The Complexity of Simultaneous Geometric Graph Embedding
TThe Complexity of Simultaneous GeometricGraph Embedding
Jean Cardinal Vincent Kusters Computer Science Department, Universit´e libre de Bruxelles (ULB), Belgium. Department of Computer Science, ETH Z¨urich, Switzerland.
Abstract
Given a collection of planar graphs G , . . . , G k on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem,or simply k -SGE, is to find a set P of n points in the plane and a bijection ϕ : V → P such that the induced straight-line drawings of G , . . . , G k under ϕ are all plane.This problem is polynomial-time equivalent to weak rectilinear realiz-ability of abstract topological graphs, which Kynˇcl (doi:10.1007/s00454-010-9320-x) proved to be complete for ∃ R , the existential theory of thereals. Hence the problem k -SGE is polynomial-time equivalent to severalother problems in computational geometry, such as recognizing intersec-tion graphs of line segments or finding the rectilinear crossing number ofa graph.We give an elementary reduction from the pseudoline stretchabilityproblem to k -SGE, with the property that both numbers k and n are lin-ear in the number of pseudolines. This implies not only the ∃ R -hardnessresult, but also a 2 Ω( n ) lower bound on the minimum size of a grid onwhich any such simultaneous embedding can be drawn. This bound istight. Hence there exists such collections of graphs that can be simultane-ously embedded, but every simultaneous drawing requires an exponentialnumber of bits per coordinates. The best value that can be extractedfrom Kynˇcl’s proof is only 2 Ω( √ n ) . Vincent Kusters is partially supported by the ESF EUROCORES programme EuroGIGA,CRP GraDR and the Swiss National Science Foundation, SNF Project 20GG21-134306. JeanCardinal is partially supported by the ESF EUROCORES programme EuroGIGA, CRP Com-PoSe.This paper previously appeared in: Journal of Graph Algorithms and Applications (JGAA),volume 19, number 1, 2015, pages 259–272 (doi:10.7155/jgaa.00356).
E-mail addresses: [email protected] (Jean Cardinal) [email protected] (Vincent Kusters) a r X i v : . [ c s . C G ] M a y Introduction
Given graphs G = ( V, E ) , . . . , G k = ( V, E k ) on n vertices, simultaneous geo-metric embedding (with mapping) or simply k -SGE, is the problem of finding apoint set P ⊂ R of size n and a bijection ϕ : V → P such that the inducedstraight-line drawings of G , . . . , G k under ϕ are all plane [7]. The correspond-ing decision problem (which we also refer to as k -SGE) simply asks whethersuch a point set exists. It is important to note that k is part of the input andcan thus depend on n . The problem 1-SGE amounts to planarity testing. Theproblem 2-SGE is typically referred to simply as SGE. Fig. 1 shows an exampleof two graphs and a 2-SGE. c de a bac dbe ac ed b Figure 1: Graphs G and G on the same vertex set and a 2-SGE of G and G .Early work on the topic focused on the existence of k -SGEs for restrictedgraph classes. The SGE problem was originally introduced by Brass et. al [7].They show that there is a pair of outerplanar graphs on the same vertex setthat does not admit a 2-SGE. Additionally, they give a triple of paths that doesnot admit a 3-SGE. The authors also show that various other classes of graphs,such as a pair of caterpillars, an extended star and a path, or two stars alwaysadmit a 2-SGE. The most recent positive result is a 2-SGE construction thatworks for generalizations of caterpillars with generalizations of stars, spidersand caterpillars [9]. The question of whether any two trees admit a 2-SGEremained open for six years, until the question was settled in the negative witha counterexample [15]. The most recent negative result gives a tree and a paththat do not admit a 2-SGE [2]. Research has since focused on other variations ofthe problem, such as simultaneous embedding with fixed edges (edges are drawnas arbitrary simple curves, but all graphs must use identical curves for identicaledges), matched drawings (vertices have fixed y -coordinates in all drawings,but may have different x -coordinates in each drawing), or partial simultaneousgeometric embedding (a limited number of vertices may be mapped to differentpoints in different drawings) [11]. The decision problem 2-SGE is NP-hard [10].See [6] for an excellent survey.The existential theory of the reals is the set of true sentences of the form ∃ ( x , . . . , x n ) : ϕ ( x , . . . , x n ), where ϕ is a ( ∧ , ∨ , ¬ )-formula over the signature(0 , , + , ∗ , <, ≤ , =) interpreted over the universe of real numbers [30]. The de-cision problem ETR asks whether a given sentence is true. The complexityclass ∃ R is defined as the set of decision problems that can be reduced to ETRin polynomial time. A problem is ∃ R -hard if it is at least as hard as everyproblem in ∃ R , i.e., if every problem in ∃ R can be reduced to it in polynomial2ime [29, 3]. A problem is ∃ R -complete if it belongs to ∃ R and is ∃ R -hard.It is known that NP ⊆ ∃ R ⊆ PSPACE: Boolean satisfiability can be encodedas a decision problem on a set of polynomial inequalities and Canny [8] gave apolynomial-space algorithm for ETR.In 2011, Kynˇcl [22] proved that weak rectilinear realizability of abstract topo-logical graphs is ∃ R -complete. Since this problem reduces to k -SGE in poly-nomial time [14] and since k -SGE belongs to ∃ R [10], it follows that k -SGE is ∃ R -complete. The k -SGE problem is therefore polynomial-time equivalent tomany other classical problems in computational geometry, such as finding therectilinear crossing number of a graph [4], recognizing unit disk graphs [25],recognizing intersection graphs of convex sets in the plane [29], recognizing in-tersection graphs of segments [21, 24], solving the Steinitz problem [26], anddeciding the realizability of linkages [19]. We refer the reader to recent work ofSchaefer for more references and examples [29, 30].Our contribution is an elementary self-contained construction showing the ∃ R -hardness of k -SGE. It involves a direct translation of the information con-tained in an arrangement of n pseudolines into a set of n planar graphs on a set V of O ( n ) vertices, in such a way that the pseudoline arrangement is stretchableif and only if the graphs can be simultaneously embedded. The main interestingfeature of this construction is that the size of V is linear in the number of pseu-dolines. This implies that for some positive instances of k -SGE, representingthe point set by encoding the coordinates of each point requires an exponentialnumber of bits . This follows from the analogous result on realizations of ordertypes by Goodman, Pollack, and Sturmfels [17]. Our result improves on Kynˇcl’sconstruction, which shows only that 2 Ω( √ n ) bits are sometimes necessary.In Section 2, we briefly recall standard results on (realizability of) order typesand (stretchability of) pseudoline arrangements. The reduction itself is givenin Section 3. Section 4 presents our results on coordinate sizes in simultaneousembeddings. Many combinatorial properties of a point set in the plane are captured by its order type . The order type of a point set P ⊂ R is the mapping χ : (cid:0) P (cid:1) →{− , , +1 } , where χ ( a, b, c ) = sign (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a x a y b x b y c x c y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The value of χ ( a, b, c ) determines whether the three points a, b, c make a left turn(+1), a right turn ( − χ ( a, b, c ) (cid:54) = 0 for all triples a, b, c , the point set is said to be in general position and χ is called uniform .Among other things, the order type encodes the convex hull of a point set andwhether two segments with endpoints in the point set intersect.3 bstract order types generalize the notion of order types on planar pointsets. Knuth [20] calls uniform abstract order types CC-systems and definesthem by the following five axioms (we write χ ( p, q, r ) instead of χ ( p, q, r ) = +1and ¬ χ ( p, q, r ) instead of χ ( p, q, r ) = − χ ( p, q, r ) = ⇒ χ ( q, r, p ).2. Antisymmetry: χ ( p, q, r ) = ⇒ ¬ χ ( p, r, q ).3. Nondegeneracy: χ ( p, q, r ) ∨ χ ( p, r, q ).4. Interiority: χ ( t, q, r ) ∧ χ ( p, t, r ) ∧ χ ( p, q, t ) = ⇒ χ ( p, q, r ).5. Transitivity: χ ( t, s, p ) ∧ χ ( t, s, q ) ∧ χ ( t, s, r ) ∧ χ ( t, p, q ) ∧ χ ( t, q, r ) = ⇒ χ ( t, p, r ).Abstract order types are connected to the well-studied mathematical field oforiented matroids. Specifically, if we consider the equivalence class where χ = − χ , then Knuth [20] proves that (equivalence classes of) uniform abstract ordertypes are in one-to-one correspondence with uniform acyclic rank-3 orientedmatroids . We refer the interested reader to [5] for more information on orientedmatroids. An abstract order type χ is realizable if there exists a point set in R with order type χ . Not all abstract order types are realizable: the smallestnon-realizable abstract order type is the well-known Pappus arrangement on 9points.Order types are closely related to pseudoline arrangements . Pseudoline ar-rangements are usually considered in the real projective plane P , where theycan be defined as simple closed curves, every pair of which meet in exactly onepoint [18]. We recall that the projective plane is the extension of the Euclideanplane by a point “at infinity” for each direction α where the lines with direction α are defined to intersect, and the line at infinity contains exactly the pointsat infinity. For an excellent introduction to projective geometry, we refer theinterested reader to [28], but we do not assume any familiarity with projectivegeometry here. Two projective pseudoline arrangements A and A (cid:48) in P are isomorphic if there is a self-homeomorphism of the projective plane that turns A into A (cid:48) .