# Approximating Spanning Trees with Low Crossing Number

###### Abstract

We present a linear programming based algorithm for computing a spanning tree of a set of points in , such that its crossing number is , where the minimum crossing number of any spanning tree of . This is the first guaranteed approximation algorithm for this problem. We provide a similar approximation algorithm for the more general settings of building a spanning tree for a set system with bounded VC dimension.

Our approach is an alternative to the reweighting technique previously used in computing such spanning trees.

## 1 Introduction

The reweighting technique is a powerful tool in computer science [AHK06]. In Computational Geometry, it was introduced by Chazelle and Welzl [CW89] who used it to compute spanning paths with low crossing number in set systems with bounded VC dimension. Welzl [Wel92] provided a tighter analysis for the case of spanning tree of points in . \Matousek[Mat92] used the reweighting technique to provide a powerful partition theorem that proved to be very useful in building range searching data-structures [AE98]. Also, Clarkson [Cla93] provided an algorithm for polytope approximation that used the reweighting technique. Brönnimann and Goodrich [BG95] realized that Clarkson’s algorithm implies a general method for solving hitting set and set cover problems in geometric settings.

Interestingly, Long [Lon01] had observed that set cover problems in geometric settings can be solved by using LP and taking a random sample (guided by the LP solution) that is an -net (a similar observation was later made by [ERS05]). In fact, such packing/covering LPs can be solved efficiently via reweighting [PST91]. Thus, one can interpret the reweighting algorithm for solving the geometric set cover problem as directly solving the associated LP.

[r]

The result of Welzl [Wel92], mentioned above, is quite intriguing. It shows that for a set of points in the plane (resp., in ) one can find a spanning tree of the points, such that any line (resp., hyperplane) crosses at most (resp., ) edges (i.e., segments) of the spanning tree. To appreciate this result, consider the point set formed by the grid . It is trivial in this case to come up with a spanning tree with a crossing number – any spanning tree of the grid points using only edges of the grid has this property, see figure on the right. Surprisingly, the result of Welzl [Wel92] implies that any point set behaves like a grid point set as far as the crossing number of the optimal spanning tree.

In this work, we establish a connection between computing spanning trees with low crossing number and LPs (i.e., linear programs); that is, we show that spanning trees with low crossing number can be computed using LP rounding.

Approximate spanning tree with the lowest crossing number.
Given a set system of finite VC dimension
, we show how to compute, in polynomial time^{1}^{1}1We make the standard assumption that solving a LP of
polynomial size takes polynomial time., a spanning tree of with crossing number
(assuming ), where is the minimal crossing
number of any spanning tree of . This is done by recursively
solving a LP relaxation and rounding it. See \secrefapproximation
for details.

Naturally, this algorithm also applies to the Euclidean case. Specially, given a set of points in one can compute, in polynomial time, a spanning tree such that every hyperplane crosses at most edges of , where is the minimum crossing number of any spanning tree of .

Surprisingly, this is the first guaranteed approximation algorithm known for this problem. In particular, achieving such an approximation is mentioned as open in the Open Problems Project (see http://maven.smith.edu/~orourke/T%O%P%P/P20.html#Problem.20).

Spanning trees in with crossings. We also modify the analysis of our algorithm (but not the algorithm itself) so that it yields worst case bound on the crossing number. Specifically, we get a polynomial time algorithm that, for a given set of points in , computes a spanning tree of such that any hyperplane in crosses at most edges of . Our proof of the correctness of the algorithm is self contained (except for a relatively easy lemma, see \lemrefcrossing:disk), and uses LP duality. We believe the new proof provides a new insight into why such trees exist. In particular, Chazelle and Welzl [CW89] and Welzl [Wel92] proofs of the existence of such spanning trees are simple but somewhat “mysterious” (at least for the author, but other people might not see the mystery).

