Spanning directed trees with many leaves

Spanning directed trees with man y lea v es ⋆ Noga Alon 1 , F edor V. F omin 2 , Grego ry Gutin 3 , Mic hael Kr ivelevic h 1 , and Saket Saurabh 2 1 Department of Mathematics, T el Aviv U n ivers ity T el Aviv 69978, Israel { nogaa,kr ivelev } @po st.tau.ac.il 2 Department of Informatics, Universit y of Bergen POB 7803, 5020 Bergen, Norwa y { fedor.fo min,saket } @ii.uib.no 3 Department of Computer Science Roy al Hollo wa y , Un ive rsit y of London Egham, Surrey TW20 0EX, UK gutin@cs.r hul.ac.uk Abstract. The Directed Maximum Leaf Out-Branch ing problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a giv en digraph with the maxim u m num b er of lea ves. I n this paper, we obtain tw o com binatorial results on the num b er of leav es in out-branchings. W e sho w that – ev ery strongly connected n -vertex digraph D with minimum in- degree at least 3 has an out - branching with at least ( n/ 4) 1 / 3 − 1 lea ves; – if a strongly connected digraph D does n ot contai n an out-branching with k leav es, then the path width of its underlying graph UG( D ) is O ( k log k ) . Moreo ver, if the digraph is acyclic, the pathwidth is at most 4 k . The last result implies th at it can b e decided in time 2 O ( k log 2 k ) · n O (1) whether a strongly connected digraph on n vertices has an out-branching with at least k leav es. On acyclic digraphs the runn ing time of our algo- rithm is 2 O ( k log k ) · n O (1) . 1 In tr o duction In this pap er, we initiate the combinatorial and alg orithmic study of a natura l generaliza tion of the well studied Maximum Leaf Sp anning Tree (MLST) problem on connected undirected gr aphs [11, 16, 19–21 , 24, 26, 32, 3 4]. Given a digraph D , a sub dig raph T of D is a n out-tr e e if T is a n oriented tree with only o ne vertex s of in-degr ee zero (calle d t he r o ot ). If T is a spanning out- tree, i.e. V ( T ) = V ( D ), then T is ca lled an out-br anching of D . The vertices of ⋆ Preliminary exten ded abstracts of this pap er hav e b een presented at FSTTCS 2007 [5] and ICALP 2007 [4] 2 N. Alon, F. V. F omin, G. Gutin, M. Krivelevic h, and S. Saurabh T of out-degr ee zer o are called le aves . The Directed Maximum L eaf Out- Branching (DMLOB) pr oblem is to find a n o ut- br anching in a given digr aph with the maximum num b er of leav es. It is well-kno wn that MLST is NP-ha rd for undir ected gr aphs [25 ], which means that DMLOB is NP-hard for symmetric dig raphs (i.e., digra phs in whic h the existence of an ar c xy implies the existence o f the arc y x ) and, th us, for strongly connected digra phs. W e can show that DMLOB is NP-hard for acyclic digraphs as fo llows: Co nsider a bipartite gr a ph G with bipartition X , Y and a vertex s 6∈ V ( G ). T o obtain a n acyclic digr a ph D from G a nd s , orie n t the edges of G from X to Y and add all a rcs s x , x ∈ X . Let B b e an out-branching in D . Then the set of leav e s of B is Y ∪ X ′ , where X ′ ⊂ X , a nd for each y ∈ Y there is a vertex z ∈ Z = X \ X ′ such that z y ∈ A ( D ). Observe that B has max im um nu mber of leav es if and only if Z ⊆ X is of minim um size among all sets Z ′ ⊆ X such that N G ( Z ′ ) = X . How ever, the problem of finding Z ′ of minimum size such that N G ( Z ′ ) = X is equiv alent to the Set Cover problem ( { N G ( y ) | y ∈ Y } is the family o f sets to cover), which is NP-hard. The combinatorial study of spanning trees with maximum num b er of leav es in undirected gr aphs has an extensive history . L inia l conjectured a round 19 87 that every connected gra ph o n n vertices with minimu m vertex degree δ has a spanning tree with a t least n ( δ − 2) / ( δ + 1) + c δ leav es, where c δ depe nds on δ . This is indeed the cas e for all δ ≤ 5 . K leitman and W es t [28] and Linial and Sturtev ant [3 1] show ed that every connected undirected graph G on n vertices with minimum degree at least 3 has a spanning tree with at least n/ 4 + 2 leav es. Griggs and W u [2 6] prov ed tha t the maximum num b er of leav es in a s panning tree is at least n/ 2 + 2 when δ = 5 and at least 2 n/ 5 + 8 / 5 when δ = 4. All these results are tight. The situation is less c le ar for δ ≥ 6; the first author observed that L inial’s conjecture is false for all la rge v a lues of δ . Indeed, the r esults in [2] imply that there are undirected g raphs with n vertices and minim um degree δ in which no