Cutwidth and Crossings

TL;DR

This work investigates the relationship between graph cutwidth and One-Sided Crossing Minimization (OSCM). It derives two key size bounds for graphs in terms of order $n$ and cutwidth $w$, and proves a sharp bound on the number of unsuited pairs, yielding an $\mathcal{O}^*(2^{|B|w})$-time algorithm for OSCM[CW]. It also analyzes crossing structures, establishing lower bounds on crossing-ratio along cycles, including a tight golden-ratio bound for 3-cycles with disjoint neighborhoods, which informs approximation questions for Greedy-type approaches. Overall, the results contribute to the parameterized complexity understanding of OSCM and suggest directions for tighter bounds and practical algorithms.

Abstract

We provide theoretical insights around the cutwidth of a graph and the One-Sided Crossing Minimization (OSCM) problem. OSCM was posed in the Parameterized Algorithms and Computational Experiments Challenge 2024, where the cutwidth of the input graph was the parameter in the parameterized track. We prove an asymptotically sharp upper bound on the size of a graph in terms of its order and cutwidth. As the number of so-called unsuited pairs is one of the factors that determine the difficulty of an OSCM instance, we provide a sharp upper bound on them in terms of the order $n$ and the cutwidth of the input graph. If the cutwidth is bounded by a constant, this implies an $\mathcal{O}(2^n)$-time algorithm, while the trivial algorithm has a running time of $\mathcal{O}(2^{n^2})$. At last, we prove structural properties of the so-called crossing numbers in an OSCM instance.

Figures (5)

• Figure 1: The function $f(c) = \frac{c(4c^2 + 4c\sqrt{c(1-c)} - 12c - 4\sqrt{c(1-c)} + 9)}{6 \sqrt{c(1-c)}}$ from the bound of Theorem \ref{['thm:size']} (ii) on $\left[0, \frac{1}{2}\right]$.
• Figure 2: Left: The choice of $a$ depending on $c$ for the minimum of $\frac{C'}{2a} + \frac{Ca}{C'} + \frac{C'^2}{6a^2}$. Right: $f(c) = \frac{c(4c^2 + 4c\sqrt{c(1-c)} - 12c - 4\sqrt{c(1-c)} + 9)}{6 \sqrt{c(1-c)}}$ (straight) from the bound of Theorem \ref{['thm:size']} (ii) compared to the improved bound $g(c)$ (dashed) on $\left[0, \frac{1}{2}\right]$; see Remark \ref{['rem:size']}.
• Figure 3: The constructed graph for $k=2$ and $n=7$.
• Figure 4: A bipartite graph $G$ with partite sets $A$ and $B$. If $\pi$ is the linear ordering of $V(G)$ induced by the $x$-coordinates of the vertices, then the bound of Theorem \ref{['thm:unsuited']} is tight for $G$ and $\pi$.
• Figure 5: Definition of the variables $A_i$ and $B_j$ for $d=5$.

Theorems & Definitions (18)

• Theorem 1: restate=thmubs
• Theorem 2: restate=thmunsuited
• Corollary 3: restate=corunsuited
• Theorem 4: restate=thmcycle
• Theorem 5: restate=thmthreecycle
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8