(Uniform) abstract order types correspond exactly to (simple) projectivepseudoline arrangements with a marked face. For straight-line arrangements,the marked face corresponds to the convex hull of the point set described by theorder type. For more background on pseudoline arrangements with a markedface and their encodings, the reader is referred to Felsner [12] (Chapter 6).By the Folkman-Lawrence topological representation theorem [13], equiv-alence classes of projective pseudoline arrangements correspond in one-to-onefashion to reorientation classes of simple rank-3 oriented matroids [5]. A pseu-doline arrangement is simple if no three pseudolines meet in the same point. Asimple projective pseudoline arrangement is stretchable if and only if it is iso-morphic to a simple arrangement of straight lines. In 1988, Mn¨ev proved that4very semialgebraic set is stably equivalent to the realization space of some rank-3 oriented matroid [26]. Furthermore it was shown that the underlying matroidcould be made uniform (see also Lemma 4 in Shor [31]). As a by-product of theseresults, the simple pseudoline stretchability problem of deciding stretchabilityof a simple projective pseudoline arrangement is ∃ R -complete [29].We define uniform order type realizability as the problem of deciding whethera given uniform abstract order type has a realization. The following lemmasummarizes the correspondence between abstract order types and pseudolinearrangements, and the polynomial-time equivalence of the realizability andstretchability problems. The order type realizability problem will be the startingpoint of our reduction. Lemma 1
Given a uniform abstract order type χ , we can compute in polyno-mial time a description of a simple projective pseudoline arrangement A witha marked face such that χ is realizable if and only if A is stretchable. Con-versely, given a simple projective pseudoline arrangement A with a marked face,we can compute in polynomial time a uniform abstract order type χ such that χ is realizable if and only if A is stretchable. ∃ R -completeness of k -SGE We first reproduce the reduction from weak rectilinear realizability due toGassner et al. [14] and then give a direct proof by reduction from the stretcha-bility problem. An abstract topological graph (AT-graph) is a pair ( G, R ) where G = ( V, E ) isa graph and R ⊆ (cid:0) E (cid:1) is a set of pairs of its edges. A straight-line drawing of G is a weak rectilinear realization of ( G, R ) if every pair of edges that cross inthe drawing is contained in R . Deciding if an AT-graph has a weak rectilinearrealization was shown to be ∃ R -complete by Kynˇcl [22].Kynˇcl proves ∃ R -hardness of weak rectilinear realizability by a reductionfrom simple pseudoline stretchability. Given a simple arrangement A of m pseudolines, Kynˇcl constructs an AT-graph ( G, R ) with G = ( V, E ) that admitsa weak rectilinear realization if and only if A is stretchable. In this construction,similar to the order forcing lemma of Kratochv´ıl and Matouˇsek [21], there isone edge associated with each pseudoline, but there is also a pair of edgescorresponding to each crossing between two pseudolines.The weak rectilinear realizability problem is closely related to the k -SGEproblem. The following equivalence is analogous to the equivalence given inTheorem 2 of [14]. Given graphs G = ( V, E ) , . . . , G k = ( V, E k ), we constructan AT-graph ( G, R ) with G = ( V, (cid:83) i E i ) and { e, f } ∈ R if and only if { e, f } (cid:54)⊆ E i for all i . Then G , . . . , G k admit a k -SGE if and only if ( G, R ) admits a weakrectilinear realization. Conversely, given an AT-graph (
G, R ) with G = ( V, E ),we construct a graph G ef = ( V, { e, f } ) for each pair of edges { e, f } (cid:54)∈ R . Then5 G, R ) admits a weak rectilinear realization if and only if the family F = { G ef |{ e, f } (cid:54)∈ R } admits an | F | -SGE.Combining Kynˇcl’s argument with this equivalence yields the following:given an arrangement A of m pseudolines, we can construct a set of k m graphs G i , each on n m vertices, such that G , . . . , G k m admit a k m -SGE if and onlyif A is stretchable. Here, k m = Θ( m ) and n m = Θ( m ). Fix any constant0 < ε ≤ m /ε isolated vertices to each graph G i . Af-ter this modification, n m = Θ( m /ε ) and thus k m = Θ( n εm ). Since this takespolynomial time, we obtain the following: Theorem 1
