Non-crossing paths with geographic constraints^††thanks: A preliminary version of this work appeared in Proc. 25th International Symposium on Graph Drawing and Network Visualization (GD 2017).

Highway, train, and river networks, airline and VLSI routing maps, information flow over the Internet, and the flow of goods and people between different regions all have one thing in common: they can be effectively visualized as a geographic network: a graph, whose embedding is fixed, but not completely. The vertices of a geographic network are restricted to lie in a prescribed region while the edges might or might not be required to follow a particular course. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings?

Many different formulations of this problem exist, which differ in aspects like the shape of the regions, the type of curve used to draw edges, and the type of graph being drawn. We study the seemingly simple variant where each region is a vertical segment. We restrict the edges to be drawn as simple curves that lie inside the convex hull of the vertical segments corresponding to the end vertices, to force edges to be more or less straight.

More formally, we are given a graph $G=(V,E)$ and, for each vertex $v\in V$, a vertical segment region $I_{v}$. For each edge $(u,v)\in E$, we define the tube $T_{uv}$ of $(u,v)$ as the convex hull of $I_{u}\cup I_{v}$. The goal is to determine if it is possible to draw each vertex $v\in V$ as a point $p_{v}\in I_{v}$, and each edge $e=(u,v)\in E$ as a path from $p_{u}$ to $p_{v}$ that is contained in $T_{uv}$, such that no two paths cross at a point interior to both. When this is possible, we say that the tube is connected. Moreover, we assume that $G$ is a matching, so we can solely focus on drawing the edges. We consider three different ways to draw the edges: straight line segments, $x$-monotone curves, and arbitrary paths. Figure 1 illustrates the problem with an example.

The problems arising when vertices must be inside prescribed (geographic) regions have not been explored that much. In the limit case of regions that are points—thus fixed vertex positions—there is still freedom on how to draw the edges, leading to a variety of optimization problems, such as drawing edges as polygonal lines with the minimum possible number of bends [16], or minimizing the total edge length used [5]. The problem changes considerably when there is a continuous set of possible positions for each vertex, as in the problem we focus on in this paper.

To the best of our knowledge, only force-directed layout methods for some particular cases of the problem have been proposed [1]. In that work, each vertex is constrained to lie inside a given polygon, and edges must be drawn as straight line segments. Abellanas et al. [1] considered two different goals: minimizing the number of edge crossings and achieving uniform edge length. However, the methods used were heuristic and of a very different nature than those proposed here.

On the more algorithmic side, there are quite a few problems, studied under many different names, that may resemble the ones studied here, in one or more aspects. Next we review the most relevant ones.

Clustered-planarity (c-planarity for short) is a concept developed for drawing clustered graphs. Essentially, a drawing of a clustered graph is called c-planar if it is a plane drawing, and the vertices of each cluster are drawn inside their own connected region of the plane (see Feng et al. [FengCE95b] for a formal definition). Thus in c-planar drawings, vertices are constrained to lie in certain regions, but these regions are not prescribed, and moreover, the restriction is on the clusters, and not on vertices. Probably the variant most related to our problem is that of fitting planar graphs to planar maps [2]. In this setting, in addition to the clustered graph, one is given a compatible polygonal map, and the goal is to determine if there is a mapping of clusters to polygons that results in a c-planar drawing. Note, however, that the polygons are not pre-assigned to clusters, and, again, that region constraints are for clusters and not for individual vertices.

A setting closer to ours has been considered in the context of data imprecision in computational geometry, where a set of imprecise points is given, meaning that each point can be drawn anywhere inside a given region. In particular, it has been shown that if the regions are vertical line segments or scaled copies of an arbitrary shape, and the paths are straight line segments, determining if one can draw a cycle without crossings is NP-hard [13]. Thus this differs from our setting mainly in the type of graph: cycles, while we focus on matchings.

Aloupis et al. [3] studied the problem of non-crossing matchings between points and geometric objects. In that problem, each edge connects a point to a geometric object or to a set of points. Aloupis et al. [3] showed that the problem is polynomially solvable in some special cases, most notably when matching a point to one of two other points, and NP-hard when the number of options increases. In particular, they showed that our problem for vertical segment regions and straight line segment paths is NP-hard, a fact that was also proved earlier in the Master’s thesis of one of the authors of the current paper [18]. The same problem with unit-size square regions, but drawing general planar graphs instead of matchings, was also shown to be NP-hard [4].

A problem similar to the second variant studied in this paper, where edges are drawn as $x$-monotone paths, is that of Manhattan-geodesic embeddings of planar graphs [8]. The goal there is to connect points with non-crossing paths that are rectilinear and $xy$-monotone. This problem was recently shown to be NP-hard [9]. This is in contrast to our problem, which we show to be solvable in polynomial time. Moreover, our problem differs from the Manhattan-geodesic embeddings problem in two important aspects: endpoints of edges are not fixed, and paths are restricted to lie inside tubes.

