Approximating the Crossing Number of Toroidal Graphs

Approximating the Crossing Numberof Toroidal GraphsPetr Hlinˇe n´y1⋆and Gelasio Salazar2⋆⋆1Faculty of Informatics,Masaryk UniversityBotanick´a68a,60200Brno,Czech Republice-mail:hlineny@fi.muni.cz2Instituto de F´ısica,Universidad Aut´o noma de San Luis Potos´ıSan Luis Potos´ı,Mexico,78000e-mail:gsalazar@ifisica.uaslp.mxAbstract.CrossingNumber is one of the most challenging algorithmicproblems in topological graph theory,with applications to graph drawingand VLSI layout.No polynomial time approximation algorithm is knownfor this NP-Complete problem.We give in this paper a polynomial timeapproximation algorithm for the crossing number of toroidal graphs withbounded degree.In course of proving the algorithm we present a newstructural“grid”theorem for toroidal graphs.Keywords:crossing number,approximation,toroidal graph,edge-width,toroidal grid.2000Math Subject Classification:05C10,05C62,68R101IntroductionWe assume the reader is familiar with the standard terminology of graph the-ory.In this paper we considerfinite graphs,with loops or multiple edges al-lowed.Some standard topological graph theory terminology is briefly introduced throughout this paper.For other related terminology and theory we refer the reader to Mohar and Thomassen[15].Here our interest lies in toroidal graphs, that is,graphs that can be embedded(that is,drawn without edge crossings)on the torus.The crossing number cr(G)of a graph G is the minimum number of pairwise edge crossings in a drawing of G in the plane(thus,a graph is planar if and only if its crossing number is0).Crossing number problems were introduced by Tur´a n,whose work in a brick factory during the Second World War led him to inquire about the crossing number of the complete bipartite graphs K m,n. It is remarkable that this long-standing particular question is still open.Not surprisingly,exact crossing numbers are in general very difficult to compute.Nowadays,computing crossing numbers has important applications in VLSI design,and,naturally,in the graph drawing area.The algorithmic decision prob-lem of crossing minimization is formulated as follows:CrossingNumberInput:A(multi)graph G and an integer k.Question:Is cr(G)≤k?(Possibly:if so,find the corresponding drawing). The problem is in NP since one could guess the optimal drawing,replace its crossings with new(degree4,subdividing)vertices,and verify planarity of the resulting graph.It has been proved by Garey and Johnson[7]that crossing minimization is NP-complete if k is a part of the input.The same assertion has been proved true later by Hlinˇe n´y[10]both for cubic graphs and for the minor-monotone version(cf.[1])of crossing number.An important,stubborn open problem is to decide whether the crossing number of graphs with bounded tree-width can be computed in polynomial time.On the positive side,a surprising result from Grohe[9]states that the cross-ing number is an FPT parameter.Unfortunately,this algorithm is not usable in practice,not even for small values of k,since its running time is double ex-ponential in k and,moreover,the“hidden constants”are very large.Regarding approximability results,the best positive general result known to date is a(poly-nomial time)log3n-approximation algorithm by Even,Guha and Schieber[6].Constant factor approximation algorithms of crossing number are known only for some particular families of graphs,such as[8]for projective graphs of bounded degree≤∆with an approximation factor4∆2,or[11]for almost planar graphs of bounded degree with an approximation factor∆.(A graph is almost planar if deleting one edge leaves it planar.)Our new contribution generalizes these ap-proximation results to all toroidal graphs which have sufficiently“dense”toroidal embeddings(large edge-width compared to∆,precisely).Theorem1.1.Given a toroidal graph G,one can construct in polynomial √O(nedges,and then redraw the affected edges of G inside the rest of the embedding in the best possible(crossing-wise)way.We prove in Section3that this approach gives a good approximation of the correct crossing number of G by exhibiting in G a special minor(a toroidal grid)which itself has crossing number close to the quantity computed in Algorithm2.3.Technical details aboutfinding this grid minor are then given in Section4.2The algorithmSince our coming algorithm works on a toroidal embedding of a given graph, wefirst resolve the task offinding it.It is widely known how to test planarity efficiently,and a strong generalization of that result by Mohar[14]claims: Theorem2.1(Mohar).For every surfaceΣthere is a linear time algorithm which,for a given graph G,eitherfinds an embedding of G onΣor returns a subgraph of G that is a subdivision of a“minimal obstacle”forΣ.In particular,this result provides us a toroidal embedding of the input graph which is known to be toroidal.The second ingredient in our approach is a