Given graphs G = ( V, E ) , . . . , G k = ( V, E k ) on n vertices, thedecision problem k -SGE is ∃ R -complete for k = Ω( n ε ) and any constant ε > . We will give an alternative proof of the result from the previous section via apolynomial-time reduction from uniform order type realizability to k -SGE. The high-level idea is to associate one graph with each point v of the uniformabstract order type χ so that a proper geometric embedding of the graph forcessome radial ordering of the other points around v . In a set of n points in theplane, the radial ordering R ( v ) associated with the point v is simply the orderin which the n − v . We will define this order up to a circular shift. This is illustrated inFig. 2(a). We call R the (counterclockwise) radial system of χ .The radial system can be inferred from the order type of the point set, andtherefore can be defined even if χ is not realizable. A way to determine theradial ordering of v is to pick another point w and first consider only the points x such that χ ( v, w, x ) = +1, that is, the points on the left of the oriented line h h h a bh (a) h h h h a b (b) Figure 2: (a) The radial ordering around point a is h , h , b, h , h (up to acircular shift) [1]. (b) In the dual, h , h intersect a from above in this orderand b, h , h intersect a from below in this order.6 w . We can then sort these points using that x < x (cid:48) whenever χ ( x, v, x (cid:48) ) = − x such that χ ( v, w, x ) = +1 and recover thecomplete radial ordering R ( v ).The radial system can also be extracted from the pseudoline arrangementcorresponding to the abstract order type. For the pseudoline v , the correspond-ing radial ordering is constructed as follows. Starting on any point on thepseudoline v , we first report the order of the successive intersections with pseu-dolines coming from the same side as the marked face, followed by the orderof the intersections with the pseudolines coming from the other side. In a Eu-clidean realization of the arrangement, this corresponds to pseudolines comingfrom above and from below , respectively. This dual definition of the radial or-derings is illustrated on Fig. 2(b). Note that the radial orderings are not thesame as the local sequences defined by Goodman and Pollack [16]. The localsequence for a pseudoline v is simply the order of the intersections with theother pseudolines, and correspond in the primal point set to a sweep with a line through the point v , instead of a ray.The relation between radial systems and order types has been studied indepth in a more general setting in a recent paper from a superset of the currentauthors [1]. It was shown in particular that the radial orderings alone are not sufficient to recover the complete order type of a point set in the plane.Furthermore, for point sets with a triangular convex hull, there can be as manyas n − Lemma 2 ([1])
Consider a realizable abstract order type χ on n points, let S be the set of counterclockwise radial orderings of the points, and let H be the setof points on the convex hull. Then the pair ( S, H ) uniquely determines χ . The proof is straightforward and involves three steps. We first recover the orderof the points on the convex hull by looking at the radial ordering of one ofthem. Next, we recover the orientation of every triple with at least one point p on the convex hull from the radial ordering of p . Finally, the orientations of theremaining triples are deduced by sweeping a ray around a point on the convexhull and then sweeping a ray around every point encountered. Before delving into the construction, we have to argue that we can assumewithout loss of generality that the convex hull of the input abstract order type χ for the realizability problem is triangular. Using Lemma 1, we can computea projective pseudoline arrangement A that is stretchable if and only if χ isrealizable, in such a way that the convex hull of χ corresponds to the markedface of A . If this face is bounded by at most three pseudolines, then we are done.Otherwise, since it is known that every projective arrangement of n pseudolines7 t t t a h bh h v h = h h = h h ∗ b a v h = h a b v v b a Figure 3: From left to right: the copy of K , a wheel graph W v for an ordertype with convex hull h , h , h , and the graph T v . In T v we have t = h = h , t = h = h , t = h = h , and t = h = h = h .has at least n triangular faces [23], we make such a