tree ha s mor e than (1 − (1 + o (1)) ln ( δ +1) δ +1 ) n leav es, where the o (1)-term tends to zero an δ tends to infinity , and this is ess en tially tight. See also [3], pp. 4-5 and [13 ] for more informatio n. In this paper we prov e an analogue of the K leitman-W est result for directed graphs: ev ery stro ngly co nnected dig raph D of order n with minim um in-degree at lea st 3 has an o ut-branching w ith at least ( n/ 4) 1 / 3 − 1 lea ves. W e do not k now whether this b ound is tigh t, how ever we show that there a re str ongly connected digraphs with minimum in-degree 3 in which every out-br anching ha s at most O ( √ n ) leav es. Unlik e its undir ected coun terpart which has a ttracted a lot of atten tio n in all algorithmic paradigms lik e approximation alg o rithms [2 4 , 3 2, 34], par a meterized algorithms [11, 19, 21], exac t expo nential time algorithms [20] and a lso combina- torial studies [16, 2 6, 28, 31], the Directed Maximum Leaf Out-Branching problem has lar gely been neglected unt il recently . The only pap er we are aw a re of is the very recen t pa pe r [18 ] tha t describ es an O ( √ opt )-approximation alg o- rithms for DMLOB . Spanning directed trees with many leav es 3 Our second combinatorial result rela tes the n um be r of leaves in a DMLOB of a directed graph D with the pathwidth of its underlying graph UG( D ). (W e po stpo ne the definition of pathwidth till the next se c tion.) If an undirected graph G contains a star K 1 ,k as a minor, then it is p o s sible to construct a spanning tree with at least k leav e s from this minor . O therwise, ther e is no K 1 ,k minor in G , and it is possible to prove that the path width of G is O ( k ). (See, e.g. [8].) Actually , a muc h more genera l r esult due to Biensto ck et a l. [10]) is that a ny undirected gra ph o f pathwidth at least k , contains all trees on k vertices a s a minor. W e prov e a result that ca n be viewed as a generalizatio n of kno wn bounds on the num b er of leav es in a spanning tree of a n undirected gra ph in terms of its path width, to str ongly c o nnected digr aphs. W e show that either a strongly connected digraph D has a DMLOB with at le ast k leaves or the pathwidth o f UG( D ) is O ( k log k ). F or a n acyclic digra ph with a DMLOB having k leaves, we prove that the pathwidth is at most 4 k . This almost matches the b ound for undirected gra phs . These combinatorial r esults are useful in the design o f parameterize d algo rithms. In para meterized algor ithms, for de c ision pro blems with input size n , a nd a parameter k , the goal is to desig n an algor ithm with runtime f ( k ) n O (1) , where f is a function of k a lone. (F or DMLOB such a pa rameter is the num b er o f leav es in the out-tr ee.) P roblems having s uch a n algorithm are sa id to b e fixed parameter tractable (FP T). The b o ok by Downey and F e llows [17] provides an int ro duction to the topic of parameterized complexity . F or rece nt dev elopments see the b o oks by Flum and Grohe [2 3] and by Niedermeier [33]. The par a meterized version of DMLOB is defined as follows: Given a digra ph D and a p ositive in tegral parameter k , do e s D con tain an out-bra nching with a t least k leav es ? W e denote the para meter ized v e rsions of DMLOB by k -DMLOB. If in the ab ov e definition we do not insist on an out-branching and ask whether there exists an out-tree with a t least k leav es , we get the par ameterized Di- rected Maximum Leaf Out-Tree problem (denoted k -DMLOT). Our c ombinatorial bo unds, c ombined with dynamic programming o n gra phs of bo unded pathwidth imply the first parameterized algorithms for k -DMLOB on strongly connected digraphs a nd acyclic digraphs. W e r e ma rk tha t the algor ith- mic results pr esented her e a ls o hold for all digra phs if we conside r k -DMLOT rather than k -DMLOB . This answers an op en ques tio n of Mike F ellows [14, 22, 27]. How ever, we mainly restr ict ourselves to k -DMLOB for cla rity and the harder challenges it p oses , and we briefly cons ider k -DMLOT only in the las t section. V ery recently , using a mo difica tion of our approa ch, Bonsma a nd Dor n [12] prov ed that either an arbitrary digraph D has an out-bra nching with at most k leav es or the pa th width of UG( D ′ ) is O ( k 3 ), where D ′ is the digr aph o btained from D by deleting all arcs not contained in any out-branching of D . The b ound O ( k 3 ) is muc h larg er than our b o unds for strongly connected a nd acyc lic di- graphs, but it s uffices to allow Bonsma a nd Dorn to show that k -DMLOB is FPT, settling ano ther op en question o f F ellows [2 2, 27]. 