well-known concept of measuring “dual density”of a graph embedding.A closed curve on a surface is contractible if it can be continuously deformed to a single point(more formally,it is freely homotopic to a constant curve).Consider now a graph G embedded on a non-planar surfaceΣ(i.e.G is a topological rather than a combinatorial object,and the embedding G itself determines the surfaceΣ).The edge-width of the em-bedding G is then defined as the length of the shortest cycle in G which is not contractible onΣ.The edge-width of a given embedding can be efficiently computed by an algorithm of Thomassen[18].A recent improvement of running time is in[4]: Theorem2.2(Cabello,Mohar).Given an embedded graph H,one can com-pute in time O(n√4.Let G o be the graph resulting from G⋆by contracting each of C⋆1and C⋆2into single vertices w1and w2.We use breadth-first search to compute the shortest path P o of lengthℓbetween w1and w2in G o.Letδbe the simple curve on R0formed by the embedding of P o in G⋆.(Henceδconnects a point onγ1to a point onγ2,andδintersectsℓedges of the original embedding¯G.)5.Let F⊆E(G)be the set of those edges in the embedding¯G which are crossed byγ,and F′⊆E(G)be the set of those crossed byδ.Hence¯G−F is actually embedded on R0,and we modify this crossing-free subdrawing (of G−F)into a new drawing˜G of the whole graph G on R0as follows: Each edge from F is newly drawn along an appropriate section ofγ1,then along wholeδ,andfinished along a section ofγ2.6.We output˜G as the resulting drawing of G.γ1γ2Fig.1.Cutting a toroidal embedding byγ,and redrawing the affected edges alongδ.√Lemma2.4.Algorithm2.3computes in polynomial time O(napproach is,of course,compatible with Theorems2.1and2.2.A great advantage here is that we can immediately work also with the dual embedded graph,simply by interchanging the roles of vertices and faces in our structure.All subsequent manipulations we need to do with the embeddings¯G and dual G∗,G⋆can easily be implemented in linear time.Since also the graph searching in Step4is done in linear time,the total computing time of our algorithm will be dominated by running times of the algorithms in Theorems2.1and2.2.The last two observations above guarantee the consistency of our topologi-cal manipulations with the embedding¯G and its dual G∗.Hence the resulting embedding˜G,implemented precisely as follows,is a valid planar drawing of the input graph G:We construct¯G−,the(planar)embedding of G−F resulting from¯G on S1by deleting the edges of F,and adding the two new faces obtained from the union of the faces dual to V(C∗)on S1without F.We then subdivide the edges of F′with k dummy vertices each,and“embed”the edges of F in¯G−as paths through these dummy vertices along the dual path P o,resulting in˜G. Since torus is an orientable surface,the cyclic order of these paths comes out the same at both ends,but we may have to“shift”the ordering by up to k/2edges which may require up to(k/2)2additional crossings between these paths,again embedded as dummy vertices.Hence the planar embedding˜G is also a drawing of G in which edge crossings occur at the≤kℓ+k2/4dummy vertices.·kℓ≥ 116⌊∆/2⌋2−o k(1)Theorem 3.2.Let G be a graph embedded in the torus,and let k and ℓbe as in Algorithm 2.3.Then G contains a minor isomorphic to the toroidal grid of size 1⌊∆/2⌋−1 ×ℓ4cr(H )[16].Now recall that the crossing number of thetoroidal grid of size q ×p ,where p ≥q ≥3,is at least 14·18·k⌊∆/2⌋−O (ℓ)≥ 1To derive Theorem 1.1from Lemma 2.4and this estimate,we do some cal-culus using Lemma 4.1kℓ+k 2/4≤kℓ+kℓ/2≤cr(G )·34∆2−o k (1) −1≤cr(G )· 6∆2+o k (1) .We can also remove o k (1)from the factor at the expense of a worse constant:Corollary 3.3.Let G be a graph embedded in the torus with edge-width k ≥10⌊∆/2⌋.Then cr(G )≥18 k2−2 ·ℓ8·k ⌊∆/2⌋≥14Finding a grid minorIn order to finish the arguments of Section 3,we have to provide a proof of Theorem 3.2,that means to find a sufficiently large toroidal grid minor in the graph G ,relatively to the numbers k,ℓcomputed in Algorithm 2.3.For that we have to carefully examine the structure and “density”of the toroidal embedding ¯Gcomputed in the first step.(Although the embedding may not be unique,the following arguments can use any one of those,and hence show all such embeddings are “similar”in a quantitative sense.)The coming arguments are rather technical,and we have to introduce some additional topological terms,and propose some useful conventions.A closed curve on a surface is simply called a loop .Two loops α,βon a surface Σare freely homotopic if αcan be continuously transformed to βon Σ.The face-width (also called representativity)of a graph H embedded on a surface Σis defined as the smallest integer k such that there exists a noncontractible loop on Σintersecting the embedding H in k points (which can all be assumed vertices of H ).Equivalently,the face-width of embedded H equals half of the edge-width of