face the marked face of A .Applying Lemma 1 in the other direction finally gives us an abstract order type χ (cid:48) with a triangular convex hull that is realizable if and only if χ is realizable.We now have all the ingredients required for the reduction. For each v ∈ V we define the wheel graph W v on V as the union of the cycle R ( v ) correspondingto the radial ordering around v , and the star connecting v to all vertices in R ( v ).The purpose of including such a graph is to encode the radial ordering R ( v ) ofthe n − v .We next create the labeled graph T v by embedding three copies of W v into theinterior faces of a copy of K , the complete graph on four vertices { t , t , t , t } ,as shown on the left in Fig. 3. We distinguish the vertices of different copiesby adding a superscript i to the vertices of copy i . The convex hull h , h , h is embedded onto t , t , t ; the convex hull h , h , h is embedded onto t , t , t ;and the convex hull h , h , h is embedded onto t , t , t . Fig. 3 shows an exam-ple of a wheel graph W v and the resulting graph T v . The graph T v has exactly3 n − W v , and not simply one, is that the abstract order type will be preservedonly provided the convex hull is the same. We will see that three copies aresufficient to guarantee that at least one of them will have the same convex hullas the one specified by the original abstract order type.Though the T v in the example is maximal planar, this is not always the case.We do, however, have the following. Lemma 3
Each T v is -connected. Proof:
Using symmetry it is easy to verify that W v is 3-connected. We willuse Menger’s theorem to prove that T v is also 3-connected. Let u be any vertexof W v . From every vertex u in W v there is a path to h , a path to h anda path to h such that the paths share no vertex other than u . This can beseen as follows. If u = v then we can reach each h i in one step. Otherwise,one path traverses the cycle in a clockwise direction, one traverses the cycle ina counterclockwise direction and one goes via v . The same holds for the copy8f K (which can also be thought of as a wheel graph). It follows immediatelythat there are three interior pairwise vertex-disjoint paths between every twovertices in T v . Hence, the lemma follows by Menger’s theorem. (cid:3) Since T v is 3-connected, all embeddings of T v are the same up to reflection andthe choice of the outer face. Let T be the set of all T v . Theorem 2
Given a abstract order type χ on a set V of n elements, we cancompute in polynomial time a set T of n graphs, each on the same set of n − vertices, such that T admits an n -SGE if and only if χ is realizable. Proof:
Suppose that χ is realizable. Let P be a labeled point set that realizes χ and let p ( v ) be the point in P that corresponds to v in χ . After possiblyreflecting P along the y -axis, the counterclockwise ray sweep around each point p ( v ) ∈ P encounters the other points of P in the order R ( v ). Hence, by con-struction of the wheel graphs, the induced straight-line drawing of each W v on P is plane. A labeled point set whose induced straight-line drawing of each T v is plane can now easily be constructed from three copies of P and affinetransformations.Conversely, suppose that T has an n -SGE ϕ and consider its convex hull.Note that the convex hull of ϕ corresponds to a mutual face of all T v : if some T v does not have a face that corresponds to the convex hull, then some vertexof T v must have been embedded in the outer face of T v in ϕ , which is impossibleby Lemma 3. If t , t , t is the (clockwise) outer face in ϕ , then the point setcorresponding to one of the three copies is a realization of χ . This can be seenas follows. If t , t , t is the outer face in this clockwise order, then the triangle h , h , h is also oriented in this clockwise order in ϕ . By Lemma 3, this trianglemust form the convex hull of each W v . Hence, any swap of two elements in anyradial ordering R ( v ) in ϕ will induce a crossing in the drawing of W v . It followsthat ϕ is consistent with all radial orderings and therefore, from Lemma 2, itinduces a realization of χ . If a face other than t , t , t was chosen to be theouter face in ϕ , say a face bounded by three vertices of copy one, then the pointset corresponding to the vertices of the second copy (or the third; both work)is a realization of χ by a similar argument. This concludes the proof. (cid:3) We showed that uniform order type realizability can be reduced in polynomialtime to k -SGE. Since uniform order type realizability is ∃ R -complete, it followsthat k -SGE is ∃ R -hard and hence ∃ R -complete by the fact that k -SGE belongsto ∃ R [10]. We finally add a suitable amount of isolated vertices as explainedin Subsection 3.1 to complete our alternative proof of Theorem 1.We define radial system realizability as the problem of deciding whether agiven system of permutations R is the radial system of a set of points in R . Observation 1