4 N. Alon, F. V. F omin, G. Gutin, M. Krivelevic h, and S. Saurabh This pa per is orga nized as follows. I n Section 2 we provide a dditional ter- minology and no ta tion as well as some well-kno wn results. W e intro duce loca lly optimal out- br anchings in Section 3. Bo unds on the num b er of leaves in max i- m um leaf out-branchings of str ongly connected and acyclic digraphs are obtained in Section 4. In Section 5 we prove upp er bo unds on the pathwidth of the un- derlying graph of strong ly connec ted and acyclic digraphs that do not contain out-branchings with at least k le av es. In Section 6 we conclude with discussions and op en problems . 2 Preliminaries Let D b e a digra ph. By V ( D ) and A ( D ) we repre s ent the vertex set and arc set of D , re spe ctively . An oriente d gr aph is a dig raph with no directed 2 -cycle. Given a subset V ′ ⊆ V ( D ) of a digraph D , let D [ V ′ ] denote the dig raph induced by V ′ . The underlying gr aph UG( D ) of D is o btained from D by omitting all orientations of ar cs and by de le ting one edg e from each r esulting pair of par allel edges. The c onne ct ivity c omp onents o f D are the sub digraphs of D induced by the vertices o f comp onents o f UG( D ). A digraph D is str ongly c onn e cte d if, for every pair x, y of vertices ther e are directed paths from x to y and fr o m y to x. A maximal strongly co nnected sub digra ph of D is called a str ong c omp onent . A vertex u of D is an in-neighb or ( out-neighb or ) of a vertex v if uv ∈ A ( D ) ( v u ∈ A ( D ), resp ectively). The in-de gr e e d − ( v ) ( out - de gr e e d + ( v )) o f a vertex v is the num b er o f its in-neighbors (out-neighbo rs). W e denote by ℓ ( D ) the ma ximum num b er of leaves in an out-tree of a dig r aph D and b y ℓ s ( D ) we denote the maximum p ossible num b er of leav es in an out- branching of a digraph D . When D has no o ut- br anching, w e write ℓ s ( D ) = 0. The following simple result gives nec essary and sufficien t conditions for a digr aph to have an out-branching. This assertio n allows us to chec k whether ℓ s ( D ) > 0 in time O ( | V ( D ) | + | A ( D ) | ). Prop ositio n 1 ([7] ). A digr aph D has an out-br anching if and only if D has a unique str ong c omp onent with no inc oming ar cs. Let P = u 1 u 2 . . . u q be a direc ted pa th in a digraph D . An arc u i u j of D is a forwar d ( b ackwar d ) ar c for P if i ≤ j − 2 ( j < i , resp ectively). Every backw a r d arc of the type v i +1 v i is called double . F or a natur al num b er n , [ n ] denotes the set { 1 , 2 , . . . , n } . A t r e e de c omp osition of a n (undirec ted) gr aph G is a pair ( X, U ) wher e U is a tr ee whose vertices w e will ca ll no des and X = ( { X i | i ∈ V ( U ) } ) is a co llection of subsets of V ( G ) such tha t 1. S i ∈ V ( U ) X i = V ( G ), 2. for each edge { v , w } ∈ E ( G ), there is a n i ∈ V ( U ) s uch that v , w ∈ X i , and 3. for each v ∈ V ( G ) the set of no de s { i | v ∈ X i } forms a subtree of U . Spanning directed trees with many leav es 5 The width of a tree decompo s ition ( { X i | i ∈ V ( U ) } , U ) equals max i ∈ V ( U ) {| X i | − 1 } . The tr e ewidth o f a gr a ph G is the minim um width ov er all tr ee decomp ositions of G . If in the definitions of a tree decompo sition and treewidth we restric t U to be a path, then we ha ve the definitions of path deco mp os ition and pa th width. W e use the notatio n tw ( G ) and pw ( G ) to denote the tree w idth a nd the pathwidth of a gr aph G . W e also need a n equiv alent definition of path width in ter ms of vertex sepa - rators with resp ect to a linear ordering of the vertices. Let G b e a gr a ph and let σ = ( v 1 , v 2 , . . . , v n ) be an ordering of V ( G ). F o r j ∈ [ n ] put V j = { v i : i ∈ [ j ] } and denote by ∂ V j all vertices o f V j that hav e neighbors in V \ V j . Setting v s ( G, σ ) = max i ∈ [ n ] | ∂ V i | , we define the vertex sep ar ation of G as v s ( G ) = min { v s ( G, σ ) : σ is an or dering of V ( G ) } . The fo llowing a ssertion is well-kno wn. It follows directly fr om the r esults of Kirousis and Papadimitriou [30] on interv al width of a g raph, see also [29]. Prop ositio n 2 ([29 , 30] ). F or any gr aph G , v s ( G ) = pw ( G ) . 