On Finding Directed Trees with Many Leaves
The Rooted Maximum Leaf Outbranching problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at least $k$ le…
Authors: Jean Daligault, Stephan Thomasse
On nding direted trees with man y lea v es Jean Daligault and Stéphan Thomassé Otob er 24, 2018 Abstrat The R OOTED M AXIMUM L EAF O UTBRANCHING problem onsists in nding a spanning direted tree ro oted at some presrib ed v ertex of a digraph with the maxim um n um b er of lea v es. Its parameterized v ersion asks if there exists su h a tree with at least k lea v es. W e use the notion of s − t n um b ering studied in [18 ℄, [5 ℄, [19 ℄ to exhibit om binatorial b ounds on the existene of spanning direted trees with man y lea v es. These om- binatorial b ounds allo w us to pro due a onstan t fator appro ximation algorithm for nding direted trees with man y lea v es, whereas the b est kno wn appro ximation algorithm has a √ OP T -fator [10 ℄. W e also sho w that R OOTED M AXIMUM L EAF O UTBRANCHING admits a quadrati k ernel, impro ving o v er the ubi k ernel giv en b y F ernau et al [12 ℄. 1 In tro dution An outbr anhing of a digraph D is a spanning direted tree in D . W e onsider the follo wing problem: R OOTED M AXIMUM L EAF O UTBRANCHING : Input : A digraph D , an in teger k , a v ertex r of D . Output : TR UE if there is an outbran hing of D ro oted at r with at least k lea v es, otherwise F ALSE. This problem is equiv alen t to nding a Conneted Dominating Set of size at most | V ( D ) | − k , onneted meaning in this setting that ev ery v ertex is rea hable b y a direted path from r . Indeed, the set of in ternal no des in an outbran hing orresp ond to a onneted dominating set. Finding undireted trees with man y lea v es has man y appliations in the area of omm uniation net w orks, see [7℄ or [23 ℄ for instane. An extensiv e litterature is dev oted to the paradigm of using a small onneted dominating set as a ba kb one for a omm uniation net w ork. R OOTED M AXIMUM L EAF O UTBRANCHING is NP-omplete, ev en restrited to ayli digraphs [2℄, and MaxSNP-hard, ev en on undireted graphs [15 ℄. 1 T w o natural w a ys to ta kle su h a problem are, on the one hand, p olynomial- time appro ximation algorithms, and on the other hand, parameterized omplex- it y . Let us giv e a brief in tro dution on the parameterized approa h. An eien t w a y of dealing with NP-hard problems is to iden tify a parameter whi h on tains its omputational hardness. F or instane, instead of asking for a minim um v ertex o v er in a graph - a lassial NP-hard optimization question - one an ask for an algorithm whi h w ould deide, in O ( f ( k ) .n d ) time for some xed d , if a graph of size n has a v ertex o v er of size at most k . If su h an algorithm exists, the problem is alled xe d-p ar ameter tr atable , or FPT for short. An extensiv e literature is dev oted to FPT, the reader is in vited to read [9 ℄, [13 ℄ and [20 ℄. Kernelization is a natural w a y of pro ving that a problem is FPT. F ormally , a kernelization algorithm reeiv es as input an instane ( I , k ) of the parameterized problem, and outputs, in p olynomial time in the size of the instane, another instane ( I ′ , k ′ ) su h that: k ′ ≤ k , the size of I ′ only dep ends of k , and the instanes ( I , k ) and ( I ′ , k ′ ) are b oth true or b oth false. The redued instane ( I ′ , k ′ ) is alled a kernel . The existene of a k erneliza- tion algorithm learly implies the FPT harater of the problem sine one an k ernelize the instane, and then solv e the redued instane G ′ , k ′ using brute fore, hene giving an O ( f ( k ) + n d ) algorithm. A lassial result asserts that b eing FPT is indeed equiv alen t to ha ving k ernelization. The dra wba k of this result is that the size of the redued instane G ′ is not neessarily small with resp et to k . A m u h more onstrained ondition is to b e able to redue to an instane of p olynomial size in terms of k . Consequen tly , in the zo ology of parameterized problems, the rst distintion is done b et w een three lasses: W[1℄-hard, FPT, p olyk ernel. A k ernelization algorithm an b e used as a prepro essing step to redue the size of the instane b efore applying some other parameterized algorithm. Being able to ensure that this k ernel has atually p olynomial size in k enhanes the o v erall sp eed of the pro ess. See [16℄ for a reen t review on k ernelization. An extensiv e litterature is dev oted to nding trees with man y lea v es in undi- reted and direted graphs. The undireted v ersion of this problem, M AXIMUM L EAF S P ANNING T REE , has b een extensiv ely studied. There is a fator 2 ap- pro ximation algorithm for the M AXIMUM L EAF S P ANNING T REE problem [ 21 ℄, and a 3 . 75 k k ernel [11 ℄. An O ∗ (1 , 9 4 n ) exat algorithm w as designed in [14 ℄. Other graph theoretial results on the existene of trees with man y lea v es an b e found in [8℄ and [22 ℄. The b est appro ximation algorithm kno wn for M AXIMUM L EAF O UTBRANCHING is a fator √ OP T algorithm [10 ℄. F rom the P arameterized Complexit y viewp oin t, Alon et al sho w ed that M AXIMUM L EAF O UTBRANCHING restrited to a wide lass of digraphs on taining all strongly onneted digraphs is FPT [ 1 ℄, and Bonsma and Dorn extended this result to all digraphs and ga v e a faster pa- rameterized algorithm [4 ℄. V ery reen tly , Kneis, Langer and Rossmanith [17 ℄ obtained an O ∗ (4 k ) algorithm for M AXIMUM L EAF O UTBRANCHING , whi h is 2 also an impro v emen t for the undireted ase o v er the n umerous FPT algorithms designed for M AXIMUM L EAF S P ANNING T REE . F ernau et al [ 12 ℄ pro v ed that R OOTED M AXIMUM L EAF O UTBRANCHING has a p olynomial k ernel, exhibiting a ubi k ernel. They also sho w ed that the unro oted v ersion of this problem ad- mits no p olynomial k ernel, unless p olynomial hierar h y ollapses to third lev el, using a breakthrough lo w er b ound result b y Bo dlaender et al [3℄. A linear k er- nel for the ayli sub ase of R OOTED M AXIMUM L EAF O UTBRANCHING and an O ∗ (3 , 7 2 k ) algorithm for R OOTED M AXIMUM L EAF O UTBRANCHING w ere exhibited in [6 ℄. This pap er is organized as follo ws. In Setion 2 w e exhibit om binatorial b ounds on the problem of nding an outbran hing with man y lea v es. W e use the notion of s − t numb ering in tro dued in [18 ℄. W e next presen t our redution rules, whi h are indep enden t of the parameter, and in the follo wing setion w e pro v e that these rules giv e a quadrati k ernel. W e nally presen t a onstan t fator appro ximation algorithm in Setion 5 for nding direted trees with man y lea v es. 2 Com binatorial Bounds Let D b e a direted graph. F or an ar ( u, v ) in D , w e sa y that u is an in- neighb our of v , that v is an outneighb our of u , that ( u, v ) is an in-ar of v and an out-ar of u . The outde gr e e of a v ertex is the n um b er of its outneigh b ours, and its inde gr e e is the n um b er of its in-neigh b ours. An outbran hing with a maxim um n um b er of lea v es is said to b e optimal . Let us denote b y maxl e af ( D ) the n um b er of lea v es in an optimal outbran hing of D . Without loss of generalit y , w e restrit ourselv es to the follo wing. W e ex- lusiv ely onsider