Generating Minimal k-Vertex Connected Spanning Subgraphs

Generating Minimal k-Vertex ConnectedSpanning SubgraphsEndre Boros1,Konrad Borys1,Khaled Elbassioni2,Vladimir Gurvich1,Kazuhisa Makino3,and Gabor Rudolf11RUTCOR,Rutgers University,640Bartholomew Road,Piscataway NJ08854-8003;({boros,kborys,gurvich,grudolf}@)2Max-Planck-Institut f¨u r Informatik,Saarbr¨u cken,Germany;(elbassio@mpi-sb.mpg.de)3Division of Mathematical Science for Social Systems,Graduate School ofEngineering Science,Osaka University,Toyonaka,Osaka,560-8531,Japan;(makino@sys.es.osaka-u.ac.jp)Abstract.We show that minimal k-vertex connected spanning sub-graphs of a given graph can be generated in incremental polynomialtime for anyfixed k.1IntroductionVertex and edge connectivity are two of the most fundamental concepts in net-work reliability theory.While in the simplest case only the connectedness of an undirected graph,that is,the presence of a spanning tree,is required,in practical applications higher levels of connectivity are often desirable.Given the possibility that the edges of the network can randomly fail the reliability of the network is defined as the probability that the operating edges provide a certain level of connectivity.Most methods computing network reliability depend on the efficient generation of all(or many)minimal subsets of network edges which guarantee the required connectivity[Cou87,Val79].In this paper we consider the problems of generating minimal k-vertex con-nected spanning subgraphs.An undirected graph G on at least k+1vertices is k-vertex connected if every subgraph of G obtained by removing at most k−1 vertices is connected.A subgraph of a graph G is spanning if it has the same vertex set as G.For afixed integer k we define the problem of generating minimal k-vertex connected spanning subgraphs as follows:Input:A k-vertex connected graph GOutput:The list of all minimal k-vertex connected spanning subgraphs of GNote that the output of the above problem may consist of exponentially many subgraphs in terms of the input size.Thus,the efficiency of generation algorithms is measured customarily in both the input and output size(see e.g., [Val79,LLK80,JP88]).An algorithm generating all elements of a family F is said2Authors Suppressed Due to Excessive Lengthto run in incremental polynomial time if generating K elements of F(or all if F has less than K elements)can be done in time polynomial in K and the size of the input,for an arbitrary integer K.Our problems include as a special case the problem of generating spanning trees(k=1),which can be solved efficiently[RT75,GM78,Mat93,ASU97].The problem of generating2-vertex connected subgraphs and its generalization for matroids has been considered in[KBB+].1.1Main ResultsWe show that this generation problem can be solved in incremental polynomial time.Theorem1.For every K we can generate K minimal k-vertex connected span-ning subgraphs of a given graph in O(K3m3n+K2m5n4+Kn k m2)time,where n=|V|,m=|E|.We remark that the running time of our algorithm depends exponentially on k.The complexity of the above problem when k is also part of the input remains an open question.1.2The X−e+Y methodIn this section we recall a technique from[KBB+05],which is a variant of the supergraph approach introduced by[SS02].Let C be a class offinite sets and for every E∈C letπ:2E→{0,1}be a monotone Boolean function,i.e.,one for which X⊆Y impliesπ(X)≤π(Y).We assume thatπ(∅)=0andπ(E)=1. LetF={X|X⊆E is a minimal set satisfyingπ(X)=1}.Our goal is to generate all sets belonging to F.We remark that for every X⊆E for whichπ(X)=1we can derive a subset Y⊆X such that Y∈F,by evaluatingπexactly|X|times.This can be accomplished by deleting one-by-one elements of