Here is a sketch of the resulting argument why such trees exist: In the plane, it is sufficient to find spanning forest that span at least vertices of (in connected components that are not singletons) and has crossing number . One can find such a (fractional) spanning graph by doing LP relaxation. The dual LP then asks (intuitively) to separate the given points into singletons by a set of lines of minimum cardinality (i.e., for any , there exists a selected line that crosses the segment ). It is not hard to show that any such set of lines need to be of size . A somewhat more involved argument (since we are dealing with a fractional solution of an LP that has some other constraints) implies that the dual LP is feasible for and its optimal solution is bounded from below. It follows that the primal LP is feasible. Now , solving the primal LP and using a straightforward rounding implies that one can compute the required spanning graph. Applying this recursively by selecting a vertex from each connected component, and overlaying the resulting spanning graphs together results in a connected graph of with crossing number . See \secrefplanar for details.

Interestingly, while the above algorithm works for any point set in , one can do slightly better in the planar case, and get a deterministic rounding scheme, see \secrefdeterministic for details.

#### Previous work.

Fekete \etal[FLM08] suggested using LP relaxation to compute a spanning tree with low crossing number. Their LP is considerably more elaborate than ours (considering all cuts), and their iterated rounding scheme seems to perform quite well in practice (although they are unable to provide a theoretical guarantee on the performance). Furthermore, they prove that computing the spanning tree with minimal crossing number is \NPHard.

#### Organization.

In \secrefapproximation we show the approximation algorithm for spanning trees with low crossing number, for the general case of a set system with low VC dimension. In \secrefplanar, we specialize this algorithm for the case of points and hyperplanes in . We discuss our results and some related open problems in \secrefconclusions.

## 2 Approximating the spanning tree with optimal crossing number

\seclabapproximation

Consider a set system of finite VC dimension . For more details on spaces with bounded VC dimension see [PA95]. For our purposes, it is sufficient that is a set of subsets of of cardinality bounded by , and this holds for any set system induced by a subset of .

For two distinct points , we will refer to the set as an \emphiedge, denoted by . An edge \emphicrosses a set if .

The \emphicrossing number of a set of edges of is the maximum number of edges of crossed by any set of .

As a concrete example of such a set system, consider a set of points in the plane, and let

The set system in this case has VC dimension , and a spanning tree of with crossing number , is a spanning tree of , drawn in the plane by straight segments, such that every line (i.e., the boundary of a halfplane) intersects at most edges of .

Assume there exists a spanning tree for with crossing number . Then, for any subset there exists a spanning tree with crossing number at most .

inherent

###### Proof.

Convert the spanning tree of , with crossing number , into a closed cycle visiting the points of , by doing an Euler tour of , using each edge of twice. The new cycle has crossing number . Next, shortcut the cycle such that it uses only elements of , by replacing each subpath (that uses inner vertices that are not in ) connecting by the edge .

This results in a cycle that visits only the vertices of and has crossing number , as such a shortcutting can only decrease the number of edges crossing a set . Indeed, consider a subpath , and observe that if crosses a set , then one of the edges in the path must also cross this set. Thus, replacing a subpath (of a cycle) by an edge reduces the crossing number of the cycle. ∎

Let denote the set of all edges of , and consider the following LP (parameterized by ).

(1) | |||||

(*) | |||||

Intuitively, this LP tries to pick as many edges as possible (i.e., indicates that we pick the edge ), such that (i) no set is being crossed more than times, and (ii) every point of participates in at least one edge that is being picked.

The above LP might not be feasible, and in such a case is not defined. Naturally, one can modify this LP to first compute the minimal value for which it is feasible, and then solve the original LP with this value of .

In particular, in this case, we can choose almost any arbitrary target function to optimize the LP for. We had chosen this one since the dual form is convenient to work with, see \secrefplanar.

Consider a set system with bounded VC dimension, where , and let be a parameter such that is feasible. Then, one can compute (in polynomial time) a set of edges , such that , and the number of connected components of the graph is (in expectation) at most . The crossing number of is with high probability.

spanning

###### Proof.

We solve the LP \EqrefQl:p:general and compute . Next, for every , if then we add to . Otherwise, if then we pick the edge into with probability .

For a set , let be its crossing number in . We have that

For a constant sufficiently large, let

Since has VC dimension , the number of sets one has to consider (i.e., the size of ) is [PA95], which implies, by the above, that the crossing number of is bounded by , with high probability.