3 Lo cally Opt imal Out-Branc hings Our bo unds ar e ba s ed on finding lo cally optimal out-bra nc hings. Given a di- graph, D a nd an out-br anching T , we call a vertex le af , link and br anch if its out-degree in T is 0, 1 and ≥ 2 res pectively . Let S + ≥ 2 ( T ) b e the set of branch vertices, S + 1 ( T ) the set of link vertices and L ( T ) the set of leav es in the tree T . Let P 2 ( T ) b e the s et of maximal paths consis ting of link vertices. By p ( v ) w e denote the p ar ent o f a vertex v in T ; p ( v ) is the unique in- neighbor of v . W e call a pair of vertices u and v siblings if they do not belo ng to the s a me pa th from the ro o t r in T . W e s ta rt with the following well known and ea sy to o bserve facts. F act 1 | S + ≥ 2 ( T ) | ≤ | L ( T ) | − 1 . F act 2 | P 2 ( T ) | ≤ 2 | L ( T ) | − 1 . Now w e define the notion of lo ca l exchange which is intensiv ely used in o ur pro ofs. Definition 3 ℓ -Arc Exchange ( ℓ -AE) optimal out-branching: An out- br anching T of a dir e cte d gr aph D with k le aves is ℓ -AE optimal if for al l ar c subsets F ⊆ A ( T ) and X ⊆ A ( D ) − A ( T ) of size ℓ , ( A ( T ) \ F ) ∪ X is either not an out-br anching, or an out-br anching with at most k le aves. In other wor ds, T is ℓ -AE optimal if it c an ’t b e tur n e d int o an out -br anching with mor e le aves by exchanging ℓ ar cs. Let us remar k, that for every fixed ℓ , an ℓ -AE optimal out-bra nch ing can b e obtained in p olyno mial time. In our pro ofs we use only 1-AE optimal out- branchings. W e need the following s imple pro per ties of 1 -AE optimal out-br anchings. 6 N. Alon, F. V. F omin, G. Gutin, M. Krivelevic h, and S. Saurabh Lemma 1. L et T b e an 1 -AE optimal out-br anching r o ote d at r in a digr aph D . Then the fol lowing holds: (a) F or every p air of siblings u, v ∈ V ( T ) \ L with d + T ( p ( v )) = 1 , ther e is no ar c e = ( u, v ) ∈ A ( D ) \ A ( T ) ; (b) F or every p air of vertic es u, v / ∈ L , d + T ( p ( v )) = 1 , which ar e on the same p ath fr om the r o ot with dist ( r, u ) < dist ( r, v ) ther e is no ar c e = ( u, v ) ∈ A ( D ) \ A ( T ) (her e dist ( r, u ) is t he distanc e t o u in T fr om the r o ot r ); (c) Ther e is no ar c ( v , r ) , v / ∈ L su ch that t he dir e cte d cycle forme d by the ( r , v ) -p ath and the ar c ( v, r ) c ont ains a vert ex x such that d + T ( p ( x )) = 1 . Pr o of. The pro of easily follows from the fact that the existence of any o f these arcs contradicts the lo cal optimality of T with resp ect to 1-AE. ⊓ ⊔ 4 Com binatorial Bounds W e star t with a lemma that allows us to obtain low er b o unds on ℓ s ( D ). Lemma 2. L et D b e a oriente d gr aph of or der n in which every vertex is of in-de gr e e 2 and let D have an out-br anching. If D has no out-tr e e with k le aves, then n ≤ 4 k 3 . Pr o of. Let us a s sume that D has no o ut-tr ee with k leaves. Consider an out- branching T o f D with p < k leaves whic h is 1-AE optimal. Let r b e the ro ot of T . W e will b ound the n umber n of vertices in T as follows. Every v ertex o f T is either a leaf, or a br anch vertex, or a link vertex. By F acts 1 and 2 we already hav e bo unds on the n umber o f lea f and branch vertices as well as the num b er of maximal paths consisting of link vertices. So to get an upper b ound o n n in terms of k , it suffices to b ound the length o f each maximal path consisting o f link vertices. Let us consider such a path P a nd let x, y b e the first and last vertices of P , resp ectively . The vertices of V ( T ) \ V ( P ) can b e partitioned into four classes as follows: ( a ) an cestor vertices : the vertices which app ea r before x on the ( r , x )-path of T ; ( b ) descendant vertices : the vertices a ppe a ring a fter the vertices o f P on paths of T starting at r and passing through y ; ( c ) si nk vertices : the v e rtices whic h are leav e s but not descendant vertices; ( d ) sp eci al vertices : none-o f-the-ab ov e v e rtices. Let P ′ = P − x , let z b e the o ut-ne ig hbor of y on T and let T z be the subtree of T ro oted at z . By Lemma 1, there are no a rcs fro m sp ecial or ancestor vertices to the path P ′ . L e t uv b e an ar c of A ( D ) \ A ( P ′ ) such that v ∈ V ( P ′ ) . Ther e are tw o po ssibilities for u : (i) u 6∈ V ( P ′ ), (ii) u ∈ V ( P ′ ) and u v