X whose removal does not change the value ofπ.To formalize this,we canfix an arbitrary linear or-der≺on elements of E,without any loss of generality,and define a mapping P roject:{X⊆E|π(X)=1}→F by P roject(X)=X Z,where Z is the lexicographicallyfirst subset of X,with respect to≺,such thatπ(X Z)=1 andπ(X (Z∪e))=0for every e∈X Z.Clearly,by trying to delete elements of X in their≺-order,we can compute P roject(X),as we remarked above,by evaluatingπexactly|X|times.We next introduce a directed graph G=(F,E)on vertex set F.We define the neighborhood N(X)of a vertex X∈F as follows N(X)={P roject((X e)∪Y)|e∈X,Y∈Y X,e},where Y X,e is defined by Y X,e={Y|Y is a minimal subset of E X satisfyingπ((X e)∪Y)=1}.In other words,for every set X∈F and for every element e∈X we extend X e in all possible minimal ways to a set X′=(X e)∪Y for whichπ(X′)=1Generating Minimal k-Vertex Connected Spanning Subgraphs3 (since X∈F,we haveπ(X e)=0),and introduce each time a directed arc from X to P roject(X′).We call the obtained directed graph G the supergraphof our generation problem.Proposition1([KBB+05]).The supergraph G=(F,E)is strongly connected.⊓⊔Since G is strongly connected by performing a breadth-first search in G we can generate all elements of F.Thus,given two procedures:–F irst(X,e),which for every X∈F and e∈X returns an element of Y X,eif Y X,e=∅and∅otherwise,–Next(Y,X,e),which return an element of Y X,e Y if Y X,e=Y and∅otherwise,the procedure T ransversal(G),defined below,generates all elements of F.T raversal(G)Find an initial vertex X0←P roject(E),initialize a queue Q=∅and a dictionary of output vertices D=∅.Perform a breadth-first search of G starting from X o:1output X0and insert it to Q and to D2while Q=∅do3take thefirst vertex X out of the queue Q4for every e∈X do5Y←∅,Y←F irst(X,e)6while Y=∅do7compute the neighbor X′←P roject((X e)∪Y)8if X′/∈D then output X′and insert it to Q and to D9add Y to Y,Y←Next(Y,X,e)Proposition2.Assume that the procedure F irst(X,e)works in time O(φ1(E)),the for every K procedure Next(Y,X,e)outputs K elements of Y X,e in timeφ2(K,E)and there is an algorithm evaluatingπin time O(γ(E)).Then T raversal(G) outputs K elements of F in time O(K2|E|2γ(E)+K2log(K)|E|2+K|E|φ2(K,E)+K|E|φ1(E)).2Proof of Theorem1In this section we apply the X−e+Y method to the generation of all minimalk-vertex connected spanning subgraphs.For a given k-vertex connected graph(V,E)we define a Boolean functionπas follows:for a subset X⊆E letπ(X)= 1,if(V,X)is k-vertex connected;0,otherwise.4Authors Suppressed Due to Excessive LengthClearlyπis monotone,π(∅)=0,π(E)=1.Then F={X|X⊆E is a minimal set satisfyingπ(X)=1}is the family of edge sets of all minimal k-vertex connected spanning subgraphs of(V,E).2.1(k−1)-separators of(V,X e)Before describing procedures F irst(X,e)and Next(Y,X,e)we need the addi-tional notions and elementary results.A k-separator of a graph is a set of k vertices whose removal(simultane-ously removing all edges adjacent to those vertices)makes the graph no longer connected.Note that a k-vertex connected graph has no k′-separators for k′<k.Let G=(V,X)be a minimal k-vertex connected spanning subgraph of a1).k-vertex connected graph(V,E)(see FigureLet e=st be an arbitrary edge of G and let W be a(k−1)-separator of G e=(V,X e).Note that W contains neither s nor t,since otherwise W would also be a(k−1)-separator of G.We denote by S W and T W the vertex sets of the components(i.e.,maximal connected subgraphs)of G e[V W]containing s and t,respectively.Claim.G e[V W]consists of two components,G e[S W]and G e[T W](see Fig-ure2).Generating Minimal k-Vertex Connected Spanning Subgraphs5Fig.2.3-separator W={5,8,9}and the corresponding3-source S W= {s,1,2,3,4,6,7}.We denote by