As for the number of connected components in the graph , observe that a point is not adjacent to any edge of with probability

by the inequality (*) in LP \EqrefQl:p:general. Let be the number of points of that are singletons in the graph . By the above, we have that . As such, the expected number of connected components in is

∎

Consider a set system with bounded VC dimension, where . Let be the minimum crossing number of any spanning tree of . Then, one can compute, in polynomial time, a spanning tree of with a crossing number .

low:crossing

###### Proof.

We set . In the th iteration, we compute the minimal for which is feasible. Next, we compute a set of edges over , using \lemrefspanning. If the number of connected components of is larger than , we repeat this iteration (we have constant probability to succeed by Markov’s inequality). Next, from each connected component of , we pick one point into . We repeat this algorithm till we remain with a single point. This algorithm performs iteration. Now, the union forms a spanning graph of , and we return any spanning tree of .

The crossing number of is bounded by the total crossing numbers of the graphs . Now, the graph has crossing number by \lemrefspanning. By \lemrefinherent, , for all . As such, the crossing number of is

When is a set of points in , we will be interested in the spanning tree having the minimal number of crossings with any hyperplane. In particular, the above result implies the following.

Let be a set of points in , and be the minimum crossing number of any spanning tree of . Then, one can compute, in polynomial time, a spanning tree of with a crossing number crosses at most edges (i.e., segments) of . . Specifically, any hyperplane in

low:crossing

## 3 Spanning tree in \TpdfRd with low crossing number

\seclabplanar

Let be a set of points in the plane in general position (i.e., no three points are colinear). Let denote the set of all partitions of into two non-empty sets, by a line that does not contain any point of . For each such partition, we select a representative line that realizes this partition. We slightly abuse notations as refers to as a set of these lines.

We are interested in the question of finding a spanning tree of such that each line of crosses at most edges of .

For a set of lines in the plane, the crossing distance between two points is the number of lines of crossed by the segment formed by these two points. Formally, for any two points , the \emphicrossing distance between them is , where is the number of lines of having and on opposite sides, and is the number of lines that contain either or . It is easy to verify that complies with the triangle inequality (as such its a pseudo-metric).

The \emphicrossing disk of radius centered at a point , is the set of all vertices of the arrangement in crossing distance at most from . We denote this “disk” by .

We need the following lemma due to Welzl [Wel92]. {lemma}[[Wel92]] Let be a parameter, be a set of lines (of size at least ) in the plane, and let be a point in the plane not contained in any line of . Then .

crossing:disk

Here is the LP \EqrefQl:p:general specialized for this planar case, and its dual LP.

We will next show that is feasible. This would imply that one can find spanning graph of with crossing number of that uses a constant fraction of the vertices (i.e., \lemrefspanning).

The LP is feasible for .

feasible

###### Proof.

Consider the dual LP above and observe that it is always feasible (for example by setting for all and for all ). Thus, if we show that is bounded from below (and thus is finite), then the strong duality theorem would imply that is feasible and equal to .

So consider a solution to this dual LP, where all the values are rational numbers. Let be the smallest integer such that if we scale all the values in the given LP solution by then they are integers. In particular, let , for all , and , for all .

Let be a set of lines, where we pick copies of into this set, for all . Formally, copies of the same line (put into ) will be a collection of , almost identical, copies of the line slightly perturbed so that these lines are in general position. Thus, is a set of lines in general position. Furthermore, the inequalities in the LP implies that, for any segment , we have that crosses

lines of .

Observe that for any pair . Otherwise, there would be a point in the plane such that and . But the triangle inequality would imply that , which contradicts the above.

By \lemrefcrossing:disk, for , the disk contains at least distinct vertices of . On the other hand, the total number of vertices in the arrangement is . We conclude that

Now, by the Cauchy-Schwarz inequality and the above, we have that

As this implies that . The claim now follows. ∎

The proof of \lemreffeasible works also in higher dimensions, where we consider points and hyperplanes in . There, one has to use Hölder’s inequality instead of the Cauchy-Schwarz inequality. Then, the LP is feasible for . \remlabeasy

Given a set of points in , one can compute (in polynomial time) a spanning tree of with crossing number at most ; that is, any hyperplane in crosses at most edges of .

