Linking disjoint segments into a simple polygon is hard

Given a family $\mathcal{S}$ of $n$ closed line segments in the plane, Simple Circuit (respectively, Simple Path) is the problem of deciding whether these segments can be linked into a simple polygon (respectively, a simple polygonal chain) by adding segments between their endpoints.

Rappaport [4] proved that Simple Circuit is NP-hard if the segments in $\mathcal{S}$ are allowed to intersect at their common endpoints, and asked whether the problem remains NP-hard when the segments are disjoint. Later, Bose, Houle, and Toussaint [3] asked whether the related problem Simple Path is NP-hard when the segments in $\mathcal{S}$ are disjoint. More recently, Tóth [6] asked about the complexity of Simple Path again, with a special interest in the case when the segments in $\mathcal{S}$ are both disjoint and axis-parallel.

In this note, we prove the following theorem:

We prove the theorem in two steps. First, we modify the construction in Rappaport’s proof of NP-hardness of Simple Circuit on not necessarily disjoint segments [4] to show that the problem remains NP-hard on disjoint segments. Next, we modify the construction further to prove that Simple Path is also NP-hard on disjoint segments. In the following, we briefly review Rappaport’s proof, which is based on a polynomial reduction from the NP-hard problem Hamiltonian Path in planar cubic graphs.

For any family $\mathcal{S}$ of closed segments in the plane, denote by $V(\mathcal{S})$ the set of endpoints of the segments in $\mathcal{S}$. For any two endpoints $p$ and $q$ in $V(\mathcal{S})$, we call the open segment $pq$ a visibility edge if it does not intersect any closed segment in $\mathcal{S}$.

Given a planar cubic graph $G$ with $n\geq 4$ vertices, the reduction [4] first obtains a rectilinear planar layout of the planar graph using an algorithm of Rosenstiehl and Tarjan [5], then constructs a family $\mathcal{S}$ of $O(n)$ segments following the rectilinear planar layout, such that $G$ admits a Hamiltonian path if and only if $\mathcal{S}$ can be linked into a simple polygon by adding visibility edges between points in $V(\mathcal{S})$. The segments in $\mathcal{S}$ are axis-parallel and interior-disjoint, but may intersect at common endpoints. Each endpoint in $V(\mathcal{S})$ is incident to at most one horizontal segment and at most one vertical segment in $\mathcal{S}$. Since the rectilinear planar layout is of width at most $2n-4$ and of height at most $n$ [5], all coordinates of endpoints in $V(\mathcal{S})$ are integers of magnitude $O(n^{2})$; indeed a closer look at the construction [4, Figure 8] shows that $V(\mathcal{S})\subseteq[1,22n^{2}]\times[1,11n]$. The reduction is hence strongly polynomial. Consequently, Simple Circuit is strongly NP-hard, on not necessarily disjoint segments.

To show that Simple Circuit remains strongly NP-hard on disjoint segments, we will transform the family $\mathcal{S}$ of interior-disjoint axis-parallel segments, which Rappaport constructed, into a family $\mathcal{S}^{\prime}$ of disjoint segments with four distinct orientations, in polynomial time, such that $\mathcal{S}$ can be linked into a simple polygon if and only if $\mathcal{S}^{\prime}$ can be linked into a simple polygon. This transformation can be viewed as a reduction from Simple Circuit on one type of input to the same problem on another type of input.

To obtain $\mathcal{S}^{\prime}$ from $\mathcal{S}$, we first scale the integer coordinates of all segment endpoints by a factor of $40$, then locally modify each intersection between a horizontal segment and a vertical segment into a gadget. The gadgets come in four variants, one for each possible orientation of an intersection; see Figure 1. For a global picture of these modifications on a rectilinear simple polygon with intersections of all four different orientations, refer to Figure 2.

By symmetry, it suffices to describe in detail only one variant of the gadget. Refer to Figure 3. For the intersection $o$ between a horizontal segment $oa$ and a vertical segment $ob$, where $o$ is the left endpoint of $oa$ and the lower endpoint of $ob$, the corresponding gadget is constructed as follows. First, separate the two segments $oa$ and $ob$ into two disjoint segments $o^{\prime}a$ and $o^{\prime\prime}b$, by splitting their common endpoint $o$, then moving one to $o^{\prime}$ and the other to $o^{\prime\prime}$. Next, add five disjoint segments $A_{1}$, $B_{1}$, $A_{2}$, $B_{2}$, and $a^{\prime}b^{\prime}$. Finally, extend the segment $A_{1}$ to a slightly longer segment $A^{\prime}_{1}$ by moving its lower endpoint down for a distance of $\delta=\frac{1}{800n^{2}}$, from $a_{1}=(8,3)$ to $a^{\prime}_{1}=(8,3-\delta)$.

In the presence of $\mathcal{S}^{\prime}$, we say that two points $p$ and $q$ can see each other if the open segment $pq$ is disjoint from all closed segments in $\mathcal{S}^{\prime}$, and we say that two segments $A$ and $B$ in $\mathcal{S}^{\prime}$ can see each other if at least one of the four pairs of endpoints, one of $A$ and one of $B$, can see each other. The gadgets we constructed have the following property of mutual invisibility:

The following lemma shows that our local transformation preserves linkability:

Recall that the coordinates of endpoints in $V(\mathcal{S})$ are integers of magnitude $O(n^{2})$. After the transformation, all endpoints in $V(\mathcal{S}^{\prime})$ except $a^{\prime}_{1}$ have integer coordinates too. We can scale all coordinates by another factor of $1/\delta=800n^{2}$, so that all endpoints in $V(\mathcal{S}^{\prime})$ including $a^{\prime}_{1}$ have integer coordinates of magnitude $O(n^{4})$, and the reduction remains strongly polynomial. Thus Simple Circuit is strongly NP-hard, even if the input segments are disjoint and have only four distinct orientations.

To prove that Simple Path is also NP-hard, we use almost the same transformation from $\mathcal{S}$ to $\mathcal{S}^{\prime}$ as before, except two changes:

1. 1.

   Increase the initial scaling factor from $40$ to $80$, and correspondingly decrease the distance $\delta$ between $a_{1}$ and $a^{\prime}_{1}$ from $\frac{1}{800n^{2}}$ to $\frac{1}{1600n^{2}}$ when constructing the gadgets.

2. 2.

   Select an arbitrary gadget as illustrated in Figure 3, and replace it by an extended gadget as illustrated in Figure 5.

The purpose of the first change is to make room for additional segments in the extended gadget. Updating the scaling factor and the distance $\delta$ together ensures that (1) still holds, and hence Lemma 1 remains valid. Clearly, the reduction remains strongly polynomial.

We have the following lemma analogous to Lemma 2:

Thus Simple Path is also NP-hard. This completes the proof of Theorem 1.