N(·)a neighborhood in the graph G e.Let W be the set of all (k−1)-separators of G e=(V,X e)and let S={S⊆V||N(S)|=k−1,s∈S,t/∈S∪N(S)}.We call an element of S a(k−1)-source.Note that the mapping W−→S W is a bijection between W and S whose inverse is S−→N(S).For two vertices u,v∈V let D u,v={S W∈S|u∈S W,v∈T W}.For an edge f=uv let D f=D u,v∪D v,uWe call a set of hyperedges whose union contains every vertex a hyperedge cover.We show that elements of Y X,e are in one to one correspondence with the minimal hyperedge covers of H X,e.Claim.Let Y⊆E X.The graph(V,X e∪Y)is k-vertex connected if and only if f∈Y D f=S.2.2Procedures F irst(X,e)and Next(Y,X,e)We describe F irst(X,e)and Next(Y,X,e),procedures generating all elements of Y X,e.6Authors Suppressed Due to Excessive LengthF irst(X,e)1construct a hypergraph H X,e on vertex set S with edge set E= {D f|f∈E X}2find a minimal hyperedge cover C of H X,e3return a set{f|D f∈C}Next(Y,X,e)1find a a minimal hyperedge cover C of H X,e not in{D f|f∈Y,Y∈Y} 2return a set{f|D f∈C}In the remainder of this section we show that we can generate minimal hy-peredge covers of H X,e efficiently.2.3Structure of(k−1)-separatorsConsider the poset L=(S,⊆)of the(k−1)-sources ordered by inclusion. Proposition3.The poset L with operations∩and∪is a lattice.We show that the ordering of(k−1)-sources in L has a natural interpretation for the corresponding(k−1)-separators.Since the graph G e is(k−1)-vertex connected,by Menger’s Theorem it contains k−1internally vertex disjoint s-t paths.Let P1=sv11...v1l1t,P2=sv21...v2l2t,...,P k−1=sv k−11...v k−1l k−1t denote such a collection of paths(seeFigure3).We denote by V P the set of all vertices belonging to the paths P1,...,P k−1.Note that not all vertices in V necessarily belong to V P.Consider a(k−1)-separator W.Since the removal of W disconnects G e,W contains at least one internal vertex from each path P i,i=1,...,k−1.As Whas k−1vertices,W={v1α(W,1),...,v k−1α(W,k−1)},whereα(W,i)is the index ofthe vertex of P i belonging to W.Claim.Let W,U∈W.S W⊆S U if and only ifα(W,i)≤α(U,i)for all i= 1,...,k−1.Lemma1.Let S W,S U be(k−1)-sources of G e.Either S W∩T U=∅or T W∩S U=∅.Proof.We partition{1,...,k−1}into sets I,J and K as follows:I={i|α(W,i)>α(U,i)},J={i|α(W,i)=α(U,i)},K={i|α(W,i)<α(U,i)}.Let C={v iα(U,i)|i∈I}∪{v iα(W,i)|i∈I}∪{v iα(W,i)|i∈J}.Observe that|C|=2|I|+|J|.We show that N(S W∩T U)⊆C.Note that V ((S W∩T U)∪C)=T W∪S U (see Figure4).Since W and U are(k−1)-separators of G e,there is no edge between S W∩T U and T W∪S U,thus N(S W∩T U)⊆C.Generating Minimal k-Vertex Connected Spanning Subgraphs7Fig.3.Internally vertex disjoint paths P1,P2,P3of G e represented by thick edges.Let D={v iα(U,i)|i∈K}∪{v iα(W,i)|i∈K}∪{v iα(W,i)|i∈J}.Similarly,we obtain that N(T W∩S U)⊆D.Suppose for contradiction that S W∩T U=∅and T W∩S U=∅.Since S W∩T U contains neither s nor t,the removal of N(S W∩T U)disconnects G.As G is k-vertex connected,we obtain k≤|N(S W∩T U)|≤|C|,thus2|I|+|J|≥k. Similarly,we have2|K|+|J|≥k.Recall that I,J and K partition{1,...,k−1}, thus k−1=|I|+|J|+|K|.Combining this with the above inequalities we obtain2((k−1)+|I|+|J|+ |K|)≥2(k+|I|+|J|+|K|),a contradiction.⊓⊔2.4Bounding The Number of(k−1)-sourcesIt is easy to see that the numebr of(k−1)-sources is at most |V|k−1 ,since each one corresponds to different(k−1)-separators.In this section we provide a better bound on this number.Corollary1.If S W and S U are incomparable in L then there exists some i∈{1,...,k−1}such that|α(W,i)−α(U,i)|=1,i.e.,the vertices v iα(W,i)andv iα(U,i)are adjacent on the path P i.Proof.Suppose on the contrary that|α(W,i)−α(U,i)|>1for all i=1,...,k−1.Then since S W and S U are incomparable,by Claim2.3there exist j,l∈{1,...,k−1}such thatα(U,j)+1<α(W,j)andα(W,l)+1<α(U,l).Thenv jα(U,j)+1∈S W∩T U,v lα(W,l)+1∈T W∩S U contradicting Lemma1.