lo opless digraphs with a distinguished v ertex of indegree 0, denoted b y r . W e assume that ther e is no ar ( u, r ) in D with u ∈ V ( D ) , and no ar ( x, y ) with x 6 = r and y an outneighb our of r , and that r has outde gr e e at le ast 2 . Throughout this pap er, w e all su h a digraph a r o ote d digr aph . Denitions will b e made exlusiv ely with resp et to ro oted digraphs, hene the notions w e presen t, lik e onnetivit y and resulting onepts, do sligh tly dier from standard ones. Let D b e a ro oted digraph with a sp eied v ertex r . The ro oted digraph D is onne te d if ev ery v ertex of D is rea hable b y a direted path ro oted at r in D . A ut of D is a set S ⊆ V ( D ) − r su h that there exists a v ertex z / ∈ S endp oin t of no direted path from r in D − S . W e sa y that D is 2- onne te d if D has no ut of size at most 1. A ut of size 1 is alled a utvertex . Equiv alen tly , a ro oted digraph is 2-onneted if there are t w o in ternally v ertex-disjoin t paths from r to an y v ertex b esides r and its outneigh b ours. W e will sho w that the notion of s − t n um b ering b eha v es w ell with resp et to outbran hings with man y lea v es. It has b een in tro dued in [ 18 ℄ for 2-onneted undireted graphs, and generalized in [5 ℄ b y Cheriy an and Reif for digraphs whi h are 2-onneted in the usual sense. W e adapt it in the on text of ro oted 3 digraphs. Let D b e a 2-onneted ro oted digraph. An r − r n um b ering of D is a linear ordering σ of V ( D ) − r su h that, for ev ery v ertex x 6 = r , either x is an outneigh b our of r or there exist t w o in-neigh b ours u and v of x su h that σ ( u ) < σ ( x ) < σ ( v ) . An equiv alen t presen tation of an r − r n um b ering of D is an injetiv e em b edding f of the graph D where r has b een dupliated in to t w o v erties r 1 and r 2 , in to the [0 , 1 ] -segmen t of the real line, su h that f ( r 1 ) = 0 , f ( r 2 ) = 1 , and su h that the image b y f of ev ery v ertex b esides r 1 and r 2 lies inside the on v ex h ull of the images of its in-neigh b ours. Su h onvex emb e ddings ha v e b een dened and studied in general dimension b y Lo v ász, Linial and Wigderson in [19 ℄ for undireted graphs, and in [ 5℄ for direted graphs. Giv en a linear order σ on a nite set V , w e denote b y ¯ σ the linear order on V whi h is the rev erse of σ . An ar uv of D is a forwar d ar if u = r or if u app ears b efore v in σ ; uv is a b akwar d ar if u = r or if u app ears after v in σ . A spanning out-tree T is forwar d if all its ars are forw ard. Similar denition for b akwar d out-tree. The follo wing result and pro of is just an adapted v ersion of [ 5℄, giv en here for the sak e of ompleteness. Lemma 1 L et D b e a 2- onne te d r o ote d digr aph. Ther e exists an r − r num- b ering of D . Pr o of : By indution o v er D . W e rst redue to the ase where the indegree of ev ery v ertex b esides r is exatly 2. Let x b e a v ertex of indegree at least 3 in D . Let us sho w that there exists an in-neigh b our y of x su h that the ro oted digraph D − ( y , x ) is 2-onneted. Indeed, there exist t w o in ternally v ertex disjoin t paths from r to x . Consider su h t w o paths in terseting N − ( x ) only one ea h, and denote b y D ′ the ro oted digraph obtained from D b y remo ving one ar ( y , x ) not in v olv ed in these t w o paths. There are t w o in ternally disjoin t paths from r to x in D ′ . Consider z ∈ V ( D ) − r − x . Assume b y on tradition that there exists a v ertex t whi h uts z from r in D ′ . As t do es not ut z from r in D and the ar ( y , x ) alone is missing in D ′ , t m ust