the vertex-face incidence graph of H [15,Proposition 5.5.4].6Remark.A beautiful result by de Graaf and Schrijver[5]precisely relates thesize of the largest guaranteed grid minor in a toroidal graph to the face-width of its toroidal embedding.It is,unfortunately,not directly usable in our context.To shorten the next arguments,we freely use all the notation introducedthroughout Algorithm2.3,and we let∆=∆(G).Lemma4.1.Let k,ℓbe as in Algorithm2.3.Thenℓ≥k/2.Proof.Seeking a contradiction,suppose thatℓ<k/2.Recall the cycle C∗and path P o from Algorithm2.3,and denote by Q1and Q2the two paths of C∗between the endvertices of P o on it.Up to symmetry,the length of Q1is at most k/2,and hence the cycle C1=Q1∪P o in the dual embedding G∗is of length less than k/2+k/2=k.On the other hand,C1is obviously noncontractibleon S1,which is a contradiction since(see Step2)the edge-width of G∗is k.⌊∆/2⌋pairwise disjoint cycles,all freelyhomotopic toγand disjoint fromγ.Proof.Let H=¯G−F be the subgraph of¯G(disjoint fromγ)which is embedded on the cylinder R0.We denote byδ′a curve on R0connecting a point onγ1 to a point onγ2,such thatδ′has fewest possible points in common with the embedding H.We claim thatδ′intersects H in p≥ℓ⌊∆/2⌋,then a small perturbation ofδ′atthe intersected vertices of H would result in a curveδ′′intersecting fewer than ℓLetδ0be a loop on the torus S1defined as follows;ifδ(see Step4of Al-gorithm2.3)forms a loop on S1,thenδ0=δ;and otherwiseδ0is the union of δand one section(no matter which)ofγbetween the ends ofδ.Consideringγandδ0as loops with a common basepoint,we define loopsδi for i=1,2,...as the composition ofδ0and i iterations ofγ.(This belongs to standard tools of7algebraic topology.Since we are not interested in the orientation of our loops,we choose it arbitrarily.)Informally,δi “winds around”exactly once in the direction of δ0and i times in the direction of γ.Lemma 4.3.Let ¯G be as in Algorithm 2.3,and k be the edge-width of ¯G .Then the embedded graph ¯G contains at least 1⌊∆/2⌋−1 pairwise disjoint cycles which are all freely homotopic to the loop δi for some fixed i .The statement could be proved using similar means as Lemma 4.2,but this time with more complications caused by the fact that the ends of analogously found paths would not match each other in general.We instead notice that the face-width of G is at least kvertex of(original)D i may be the end of at most two bad ears throughout the process for the same reason.Tofinish the proof,we are left with the task tofind a corresponding collection C′1,C′2,...,C′q in H such that each C′j hits once each D′i.For chosen j,we define a sequence a(i),i=1,...,s of integers such that|a(i+1)−a(i)|=1andC j intersects all the D′-cycles in the cyclic order D′a(1),D′a(2),...,D′a(s)(indicesmodulo s).Notice that a(i+1)=a(i);specifically,if C j has more vertices“consecutively”in common with D′a(i),there is just one entry a(i)witnessingthis in the sequence.If1≤i≤s is such that a(i−1)>a(i)and a(i+1)>a(i), then wefind the largest index i′<i such that a(i′)=a(i)(considering the cyclic ordering of indices),and we replace this section of C j,which forms an earof D′a(i)embedded to the right of D′a(i),with the appropriate section of D′a(i).(Informally,i′is the index of the immediately preceding intersection of C j withD′a(i)to the left of i.Such i′exists since C j is homotopic toδi on S1.)Iterating this process,which is clearlyfinite,we get the desired collection C′1,...,C′q.On a high level of formality,we should note that we transparently modify the cycles C j and their sequences a()throughout this process.Why the resulting cycles C′1,C′2,...,C′q in H,which are still homotopic toδi,are pairwise disjoint?Assume,for a contradiction,that C′j and C′m intersect.Since C j has been disjoint from C m,a vertex in the intersection of C′j and C′m must belong to a cycle D′t.However,that means,by the Jordan curve theorem,there has been an ear P of D′t contained in C j,up to symmetry,and P embedded to the left of D′t. (Informally,sections of the cycles C j and C m“overlap”each other along D′t,and one of them has to“bypass”the other one to the left of D′t.)By the left-most property,we have that P intersects D′t−1.We now look at the a()sequence of the cycle C j in this place a(i)=t,and notice that the ear P certifies a(i−1)=t−1. This violates the above condition a(i−1)>a(i)=t at the moment the section of C j preceding P has been replaced with a section of D′t,a contradiction.So contracting,for every pair i,j as above,the intersection path of C′j and D′i into a vertex creates a subdivision of a q×p toroidal grid in H,and hence also in G.4a bit,the most promising direction is tofind an improved inequality for thecrossing number of a minor,suited to our special case(or to use cubic grids since crossing number is monotone for cubic minors).On the other hand,if one9wanted to improve(or to remove it completely if possible)the1。