A tighter insertion-based approximation of the crossing number

Abstract

Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of $G+F$ with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F. We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee—depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph $G+F$, while computing the crossing number of $G+F$ exactly is NP-hard already when $|F|=1$. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.

Appendix: Ziegler’s proof of NP-hardness of MEI

In his PhD-thesis (Ziegler 2001), Ziegler showed that MEI is NP-hard (the corresponding decision problem is NP-complete) for general k. Since this thesis is somewhat hard to obtain, we reproduce a slightly simplified version of his proof.

Theorem 6.1

(Ziegler 2001) Given a graph G, a set F of unordered vertex-pairs, and an integer b, it is NP-complete to decide whether there is a planar drawing D of G such that we can insert an edge vw for each vertex pair $\{v,w\}\in F$ into D with overall at most b crossings.

Proof

NP-membership is trivial and it hence remains to show NP-hardness. We use a reduction from

Fixed Linear Crossing Number (FLCN):

Given a graph $H=(V,E)$, a 1-to-1 function $f:V\rightarrow \{1,2,\ldots ,|V|\}$, and an integer $\ell$. Does there exist an f-linear drawing of H with at most $\ell$ crossings?

Thereby, an f-linear drawing is one where all vertices are placed on a horizontal line, each vertex v at coordinate f(v), and each edge is either drawn completely above or completely below that line. It was shown in Masuda et al. (1990) that Flcn is NP-complete.

Let $(H=(V,E),f,\ell )$ be an instance to FLCN. We will construct a corresponding MEI instance $(G=(W,E^*),F,b)$ of size polynomial in $|V(H)+E(H)|$ which is a yes-instance for MEI if and only if $(H,f,\ell )$ is a yes-instance for FLCN. The key idea is to build a rigid graph G that models the restrictions of f-linear drawings, into which we then have to insert the original edges E. Observe that FLCN can only be hard if $\ell <|E|^2$.

Label the vertices $V=\{v_1,\ldots ,v_{n}\}$ such that $f(v_i)=i$ and $n=|V|$. We may assume w.l.o.g. that $n\ge 3$. We start with constructing a graph $G'=(W,E')$ on $n+2$ vertices where $W:=V\cup \{w_a,w_b\}$ and $E':=\{v_iv_{i+1} : 1\le i<n\} \cup \{v_1w_a, v_nw_a, v_1w_b, v_nw_b, w_aw_b\}$. Observe that $G'$ is planar and (since its SPR-tree consists of one R- and one S-node) allows only a unique embedding (and its mirror). We obtain G from $G'$ by replacing each edge in $E'$ by $|E|^2$ parallel edges. Up to ordering the multiple edges amongst its peers, G still allows only a unique embedding.

Now, set $b:=\ell$ and $F:=E$, i.e., we want to insert the edges of H into planar G. We can assume w.l.o.g. that E contains no edges $v_iv_i$ or $v_iv_{i+1}$ for any i. G has exactly four faces with more than two incident vertices: let $\varphi _1$ ($\varphi _2$) be the face incident to exactly $\{w_a,w_b,v_1\}$ ($\{w_a,w_b,v_n\}$, respectively). Let $\varphi _a$ ($\varphi _b$) be the face incident to all of V and $w_a$ ($w_b$, respectively). To go from one of these four faces to another, we would always have to cross at least $|E|^2$ edges (a parallel bunch), which is infeasible when asking for a solution with at most $b=\ell <|E|^2$ crossings. No edge of $F=E$ will be placed in $\varphi _1$ or $\varphi _2$ as they are only incident to one vertex of V. Hence each edge is either completely within $\varphi _a$ or $\varphi _b$, and the equivalence with being above or below the horizontal line of an f-linear drawing follows. $\square$