is ba ckw ard for P ′ (there are no forward ar cs for P ′ since T is 1-AE optimal). Note tha t every vertex of type (i) is either a descendant vertex or a sink. Observe also that the backw ar d a rcs for P ′ form a vertex-disjoint collection of out-trees with ro o ts at Spanning directed trees with many leav es 7 vertices that a re not terminal vertices of backward arcs for P ′ . Thes e r o ots ar e terminal vertices of arcs in which first vertices a re descendant v er tices or sinks. W e denote by { u 1 , u 2 , . . . , u s } and { v 1 , v 2 , . . . , v t } the sets of vertices on P ′ which hav e in-neighbors that are desc endant vertices and sinks, resp ectively . L e t the out-tree formed by backw ar d arcs for P ′ ro oted at w ∈ { u 1 , . . . , u s , v 1 , . . . , v t } be denoted by T ( w ) and let l ( w ) denote the num b er of leav es in T ( w ) . Observe that the following is an out-tree r o oted at z : T z ∪ { ( in ( u 1 ) , u 1 ) , . . . , ( in ( u s ) , u s ) } ∪ s [ i =1 T ( u i ) , where { in ( u 1 ) , . . . , in ( u s ) } are the in-neighbors of { u 1 , . . . , u s } on T z . This out- tree has at leas t P s i =1 l ( u i ) leaves a nd, thus, P s i =1 l ( u i ) ≤ k − 1 . Let us deno te the subtree o f T r o oted a t x by T x and let { i n ( v 1 ) , . . . , in ( v t ) } be the in-neighbors of { v 1 , . . . , v t } on T − V ( T x ). Then we ha ve the following out-tree: ( T − V ( T x )) ∪ { ( in ( v 1 ) , v 1 ) , . . . , ( in ( v t ) , v t ) } ∪ t [ i =1 T ( v i ) with at lea st P t i =1 l ( v i ) leav es . T hus, P t i =1 l ( v i ) ≤ k − 1 . Consider a path R = v 0 v 1 . . . v r formed by backw a r d arcs. Observe that the arcs { v i v i +1 : 0 ≤ i ≤ r − 1 } ∪ { v j v + j : 1 ≤ j ≤ r } for m an out-tree with r leav es, wher e v + j is the out-neighbor of v j on P . Th us, there is no path of backw ar d arcs of length more than k − 1 . Every out-tree T ( w ), w ∈ { u 1 , . . . , u s } has l ( w ) leaves and, thus, its arcs can be dec omp o sed in to l ( w ) paths, each of le ng th at mo st k − 1. Now we can bound the num b er of ar cs in all the trees T ( w ), w ∈ { u 1 , . . . , u s } , as follows: P s i =1 l ( u i )( k − 1) ≤ ( k − 1) 2 . W e can similarly bo und the num b er of ar cs in a ll the trees T ( w ), w ∈ { v 1 , . . . , v s } by ( k − 1) 2 . Recall that the vertices of P ′ can be either ter minal vertices of backward arcs for P ′ or vertices in { u 1 , . . . , u s , v 1 , . . . , v t } . Observe that s + t ≤ 2( k − 1) since P s i =1 l ( u i ) ≤ k − 1 and P t i =1 l ( v i ) ≤ k − 1 . Thu s, the num b er of vertices in P is bo unded fro m ab ove by 1 + 2( k − 1 ) + 2( k − 1 ) 2 . Therefor e, n = | L ( T ) | + | S + ≥ 2 ( T ) | + | S + 1 ( T ) | = | L ( T ) | + | S + ≥ 2 ( T ) | + X P ∈ P 2 ( T ) | V ( P ) | ≤ ( k − 1) + ( k − 2 ) + (2 k − 3)(2 k 2 − 2 k + 1 ) < 4 k 3 . Thu s, we conclude that n ≤ 4 k 3 . ⊓ ⊔ Theorem 4. L et D b e a str ongly c onne cte d digr aph with n vertic es. (a) If D is an oriente d gr aph with minimum in-de gr e e at le ast 2, then ℓ s ( D ) ≥ ( n/ 4) 1 / 3 − 1 . 8 N. Alon, F. V. F omin, G. Gutin, M. Krivelevic h, and S. Saurabh (b) If D is a digr aph with minimum in-de gr e e at le ast 3, t hen ℓ s ( D ) ≥ ( n/ 4) 1 / 3 − 1 . Pr o of. Since D is s tr ongly connected, we have ℓ ( D ) = ℓ s ( D ) > 0 . Let T b e an 1-AE optimal out-branching o f D with maximum num b er o f leav es. (a) Delete some arcs from A ( D ) \ A ( T ), if needed, such that the in-degree of each vertex of D b ecomes 2. Now the inequa lit y ℓ s ( D ) ≥ ( n/ 4) 1 / 3 − 1 follows from Lemma 2 and the fa ct that ℓ ( D ) = ℓ s ( D ). (b) Let P b e the path formed in the pr o of of Lemma 2. (Note that A ( P ) ⊆ A ( T ).) Delete ev ery double a rc of P , in case there are an y , and delete some more arcs from A ( D ) \ A ( T ), if needed, to ensure that the in-degree of each vertex of D b ecomes 2. It is not difficult to see that the pro of of Lemma 2 remains v alid for the new digraph D . No w the inequa lit y ℓ s ( D ) ≥ ( n/ 4) 1 / 3 − 1 follows from Lemma 2 and the fact that ℓ ( D ) = ℓ s ( D ). ⊓ ⊔ Remark 5 It is e asy to se e that The or em 4 holds also for acyclic digr aphs D with ℓ s ( D ) > 0 . While we do not know whether the b ounds of Theo rem 4 are tight, w e can show that no line a r b ounds are p ossible. The following re s ult is formulated for Part (b) of Theorem 4, but a similar result holds fo r Part (a) a