Recognition of point visibility graphs on a grid

We now prove that the recognition problem for visibility graphs on a grid is decidable if and only if the existential theory of the rationals is decidable . The computational complexity of answering the question “Does this object have a realization on a grid?”

(rational realization problem) is unknown for various types of objects. Most prominently, it is unknown for polytopes and oriented matroids and (non-simple) order types and thus (non-simple) line arrangements. Matiyasevich [Mat70] showed that the existential theory of the integers is undecidable by giving a negative solution to Hilbert’s tenth problem: Deciding whether a diophantine equation has a solution is undecidable. This cannot be applied to a grid realization of a PVG, since a realization of a PVG with rational coordinates, which can be obtained by a rational solution of the inequality system, leads to a grid realization by scaling. Hence for those geometric realizations on the grid the decidability of Hilbert’s tenth problem over the rationals is of interest.

Grünbaum [Grü72] conjectured in 1972 that there is no algorithm that enumerates all arrangements in the rational projective plane, which is equivalent to the recognition problem of order types that can be represented on a grid. This conjecture is still open.

Poonen [Poo09] tried to define the integers using rational logic, i.e., to give a formula F(w)over the rationals that is satisfiable if and only ifwis an integer. These formulas are known, but they involve existential and universal quantifiers. The existence of such a formula using only existential quantifiers would show the undecidability of the existential theory of the rationals.

Similar to work a of Sturmfels [Stu87] for oriented matroids and polytopes, we show the following theorem.

Theorem 5.9. The realization problem for visibility graphs of points on a grid is de-cidable if and only if the existential theory of the rationals is dede-cidable.

Before proving this theorem, we point out a connection of this question to finding an upper bound on the grid size of a PVG that is realizable on a grid.

Corollary 5.10. Assume the recognition problem for PVG on a grid is undecidable.

Then there is no computable functionf :N→Nsuch that each PVG withnpoints that is realizable on a grid can be drawn on a grid of sizef(n)×f(n).

Proof. We suppose that a computable function f : N → N exists, such that every PVG that is realizable on a grid with n vertices can be represented on a grid of size f(n) ×f(n). Using this function, we can give an algorithm that decides whether a graphGwith|V(G)|=nhas a realization as a PVG on a grid.

We first compute f(n), then for eachx ∈[f(n)]²ⁿ we check whether Gis the PVG of the point set (x(v₁), y(v₁), . . . , x(v_n), y(v_n)) = x. If there is such an x, the algorithm returns the realization, otherwise no realization exists.

This algorithm is clearly an effective decision procedure, and thus the recognition problem for PVG on a grid is decidable – a contradiction to the assumption.

If we try to use the reduction of Theorem 5.6 in order to prove Theorem 5.9, the main obstacle is that we cannot reduce to a strict inequality system. This leads to an open realization space, and we can always find a rational point in an non-empty open set.

If we do not have an open realization space we do not know if we can perturb the lines/points as in the proof of Theorem 5.6 to get rid of collinearities of points of the PVG that do not lie on a common line of the arrangement. However, since we are only concerned with the decidability of the problem, we use a (non-polynomial) computable reduction.

Proof of Theorem 5.9. We give a computable reduction from the rational realization problem for (non-simple) line arrangements to the rational realization problem for PVGs. Given a line arrangement L of n lines, we construct a finite set of graphs GL, such that L has a rational realization if and only if at least one graph in GL has a rational PVG realization.

We construct the set of graphsGLas follows. We want to encodeLin a fan. Thus, we pick one unbounded facef_pofLas the face to place the pointp, the origin of all rays of the fan. Since we do not know the circular order of the vertices of Laroundpwe create a set of graphs GAfor each possible circular order A of vertices ofL aroundp, namely for each possible allowable sequence A withp in the fixed unbounded face f_p. The set of graphs GAcontains possible fans with circular orderA of the vertices ofLaroundp: Depending on a realization ofL (withA) the visibility of two points of the fan that do not lie on the same line ofLor the same ray of the fan can be blocked by another point.

For each combination of additional blocked edges we create a single graph. Therefore, note that the only edges we require to keep the unique representation (all lines ofLare preserved) of a fan (Lemma 5.5) are the edges of one vertex to the vertices on different rays on the segment s1 (the segment that is added closest to p in the construction of the fan), and the edges of the path on one ray and the path on the segment s1. All otheroptional edges can be removed (which may lead to a non-realizable graph) to keep the points on the lines of L collinear in each PVG representation of the fan. For each subset S of optional edges, the set of graphs GA contains the fans forcing the circular order A aroundp, where exactly the optional edges contained in S are also contained in the fan.

Now, L has a rational realization if and only if there is a graph G ∈ GL that has a rational PVG realization:

First assume L has a rational realization R. From R we construct a fan as before by placing p in the same unbounded face. The PVG realization obtained by this fan

5.4. Point visibility graphs on a grid construction is a rational realization of some graphGcontained in GA, whereA is the circular order aroundp in this realization.

On the other hand, a rational PVG realization of one of the graphs in GL leads to a rational realization ofL, since each graph in GL preserves the collinearities of points on Lin each PVG realization.

The set GL can be computed from L. Thus, an effective decision algorithm for the rational realization problem for PVGs leads to an effective decision algorithm for the rational realization problem for line arrangements by applying the algorithm on each graph inGL. This shows that if the rational realization problem for PVGs is decidable, then the rational realization problem for line arrangements is decidable. On the other hand, we can encode the realization problem for PVGs in an existential formula as done in [GR14]. A rational solution of this formula leads to a rational realization of the PVG, because the variables encode the coordinates of the points.