ut x and not y from r in D ′ . Whi h is a on tradition, as there are t w o in ternally disjoin t paths from r to x in D ′ . By indution, D ′ has an r − r n um b ering, whi h is also an r − r n um b ering for D . Hene, let D b e a ro oted digraph, where ev ery v ertex b esides r has indegree 2 . As r has indegree 0, there exists a v ertex v with outdegree at most 1 in D b y a oun ting argumen t. If v has outdegree 0, then let σ b e an r − r n um b ering of D − v , let u 1 and u 2 b e the t w o in-neigh b ours of v . Insert v b et w een u 1 and u 2 in σ to obtain an r − r n um b ering of D . Assume no w that v has a single outneigh b our u . Let w b e the seond in-neigh b our of u . Let D ′ b e the graph obtained from D b y on trating the ar ( v , u ) in to a single v ertex uv . As D ′ is 2-onneted, onsider b y indution an r − r n um b ering σ of D ′ . Replae uv b y u . It is no w p ossible to insert v b et w een its t w o in-neigh b ours in order to mak e it so that u lies b et w een v and w . Indeed, assume without loss of generalit y that w is after uv in σ . Consider the smallest in-neigh b our t of v in σ . As σ 4 is an r − r n um b ering of D ′ , t lies b efore uv in σ . W e insert v just after t to obtain an r − r n um b ering of D . Note that an r − r n um b ering σ of D naturally giv es t w o ayli o v ering sub digraphs of D , the ro oted digraph D | σ onsisting of the forw ard ars of D , and the ro oted digraph D | ¯ σ onsisting of the ba kw ard ars of D . The in tersetion of these t w o ayli digraphs is the set of out-ars of r . Corollary 1 L et D b e a 2- onne te d r o ote d digr aph. Ther e exists an ayli onne te d sp anning sub digr aph A of D whih ontains at le ast half of the ar s of D − r . Let G b e an undireted graph. A vertex over of G is a set of v erties o v ering all edges of G . A dominating set of G is a set S ⊆ V su h that for ev ery v ertex x / ∈ S , x has a neigh b our in S . A str ongly dominating set of G is a set S ⊆ V su h that ev ery v ertex has a neigh b our in S . Let D b e a ro oted digraph. A str ongly dominating set of D is a set S ⊆ V su h that ev ery v ertex b esides r has an in-neigh b our in S . W e need the follo wing folklore result: Lemma 2 A ny undir e te d gr aph G on n verti es and m ar s has a vertex over of size n + m 3 . Pr o of : By indution on n + m . If there exists a v ertex of degree at least 2 in G , ho ose it in the v ertex o v er, otherwise ho ose an y non-isolated v ertex. Lemma 3 L et G b e a bip artite gr aph over A ∪ B , with d ( a ) = 2 for every a ∈ A . Ther e exists a subset of B dominating A with size at most | A | + | B | 3 . Pr o of : Let G ′ b e the graph whi h v ertex set is B , and where ( b, b ′ ) is an ar if b and b ′ share a ommon neigh b our in A . The result follo ws from Lemma 2 sine G ′ has | A | ars and | B | v erties. Corollary 2 L et D b e an ayli r o ote d digr aph with l verti es of inde gr e e at le ast 2 and with a r o ot of outde gr e e d ( r ) ≥ 2 . Then D has an outbr anhing with at le ast l + d ( r ) − 1 3 + 1 le aves. Pr o of : Denote b y n the n um b er of v erties of D . F or ev ery v ertex v of indegree at least 3, delete inoming ars un til v has indegree exatly 2. Sine D is ayli, it has a sink s . Let Z b e the set of v erties of indegree 1 in D , of size n − 1 − l . Let Y b e the set of in-neigh b ours of v erties of Z , of size at most n − 1 − l . Let A ′ b e the set of v erties of indegree 2 dominated b y Y . Let B = V ( D ) − Y − s . Let A b e the set of v erties of indegree 2 not dominated b y Y . Note that Y annot ha v e the same size as Z ∪ A ′ . Indeed, Z on tains the outneigh b ours of r , and hene Y on tains r , whi h has outdegree at least 2. More preisely , | Y | + d ( r ) − 1 ≤ | Z ∪ A ′ | . As B = V ( D ) − Y − s and A = V ( D ) − A ′ − Z − r , 5 Figure 1: The "b oloney" graph D 6 w e ha v e that | B | ≥ | A | + d ( r ) − 1 . Moreo v er, as Y has size at most n − 1 − l , w e ha v e that | B | ≥ l . Consider a op y A 1 of A and a op y B 1 of B . Let G b e the bipartite graph with v ertex bipartition ( A 1 , B 1 ) , and where ( b, a ) , with a ∈ A 1 and b ∈ B 1 , is an edge if ( b, a ) is an ar in D . By Lemma 3 applied to G , there exists a set X ⊆ B of size at most | A | + | B | 3 ≤ 2 | B |− ( d ( r ) − 1) 3 whi h dominates A in D . The set C = X ∪ Y strongly dominates V ( D ) − r in D , and has size at most | X | + | Y | ≤ 2 | B |− ( d ( r ) − 1) 3 + | Y | = | B | + | Y | − | B | + d ( r ) − 1 3 . As | Y | + | B | = n − 1 and | B | ≥ l , this yields | X ∪ Y | ≤ n − 1 − l + d ( r ) − 1 3 . As D is ayli, an y set strongly dominating V − r on tains r and is a onneted dominating set. Hene there exists an outbran hing T of D ha ving a subset of C as in ternal v erties. T has at least l + d ( r ) − 1 3 + 1 lea v es. This b ound is tigh t up to one leaf. The ro oted digraph D k depited in Figure 1 is 2-onneted, has 3 k − 2 v erties of indegree at least 2, d ( r ) = 3 and maxl e af ( D k ) = k + 2 . Finally , the follo wing om binatorial b ound is obtained: Theorem 1 L et D b e a 2- onne te d r o ote d digr aph with l verti es of inde gr e e at le ast 3. Then maxl e af ( D ) ≥ l 6 . Pr o of : Apply Corollary 2 to the ro oted digraph with the larger n um b er of v er- ties of indegree 2 among D σ and D ¯ σ . An ar is simple if do es not b elong to a 2-iruit. A v ertex v is ni e if it is iniden t to a simple in-ar. The seond om binatorial b ound is the follo wing: Theorem 2 L et D b e 2- onne te d r o ote d digr aph. Assume that D has l ni e verti es. Then D has an outbr anhing with at le ast l 24 le aves. Pr o of : By Lemma 1 , w e onsider an r − r n um b ering σ of D . F or ev ery nie v ertex v (iniden t to some in-ar a ) with indegree at least three, delete inoming ars of v dieren t from a un til v has only one inoming forw ard ar and one inoming ba kw ard ar. F or ev ery other v ertex of indegree at least 3 in D , delete 6 inoming ars of v un til v has only one inoming forw ard ar and one inoming ba kw ard ar. A t the end of this pro ess, σ is still an r − r n um b ering of the digraph D , and the n um b er of nie v erties has not dereased. Denote b y T f the set of forw ard ars of D , and b y T b the set of ba kw ard ars of D . As σ is an r − r n um b ering of D , T f and T b are spanning trees of D whi h partition the ars of D − r . The ruial denition is the follo wing: sa y that an ar uv of T f (resp. of T b ), with u 6 = r , is tr ansverse if u and v are in omp ar able in T b (resp. in T f ), that is if v is not an anestor of u in T b (resp. in T f ). Observ e that u annot b e an anestor of v in T b (resp. in T f ) sine T b is ba kw ard (resp. T f is forw ard) while uv is forw ard (resp. ba kw ard) and u 6 = r . Assume without loss of generalit y that T f on tains more transv erse ars than T b . Consider no w an y planar dra wing of the ro oted tree T b . W e will mak e use of this dra wing to dene the follo wing: if t w o v erties u and v are inomparable in T b , then one of these v erties is to the left of the other, with resp et to our dra wing. Hene, w e an partition the transv erse ars of T f in to t w o subsets: the set S l of transv erse ars uv for whi h v is