, and transitive acyclic digraphs, and they derive explicit formulas for maximal arrangements in these cases. Overall, the work broadens understanding of linear arrangements in directed graphs, offering both hardness results and constructive algorithms for notable graph families.

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.

Figures (4)

• Figure 1: Gadgets for clauses $(a \lor \overline{b}), (c \lor {d}), (\overline{e} \lor \overline{f}), (x), (\overline{y})$.
• Figure 2: Planar 3SAT to restricted Planar 3SAT tippenhauer2016
• Figure 3: Construction of a reduction digraph for all clauses containing some variable $a$ in Restricted Planar 3SAT. The clauses in Restricted Planar 3SAT are $(a \lor b \lor c), (a \lor \overline{x}), (\overline{a} \lor y)$. The associated clauses in restricted Planar 2SAT are $(a), (b), (c), (d), (\overline{a} \lor \overline{b}), (\overline{b} \lor \overline{c}), (\overline{c} \lor \overline{a}), (a \lor \overline{d}), (b \lor \overline{d}), (c \lor \overline{d}); (a \lor \overline{x}); (\overline{a} \lor y)$.
• Figure 4: Flow of FindSignatures $(G,f)$ when solving for the above orientation of a path of length three.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• proof
• proof
• Theorem 3.1
• proof