Finally, the most relevant previous work in our context deals with the non-crossing connector problem, studied by Kratochvíl and Ueckerdt [11]: given $m$ sets of points $P_{i}$, $1\leq i\leq m$, and a region $R_{i}$ (with $P_{i}\subset R_{i}$) for each $i$, the goal is to compute a curve inside each region $R_{i}$ that goes through all the points in $P_{i}$ and no two curves cross. Kratochvíl and Ueckerdt [11] showed that non-crossing connectors always exist if the regions are pseudo-disks (i.e., the boundaries of any two regions intersect in at most two points). Their proof proceeds constructively, providing a method to connect the points within each region while avoiding crossings with other connections. For regions that are not pseudo-disks, existence can be decided in polynomial time only for a few cases, while in general the problem is NP-complete. An important difference with our setting is that all given points $P_{i}$ in each region must be connected.¹¹1Confusingly, Kratochvíl and Ueckerdt [11] quoted one of the current authors as stating that our problem is NP-complete for monotone paths, which is not the case. Also, they claimed that in the aforementioned Master’s thesis [18] it was proven that the problem is NP-complete for unit segments, but that is not the case either: the reduction there requires segments of several lengths. Still, we will build on top of this result for the case of arbitrary paths, presented in Section 4.

In this section we show that, if the edges must be drawn as straight line segments, the problem is NP-complete, even if all the segments defining the tubes have unit length. Let $V=\{v_{1},\ldots,v_{2n}\}$ and let $I_{i}$ be the unit length vertical segment associated with $v_{i}$. For convenience we assume that there is an edge between $v_{2i-1}$ and $v_{2i}$ ($1\leq i\leq n$), and let $T_{i}$ be the corresponding tube.

We prove NP-hardness by reduction from Rectilinear planar 3-SAT [10]. An instance of this NP-complete problem consists of a 3-SAT formula and a rectilinear embedding of the graph associated to the formula. In the embedding all variable vertices lie on a straight line, and clauses are represented as horizontal line segments with at most three vertical line segments that connect to the variables appearing in the clause. See Figure 2 for an illustration of four variables and three clauses. The reduction relies on the following gadgets for variables and clauses.

With this construction we obtain the desired NP-hardness reduction. The following lemma is a direct consequence of the construction.

It remains to show that the problem is in NP.

In this section we consider edges drawn as $x$-monotone paths. A path is said to be $x$-monotone if no vertical line intersects the path more than once, see Figure 5 for an illustration.

We start with some observations that hold for arbitrary paths. We say that two tubes fully cross if the vertical segments are completely disjoint from the other tube, and the intersection of the two tubes is nonempty (see Figure 6). The first basic observation is that whenever two tubes fully cross, no solution can exist. Therefore we assume from now on that no two tubes fully cross. The most interesting cases occur when two tubes intersect, without fully crossing. This necessarily happens because (at least) one of the vertical segments of a tube intersects the other tube. Figure 7 shows examples of such situations. We distinguish between single intersections, where only one tube segment intersects another tube, or double intersections, where two different segments intersect another tube (either both from the same tube, or one from each).

Single intersections locally induce a vertical order between the paths in any solution. For instance, in the situations in Figure 7(a), the red tube can be considered above the blue one, because in any solution the red path will be above the blue one at the $x$-coordinate equal to that of the vertical segment creating the intersection. On the other hand, no such order exists for a double intersection. Indeed, in any double intersection there are solutions with both orders of the paths in the tubes, see Figure 7(b). Based on this we define the order graph.

If we allow edges to be drawn by arbitrary paths, then a cycle in the order graph can sometimes be realized, as shown in Figure 9(a)—the cycle here is blue $\rightarrow$ red $\rightarrow$ green $\rightarrow$ blue. However, that is not always the case, as the example in Figure 9(b) shows.

Nevertheless, if we disallow double intersections, then we can still decide in polynomial time whether a solution exists. The key idea is to use a result by Kratochvíl and Ueckerdt [11] that states that if the regions (in our case, tubes) form a set of pseudo-disks, then there is always a solution. Two tubes with a single intersection may not be pseudo-disks, but we can try to convert them into pseudo-disks by cutting off parts that cannot be used in any solution. This leads to a procedure that allows us to determine in polynomial time if a solution exists.

In this work we began the study of a fundamental problem for geographic networks: deciding if a given network can be drawn without crossings. We have focused on a simple variant where regions are vertical segments, and the graph is a matching. Despite the apparent simplicity of the problem, it turned out to be very challenging, with varying problem complexities depending on the way in which edges can be drawn. There are multiple directions in which this work can be continued.

To begin with, in the restricted setting studied here, the arguably most important variant of arbitrary paths is still not fully solved: our algorithm only works in the absence of double intersections. If we allow double intersections, the problem remains open. However, we expect the problem to remain polynomial also in this case. Furthermore, we conjecture that the problem admits a Helly-type property, implying a polynomial time algorithm.

A whole set of open questions arises when the setting is made more general. A first step would be, for example, to consider tubes defined by line segments with more than one orientation (say, using $k>1$ different slopes). This would probably require a different approach. Then it is most interesting to understand what is the simplest type of 2-dimensional region that can be solved in polynomial time when the graph is matching, as in this work. Going in a different direction, it would also be worthwhile to consider more general families of graphs, such as paths or trees, even if it is for tubes defined by vertical segments.