s well. Theorem 6. F or e ach t ≥ 6 ther e is a st r ongly c onne cte d digr aph H t of or der n = t 2 + 1 with minimum in-de gr e e 3 such that 0 < ℓ s ( H t ) = O ( t ) . Pr o of. Let V ( H t ) = { r } ∪ { u i 1 , u i 2 , . . . , u i t | i ∈ [ t ] } and A ( H t ) =  u i j u i j +1 , u i j +1 u i j | i ∈ [ t ] , j ∈ { 0 , 1 , . . . , t − 3 }  [  u i j u i j − 2 | i ∈ [ t ] , j ∈ { 3 , 4 , . . . , t − 2 }  [  u i j u i q | i ∈ [ t ] , t − 3 ≤ j 6 = q ≤ t  , where u i 0 = r fo r every i ∈ [ t ] . It is ea sy to chec k that 0 < ℓ s ( H t ) = O ( t ) . ⊓ ⊔ 5 P athwidth of underlying graphs and parameterized algorithms By Prop ositio n 1, an acy c lic digra ph D has an out-branching if and only if D po ssesses a single vertex of in-deg ree zero. Theorem 7. L et D b e an acyclic digr aph with a single vertex of in-de gr e e zer o. Then either ℓ s ( D ) ≥ k or the underlying undir e cte d gr aph of D is of p athwidth at most 4 k and we c an obtain t his p at h de c omp osition in p olynomial time. Pr o of. Assume that ℓ s ( D ) ≤ k − 1. Cons ide r a 1 - AE optimal out-branching T of D . Notice that | L ( T ) | ≤ k − 1 . Now remove a ll the leav es and branch v ertices from the tree T . The remaining vertices form ma x imal directed pa ths consis ting Spanning directed trees with many leav es 9 of link vertices. Delete the first vertices of all pa ths. As a re s ult we o btain a collection Q of directed paths. Let H = ∪ P ∈Q P . W e w ill show that ev er y arc uv with u, v ∈ V ( H ) is in H . Let P ′ ∈ Q . As in the pro of of Lemma 2, we see that there are no forward arcs for P ′ . Since D is acyclic, ther e are no backw ar d arcs for P ′ . Suppos e uv is an arc o f D such that u ∈ R ′ and v ∈ P ′ , where R ′ and P ′ are distinct pa ths from Q . As in the pro of of Lemma 2, we see that u is either a sink or a descendent vertex for P ′ in T . Since R ′ contains no sinks of T , u is a descendent vertex, which is imp ossible as D is acyclic. Thus, we hav e proved that pw (UG( H )) = 1 . Consider a path decomp osition of H o f width 1. W e ca n o btain a path de- comp osition of UG( D ) by adding a ll the vertices of L ( T ) ∪ S + ≥ 2 ( T ) ∪ F ( T ), wher e F ( T ) is the s et o f first v er tices of maxima l directed paths consisting o f link ver- tices of T , to eac h o f the bags of a path decomp osition of H o f width 1. Observe that the pathwidth of this decomp osition is bo unded from ab ov e b y | L ( T ) | + | S + ≥ 2 ( T ) | + | F ( T ) | + 1 ≤ ( k − 1) + ( k − 2) + (2 k − 3) + 1 ≤ 4 k − 5 . The b ounds on the v arious sets in the ineq uality above follo ws from F acts 1 and 2. This proves the theor em. ⊓ ⊔ Corollary 1. F or acyclic digr aphs, the pr oblem k -DMLOB c an solve d in time 2 O ( k log k ) · n O (1) . Pr o of. The pro of of The o rem 7 can b e easily tur ned into a po lynomial time algorithm to either build a n out-branching of D with at lea s t k leav es or to show that pw (UG( D )) ≤ 4 k a nd provide the corres p o nding path decomp ositio n. A standard dynamic progra mming ov e r the path (tree) decompos ition (see e.g. [6 ]) gives us an algorithm of running time 2 O ( k l og k ) · n O (1) . ⊓ ⊔ The following simple lemma is well kno wn, see, e.g., [15]. Lemma 3. L et T = ( V , E ) b e an undir e cte d tr e e and let w : V → R + ∪ { 0 } b e a weight fun ction on its vertic es. Ther e exists a vertex v ∈ T s u ch that the weight of every subtr e e T ′ of T − v is at most w ( T ) / 2 , wher e w ( T ) = P v ∈ V w ( v ) . Let D b e a strong ly connected digr aph with ℓ s ( D ) = λ and let T b e an out- branching of D with λ lea ves. Consider the following decomp osition of T (called a β - de c omp osition ) which will b e useful in the pro of of Theo rem 8. Assign weigh t 1 to all le av es o f T and weigh t 0 to a ll non- le aves of T . B y Lemma 3 , T has a vertex v s uch that each co mpo nen t of T − v has at most λ/ 2 + 1 leav es (if v is not the r o ot and its in-neig h b o r v − in T is a link v ertex, then v − bec omes a new leaf ). Le t T 1 , T 2 , . . . , T s be the comp onents of T − v and let l 1 , l 2 , . . . , l s be the num b e rs of leav es in the comp onents. Notice that λ ≤ P s i =1 l i ≤ λ + 1 (we may get a new lea f ). W e may assume that l s ≤ l s − 1 ≤ · · · ≤ l 1 ≤ λ/ 2 + 1 . Let j b e the first index suc h that P j i =1 l i ≥ λ 2 + 1 . Consider t wo ca ses: (a) l j ≤ ( λ + 2) / 4 and (b) l j > ( λ + 2) / 4. In Case (a), we hav e λ + 2 2 ≤ j X i =1 l i ≤ 3( λ + 2) 4 and λ − 6 4 ≤ s X i = j + 1 l i ≤ λ 2 . 