⊓⊔8Authors Suppressed Due to Excessive LengthFig.4.(k−1)-separators W and U.Black nodes are vertices of C.The width of a poset is the size of its largest antichain.We show that the width of L is bounded.Proposition4.The width of L is at most2k−1.Proof.We associate to every(k−1)-separator W a0-1vectorπ(W)=(α(W,1) mod2,...,α(W,k−1)mod2).By Corollary1,if two(k−1)-separators W, U are incomparable,there exists some i∈{1,...,k−1}such that|α(W,i)−α(U,i)|=1,implyingπ(W)=π(U).Since the number of different0-1vectors of length k−1is2k−1,every an-tichain in P has size at most2k−1.⊓⊔Corollary2.For everyfixed k the number of(k−1)-sources is O(|V|).2.5Generating Minimal Hyperedge Covers of H X,eIn this section we reduce the problem of generating minimal hyperedge cov-ers of H X,e to the problem of generating minimal transversals of2-conformal hypergraphs.For the latter problem the algorithm is provided in[BEGK04].A transversal is a set of vertices intersecting every hyperedge.A hyper-graph isδ-conformal if its transpose isδ-Helly(see[Ber89]for other equivalent definitions).Generating Minimal k-Vertex Connected Spanning Subgraphs9 First we show that the hypergraphs H X,e are2-Helly.Claim.Either D u,v=∅or D v,u=∅for all u,v∈V.Proof.Suppose on the contrary that we have S W∈D u,v and S U∈D v,u.Then u∈S W∩T U and v∈T W∩S U,contradicting Lemma1.⊓⊔Claim.D f is a sublattice of L(see Figure5).Note that minimal hyperedge covers of H X,e are minimal transversals of H T X,e.An algorithm from[BEGK04]generates K minimal transversals ofδ-conformal hypergraph in O(K2i2j+Kiδ+2jδ+2),where i and j are the numeber of vertices and number of hyperedges,respectively.2.6ComplexityIn this section we analyze the complexity of T raversal(G).Let n=|V|,m=|E|. Since G is k-vertex connected we have m≥n.Note thatπ(X)can be evaluated in O(k3|V|2)time[CKT93],thusγ(E)=n2.10Authors Suppressed Due to Excessive LengthClaim.For every X∈F and e∈X the hypergraph H X,e=(S,E)has O(n)vertices and O(m)edges and it can be constructed in O(n k m)time.Proof of Claim6:By Corollary2the number of vertices of H X,e=(S,E)is at most O(n).The number of edges of H X,e=(S,E)is exactly|E X|≤msince we add an edge to H X,e=(S,E)for every edge of E X.To construct H X,e=(S,E)wefirst need tofind all(k−1)-sources and (k−1)-separators.We can check if after removing a given set of k−1verticesthe graph G is still connected in O(n+m)time using,e.g.,depthfirst search.Thus we canfind all(k−1)-separators by repeating the above procedure for every(k−1)-element subset of V.The number of such subsets is n k−1 ≤n k−1. Thus we can compute all(k−1)-sources and(k−1)-separators in O(n k−1m)time.To add edges we need to check for every f∈E X and every(k−1)-separator W if S W belongs to D f,which can be done in O(n)time for each pair f and W.Thus the complexity of constructing edges of H is O(n2m).⊓⊔Since we canfind a minimal transversal of H T X,e in O(|E|)time and by Claim2.6we haveφ1(E)=n k m.Recall thatφ2(K,E)=K2m2n+Km4n4 (see Section2.5).Thus by Proposition2the complexity of T raversal(G)is O(K3m3n+K2m5n4+Kn k m2).References[ASU97] A.Tamura A.Shioura and T.Uno.An optimal algorithm for scanning all spanning trees of undirected graphs.SIAM Journal on Computing,26(3):678–692,1997.[BEGK04] E.Boros,K.Elbassioni,V.Gurvich,and L.Khachiyan.Generating max-imal independent sets for hypergraphs with bounded edge-intersections.InMartin Farach-Colton,editor,LATIN2004:Theoretical Informatics,6thLatin American Symposium,volume2976of Lecture Notes in ComputerScience,pages488–498,Buenos Aires,Argentina,April2004.Springer. 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