An algorithm for estimating the crossing number of dense graphs, and continuous analogs of the crossing and rectilinear crossing numbers

TL;DR

The paper delivers a near-optimal approach to estimating the crossing number for dense graphs by combining a deterministic $n^{2+o(1)}$-time algorithm with a randomized polynomial-time process, achieving an additive error $o(n^4)$ and, in dense cases, a $(1+o(1))$-approximation. It develops a robust toolkit—edge-weighted crossing numbers, cut distance, and Frieze-Kannan regularity—then proves a stability result showing that graphs close in cut distance have close crossing numbers, enabling the reduction to a smaller quotient graph and a blow-up argument. A major conceptual advance is the introduction of crossing-density and rectilinear crossing-density for graphons, establishing their estimability and continuity with respect to the cut norm and linking dense graph limits to continuous analogs of crossing numbers. The work thus bridges discrete graph theory and graphon theory, providing a principled framework for studying the crossing behavior of dense graphs and laying groundwork for further questions about minimum densities and extensions to other crossing notions.

Abstract

We present a deterministic $n^{2+o(1)}$-time algorithm that approximates the crossing number of any graph $G$ of order $n$ up to an additive error of $o(n^4)$. We also provide a randomized polynomial-time algorithm that constructs a drawing of $G$ with $\text{cr}(G)+o(n^4)$ crossings. These results yield a $1+o(1)$ approximation algorithm for the crossing number of dense graphs. Our work complements a paper of Fox, Pach and Súk, who obtained similar results for the rectilinear crossing number. The results of Fox, Pach and Súk and in this paper imply that the (normalized) crossing and rectilinear crossing numbers are estimable parameters. Motivated by this, we introduce two graphon parameters, the \textit{crossing density} and the \textit{rectilinear crossing density}, and we prove that, in a precise sense, these are the correct continuous analogs of the crossing and rectilinear crossing numbers of graphs.

Figures (4)

• Figure 1: A small portion of $C_0$ has been highlighted in red. Assuming that the vertex $v$ of $G$ has been assigned to the shaded region, we alter the triangulation and the cycle around a small neighborhood of $v$ so that this point belongs to the interior of the shaded region defined by $C_0'$. This increases the number of nodes of the cycle (and the whole graph) by less than the degree of $v$ in $T_0$.
• Figure 2: The region $R_i$ is shaded. A section of $C_i'$ (red) is contained in the boundary of $R_i$, and the two adjacent arcs of $C_i'$ (also in red) lie inside $R_i$. For every vertex of $T_i'$ in this section, consider the edges of $T_i'$ that are incident to it, lie in $R_i$, and are contained in a curve that represents an edge of $G$ in $\mathcal{D}$. Note that, since no vertex of $G$ belongs to $C_i$, there are at most two edges of $G$ going through any of these vertices. Now, modify $C_i'$ as shown above. This increases the number of nodes in the curve by at most three times the number of nodes that conform the original section of $C_i$.
• Figure 3: On the right, we have the curve that represents $(u',v')$ in $\mathcal{D}$. Setting $a(u,v)=p_4$ and $b(u,v)=p_3$ and proceeding as described in the proof, we arrive at a curve that connects $u$ and $v$. Note that we could have also chosen $a(u,v)=p_1$ and $b(u,v)=p_2$, or $a(u,v)=p_5$ and $b(u,v)=p_6$.
• Figure 4: Edges that share entire non-extremal sections can be modified as shown above. It is important that these modifications be carried out so that any two extremal sections cross at most once and the number of crossings between the extremal sections in $r_i$ and the edges of $G_1$ that do not have an endpoint in $r_i$ does not increase (the same goes for $r_j$).

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Lemma 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 3.1
• proof
• Claim 3.2