Radial system realizability is ∃ R -complete. Proof:
We prove that radial system realizability is polynomially equivalent touniform order type realizability. 9onsider an abstract order type χ . Using the method described at the be-ginning of this section, compute an abstract order type χ (cid:48) from χ in polynomialtime such that χ (cid:48) is realizable if and only if χ is realizable and χ (cid:48) has a triangularconvex hull. Compute the radial system R of χ (cid:48) . Since χ (cid:48) has a triangular convexhull, χ (cid:48) is the only abstract order type with radial system R (Theorem 1 in [1]).Hence, R is realizable if and only if χ is realizable.Conversely, consider a system of permutations R on a set of n elements. Wecompute in polynomial time the set T ( R ) of at most n − R as their radial system (Theorem 1 and Corollary 1in [1]). Then R is realizable if and only if at least one uniform abstract ordertype in T ( R ) is realizable. (cid:3) Many graph drawing questions involve drawing graphs on a small grid. Ourconstruction gives insight on the following simple problem: given a collection of k graphs on n vertices which admit a k -SGE, can we provide any guarantee onthe size of the largest grid on which we can embed them?Since the decision problem is ∃ R -hard, it is unlikely that simultaneous ge-ometric embeddings can all be drawn on a small grid. In fact, showing thatall collections of k = poly( n ) graphs that admit a k -SGE, admit a k -SGE on agrid of size at most exponential in a polynomial in n would directly imply that ∃ R = N P , since the drawing could be encoded using a polynomial number ofbits and used as a certificate.Our construction implies the following lower bound on the size of the smallestgrid required for a k -SGE. Theorem 3
There exist collections of k graphs on n = Θ( k ) vertices that admita k -SGE, every k -SGE of which requires a grid of size Ω( n ) . Proof:
We use a well-known construction due to Goodman, Pollack, and Sturm-fels [17]. They construct a set of k points in the plane such that every realizationof its order type requires a grid of size 2 Ω( k ) . This construction implements aniterative squaring procedure using the multiplication gadget from Von Staudt’salgebra of throws [27, 25].Let χ be an order type on such a set of k points requiring a doubly exponential-size grid. By Theorem 2, we can construct graphs T , . . . , T k where each T i has n = 3 k − P that admits an k -SGE of T , . . . , T k , where | P | = n , will contain a copy of a realization of χ . By defini-tion of χ , P cannot be represented with points of integer coordinates smallerthan 2 Ω( n ) . (cid:3) In the same paper, Goodman, Pollack, and Sturmfels [17] prove that every real-izable order type in general position has a realization with coordinates bounded10y 2 O ( n ) . If a set of graphs admits a k -SGE, then the resulting point set canperturbed into general position without introducing any crossing. Finally, sincethe order type of a point set determines whether two segments cross, it followsthat a k -SGE never requires coordinates larger than 2 O ( n ) . Hence, Theorem 3is tight.This is a significant improvement compared to what can be extracted fromthe construction of Kynˇcl [22]. In the latter, an arrangement of m pseudolinesis realized via the simultaneous geometric embedding of graphs on a set of k m graphs on n m = Θ( m ) vertices. Hence although the coordinates may havevalue 2 Ω( m ) , this is only 2 Ω( √ nm ) . We gave an alternative proof for the ∃ R -hardness of k -SGE and we showed thata k -SGE may sometimes need a grid of size 2 Ω( n ) . Our hardness proof relies onchoosing k = Ω( n ε ), and it is not clear how to weaken this requirement. Thecomplexity of the cases k = O (log n ) and in particular k = 2 are still open,including whether these problems are in NP. Acknowledgments.
This work was initiated during a visit of the second au-thor in Brussels, supported by the EUROCORES programme EUROGIGA,CRP ComPoSe. The authors gratefully acknowledge discussions with StefanLangerman, who gave insightful comments on preliminary versions of the re-sults. The authors would like to thank Marcus Schaefer for pointing out therelevance of [22] and the anonymous reviewers for their helpful suggestions.
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