###### Proof.

By \lemreffeasible and \remrefeasy, is feasible, for . As such, by \lemrefspanning, we can compute a set of edges that engages a constant fraction of the points of , and it has crossing number . Using the algorithm of \thmreflow:crossing generates a spanning tree with crossing number . Namely, the resulting spanning tree has crossing number . ∎

[Connection to separating/hitting and packing LPs.] It is interesting to consider the LP that just tries to separate all points of from each other. It looks similar to our dual LP while being simpler.

Now, a similar scaling argument to the one used in the proof of \lemreffeasible implies that . Namely, any fractional set of lines separating points in the plane is of size . Naturally, this argument works also in higher dimensions, where a fractional set of hyperplanes of size is required to separate a set of points in .

Observe, that since the LP is more restrictive than this LP, we conclude that for any feasible solution to it holds that .

The dual to the above LP is the packing LP that tries to pick as many fractional edges as possible, while no line crosses edges with total value exceeding . While this is similar to our primal LP , it is not clear how to round it, since we do not have the guarantee that every point has sufficient number of edges attached to it in the fractional solution.

### 3.1 A Deterministic algorithm for the planar case

\seclabdeterministic

Interestingly, at least in the planar case, one can do the rounding deterministically.

Let be a set of points in the plane and a parameter , such that is feasible. Then, one can compute, in polynomial deterministic time, a set of edges , such that (i) the crossing number of is , and (ii) the number of connected components in is .

planar

\figlabmodified:l:p

###### Proof.

Instead of computing we slightly modify the LP so that it finds the “shortest” such solution. The resulting modified LP is depicted in \figrefmodified:l:p.

[r]

Let be the set of all the edges in the solution to this LP such that . We claim that this set of edges is planar. Indeed, if two such segments and intersect, then consider two opposing edges and of the quadrant formed by the convex hull of the endpoints of and , see figure on the right.

We have that , which implies that, for sufficiently small, the solution , , , is feasible, as the total value of the edges attached to a vertex does not change, and the crossing number of any line does not increase by this change, as can be easily verified. But this implies that there is a feasible solution with a better target value (specifically, the target value goes down by ). A contradiction.

Thus is a planar graph where each point of has at least one edge attached to it. Furthermore, the average degree in a planar graph is at most , which implies that at least half of the points of have degree at most in . But each such point , must have an edge attached to it, such that , because of (*) in the LP \EqrefQsecond:l:p.

Thus, scale the LP solution by a factor of and pick all the edges with into the set . We get a set that is adjacent to at least half of the points of , and its crossing number is at most . ∎

The planarity argument in the above proof of \lemrefplanar is similar to the one used by Fekete \etal[FLM08] – they use it to argue that there is one heavy edge, while we use it to argue that there are many heavy edges.

Let be a set of points in the plane. One can compute, in deterministic polynomial time, a spanning tree of with a crossing number , where is the minimum crossing number of any spanning tree of .

low:crossing:2

## 4 Conclusions

\seclabconclusions

We presented an approximation algorithm for computing a spanning tree with low crossing number. The new algorithm relies on a natural LP relaxation of the problem and a straightforward rounding scheme.

Interestingly, our approach enables us to provide a direct proof to the existence of such spanning trees in . This is, as far as we know, the first algorithm for this problem that avoids using the reweighting technique. Intuitively, our algorithm (together with previous results [Lon01]) suggests that reweighting in geometric settings can sometimes be replaced by LP rounding. This is a significant feature, as LPs are considerably more general and flexible tool than reweighting. For example, using our algorithm, we can add other constraints to the LP; e.g., we can insist that some certain cuts would have significantly lower crossing number than some other cuts. In particular, it is not clear how one can incorporate such considerations into a reweighting algorithm computing spanning trees with low crossing number.

One interesting open problem, is to compute spanning trees with
*relative* crossing number using the new LP approach. Here,
given a point set in , one would like to compute a
spanning tree such that if a halfspace contains points
of then it boundary plane crosses (say) edges of . Such a result is known in the plane
[AHS07], but the problem is open in higher dimensions.

## Acknowledgments

The author would like to thank Chandra Chekuri, Sá\sindor Fekete, Jirka \Matousek, and Emo Welzl for helpful discussions on the problems studied in this paper.

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