10 N. Alon, F. V. F omin, G. Gutin, M. Krivelevic h, and S. Saurabh In Case (b), we hav e j = 2 and λ + 2 4 ≤ l 1 ≤ λ + 2 2 and λ − 2 2 ≤ s X i =2 l i ≤ 3 λ + 2 4 . Let p = j in Cas e (a) and p = 1 in Case (b). Add to D and T a c opy v ′ of v (with the same in- and out-neighbor s). Then the num b er of leaves in each of the out-trees T ′ = T [ { v } ∪ ( ∪ p i =1 V ( T i ))] and T ′′ = T [ { v ′ } ∪ ( ∪ s i = p +1 V ( T i ))] is betw een λ (1 + o (1)) / 4 and 3 λ (1 + o (1)) / 4. Observe that the vertices of T ′ hav e at most λ + 1 out-neighbor s in T ′′ and the vertices of T ′′ hav e at mos t λ + 1 out-neighbors in T ′ (w e add 1 to λ due to the fact that v ‘b elongs’ to b o th T ′ and T ′′ ). Similarly to deriv ing T ′ and T ′′ from T , we can obtain tw o out-tre e s from T ′ and t wo out-tree s from T ′′ in which the num b ers of leaves are approximately betw een a quar ter and three qua rters o f the n um be r of leaves in T ′ and T ′′ , resp ectively . Obse rve that after O (log λ ) ‘dividing’ steps, we will end up with O ( λ ) o ut-trees with just one leaf, i.e., dire cted pa ths. T hes e paths contain O ( λ ) copies of vertices of D (suc h as v ′ ab ov e). After deleting the copies, w e obtain a collection of O ( λ ) disjoin t directed paths covering V ( D ). Theorem 8. L et D b e a str ongly c onne cte d digr aph. Then either ℓ s ( D ) ≥ k or the underlying undir e cte d gr aph of D is of p athwidth O ( k log k ) . Pr o of. W e ma y assume that ℓ s ( D ) < k . Let T be b e a 1-AE o ptimal out- branching. Co nsider a β -deco mpo s ition of T . The deco mpo sition pro cess c a n be viewed as a tree T roo ted in a no de (asso cia ted with) T . The children of T in T are no des (asso ciated with) T ′ and T ′′ ; the leav e s of T ar e the directed paths of the decomp ositio n. The first layer of T is the no de T , the se c ond layer ar e T ′ and T ′′ , the thir d layer are the c hildren o f T ′ and T ′′ , etc. In wha t follows, we do not distinguish b etw een a no de Q of T and the tree asso ciated with the no de. Assume that T has t lay ers. Notice that the la st layer consists of (some) leav es of T and that t = O (log k ), whic h was prov ed ab ov e ( k ≤ λ − 1). Let Q b e a no de o f T at layer j . W e will prove that pw (UG ( D [ V ( Q )])) < 2( t − j + 2 . 5 ) k (1) Since t = O (log k ), (1) for j = 1 implies that the underly ing undirected graph of D is of pathwidth O ( k lo g k ). W e firs t prove (1) fo r j = t when Q is a path from the decomp osition. Le t W = ( L ( T ) ∪ S + ≥ 2 ( T ) ∪ F ( T )) ∩ V ( Q ) , where F ( T ) is the set o f first vertices of maximal paths of T consisting of link vertices. As in the pro of of Theor em 7, it follows from F acts 1 and 2 that | W | < 4 k . O btain a digr aph R b y deleting from D [ V ( Q )] all arcs in which a t least one end-v ertex is in W a nd whic h are not arcs of Q . As in the pr o of of Theo r em 7, it fo llows from Lemma 1 and 1- AE opti- mality of T that there are no forward arcs for Q in R . Let Q = v 1 v 2 . . . v q . F or Spanning directed trees with many leav es 11 every j ∈ [ q ], le t V j = { v i : i ∈ [ j ] } . If for some j the set V j contained k vertices, say { v ′ 1 , v ′ 2 , · · · , v ′ k } , having in-neighbors in the set { v j +1 , v j +2 , . . . , v q } , then D would contain an out-tr ee w ith k leaves formed b y the path v j +1 v j +2 . . . v q to- gether with a bac kward a rc terminating a t v ′ i from a vertex on the pa th for each 1 ≤ i ≤ k , a contradiction. Thus v s (UG( D 2 [ P ])) ≤ k . By Prop os ition 2, the pathwidt h of UG( R ) is at most k . Let ( X 1 , X 2 , . . . , X s ) be a path decomp osition of UG( R ) of width at most k . Then ( X 1 ∪ W , X 2 ∪ W , . . . , X s ∪ W ) is a path decomp osition of UG( D [ V ( Q )]) of width less tha n k + 4 k . Thus, pw (UG ( D [ V ( Q )])) < 5 k (2) Now as s ume that we hav e prov ed (1) for j = i a nd show it fo r j = i − 1. Let Q be a no de o f lay er i − 1. If Q is a leaf o f T , we a re done b y (2). So , we may assume that Q has children Q ′ and Q ′′ which a re no des of layer i . In the β -de c omp o sition of T given b efore this theorem, we saw that the vertices o f T ′ hav e at most λ + 1 out-neigh bo rs in T ′′ and the vertices of T ′′ hav e at most λ + 1 out-neighbors in T ′ . Similarly , we can see that (in the β - decomp osition of this pro of ) the vertices of Q ′ hav e at most k o ut-neighbors in Q ′′ and the vertices of Q ′′ hav e at mos t k out-neighbors in Q ′ (since k ≤ λ − 1). Let Y denote the set o f the above-men tioned out- ne ig hbors on Q ′ and Q ′′ ; | Y | ≤ 2 k . Delete from D [ V ( Q ′ ) ∪ V ( Q ′′ )] all ar cs in which at least o ne end-vertex is in Y and whic h do not b elong to Q ′ ∪ Q ′′ Let G denote the o btained digraph. Observe that G is disconnected and G [ V ( Q ′ )] and G [ V ( Q ′′ )] ar e compone nts of G . Thus, pw (UG( G )) ≤ b , where b = max { pw (UG ( G [ V ( Q ′ )])) , pw (UG ( G [ V ( Q ′′ )])) } < 2 ( t − i + 4 . 5 ) k (3) Let ( Z 1 , Z 2 , . . . , Z r ) be a path deco mpo sition o f G o f width at most b. Then ( Z 1 ∪ Y , Z 2 ∪ Y , . . . , Z r ∪ Y ) is a path deco mpo s ition of UG( D [ V ( Q ′ ) ∪ V ( Q ′′ )]) of width at mos t b + 2 k < 2( t − i + 2 . 5 ) k . ⊓ ⊔ Similar to the pro of of Corolla r y 1, we obtain the following: Corollary 2. F or a str ongly c onne cte d digr aph D , the pr oblem k -DMLOB c an b e solve d in time 2 O ( k log 2 k ) · n O (1) . 6 Discussion and Op en Problems In this pap er , we initiated the alg orithmic and combinatorial study of the Di- rected Maximum Leaf Out-Branching pro blem. In particular , w e show ed that for every strongly connected digr aph D of order n and with minim um in- degree at least 3, ℓ s ( D ) = Ω ( n 1 / 3 ). An interesting open combinatorial question here is whether this b ound is tight. If it is not, it would b e in teresting to find the maximum n um b e r r such that ℓ s ( D ) = Ω ( n r ) for every strongly connected digraph D of order n and with minimum in- degree at least 3. It follows from our results that 1 3 ≤ r ≤ 1 2 . 12 N. Alon, F. V. F omin, G. Gutin, M. Krivelevic h, and S. Saurabh W e also pr ovided an algor ithm of time complexity 2 O ( k log 2 k ) · n O (1) which solves the k -DMLOB pro blem for a strongly connected digraph D . The algo - rithm is based on a co m binatorial b ound on the pathwidth of the under lying graph of D . Instead of using results fro m Section 5, one can use Bo dlae nder ’s algorithm [9] computing (for fixed k ) tree decomp osition o f width k (if such a decomp osition exists) in linear time. Combin ed w ith o ur com binatorial bo unds this yields a linea r time algorithm for k -DMLOB (for a str o ngly connected di- graphs). How ever, the expo ne ntial dep endence of k in B o dlaender’s algor ithm is c k 3 for some lar g e constant c . Finally , let us obser ve that while our results are for strongly connected di- graphs, they can b e extended to a larger class of digraphs. Notice that ℓ ( D ) ≥ ℓ s ( D ) for e a ch dig raph D . Let L b e the family of digraphs D for which either ℓ s ( D ) = 0 or ℓ s ( D ) = ℓ ( D ). The following a ssertion shows that L includes a large n umber digra phs including all str ongly connected digraphs a nd a cyclic di- graphs (and, also, the well-studied classes of se mico mplete m ultipartite digra phs and quasi-tra nsitive digraphs, se e [7 ] for the definitions). Prop ositio n 3 ([5] ). Supp ose that a digr aph D satisfies the fol lowing pr op erty: for every p air R and Q of distinct str ong c omp onents of D , if ther e is an ar c fr om R to Q then e ach vertex of Q has an in-neighb or in R . Then D ∈ L . Let B b e the family o f digra phs that contain out-br anchings. The results of this pap er proved for str o ngly connected digraphs can b e extended to the class L ∩ B o f digraphs since in the pro ofs we use only the following prop erty of strongly connected dig raphs D : ℓ s ( D ) = ℓ ( D ) > 0. F or a digraph D a nd a vertex v , let D v denote the subdig raph of D induced by all vertices reachable fro m v . Using the 2 O ( k log 2 k ) · n O (1) algorithm fo r k - DMLOB on digra phs in L ∩ B a nd the facts that (i) D v ∈ L ∩ B for each digr aph D and v ertex v and (ii) ℓ ( D ) = max { ℓ s ( D v ) | v ∈ V ( D ) } (for details, see [5]), we can o btain a n 2 O ( k log 2 k ) · n O (1) algorithm for k -DMLOT on al l digr aphs. F or acyclic digra phs, the running time ca n be reduced to 2 O ( k l og k ) · n O (1) . Ac kno wledgeme n ts. Resear ch of N. Alon and M. Krivelevic h was suppo rted in pa r t by USA-Israeli BSF gra nt s and b y grants from the Israel Scie nce F oun- dation. Research of F. 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