to the left of u , and the set S r of transv erse ars uv for whi h v is to the righ t of u . Assume without loss of generalit y that | S l | ≥ | S r | . The digraph T b ∪ S l is an ayli digraph b y denition of S l . Moreo v er, it has | S l | v erties of indegree t w o sine the heads of the ars of | S l | are pairwise distint. Hene, b y Corollary 2 , T b ∪ S l has an outbran hing with at least | S l | + d ( r ) − 1 3 + 1 lea v es, hene so do es D . W e no w giv e a lo w er b ound on the n um b er of transv erse ars in D to b ound | S l | . Consider a nie v ertex v in D , whi h is not an outneigh b our of r , and with a simple in-ar uv b elonging to, sa y , T f . If uv is not a transv erse ar, then v is an anestor of u in T b . Let w b e the outneigh b or of v on the path from v to u in T b . Sine uv is simple, the v ertex w is distint from u . No path in T f go es from w to v , hene v w is a transv erse ar. Therefore, w e pro v ed that v (and hene ev ery nie v ertex) is iniden t to a transv erse ar (either an in-ar, or an out-ar). Th us there are at least l − d ( r ) 2 transv erse ars in D . Finally , there are at least l − d ( r ) 4 transv erse ars in T f , and th us | S l | ≥ l − d ( r ) 8 . In all, D has an outbran hing with at least l 24 lea v es. As a orollary , the follo wing result holds for orien ted graphs (digraphs with no 2-iruit): Corollary 3 Every 2- onne te d r o ote d oriente d gr aph on n verti es has an out- br anhing with at le ast n − 1 24 le aves. 3 Redution Rules W e sa y that P = { x 1 , . . . , x l } , with l ≥ 3 , is a bip ath of length l − 1 if the set of ars adjaen t to { x 2 , . . . , x l − 1 } in D is exatly { ( x i , x i +1 ) , ( x i +1 , x i ) | i ∈ { 1 , . . . , l − 1 } } . 7 T o exhibit a quadrati k ernel for R OOTED M AXIMUM L EAF O UTBRANCHING , w e use the follo wing four redution rules: (0) If there exists a v ertex not rea hable from r in D , then redue to a trivially F ALSE instane. (1) Let x b e a utv ertex of D . Delete v ertex x and add an ar ( v , z ) for ev ery v ∈ N − ( x ) and z ∈ N + ( x ) − v . (2) Let P b e a bipath of length 4. Con trat t w o onseutiv e in ternal v erties of P . (3) Let x b e a v ertex of D . If there exists y ∈ N − ( x ) su h that N − ( x ) − y uts y from r , then delete the ar ( y , x ) . Note that these redution rules are not parameter dep enden t. Rule (0) only needs to b e applied one. Observ ation 1 L et S b e a utset of a r o ote d digr aph D . L et T b e an outbr anh- ing of D . Ther e exists a vertex in S whih is not a le af in T . Lemma 4 The ab ove r e dution rules ar e safe and an b e he ke d and applie d in p olynomial time. Pr o of : (0) Rea habilit y an b e tested in linear time. (1) Let x b e a utv ertex of D . Let D ′ b e the graph obtained from D b y deleting v ertex x and adding an ar ( v , z ) for ev ery v ∈ N − ( x ) and z ∈ N + ( x ) − v . Let us sho w that maxleaf ( D ) = maxleaf ( D ′ ) . Assume T is an outbran hing of D ro oted at r with k lea v es. By Observ ation 1, x is not a leaf of T . Let f ( x ) b e the father of x in T . Let T ′ b e the tree obtained from T b y on trating x and f ( x ) . T ′ is an outbran hing of D ′ ro oted at r with k lea v es. Let T ′ b e an outbran hing of D ′ ro oted at r with k lea v es. N − ( x ) is a ut in D ′ , hene b y Observ ation 1 there is a non-empt y olletion of v erties y 1 , . . . , y l ∈ N − ( x ) whi h are not lea v es in T ′ . Cho ose y i su h that y j is not an anestor of y i for ev ery j ∈ { 1 , . . . , l } − { i } . Let T b e the graph obtained from T ′ b y adding x as an isolated v ertex, adding the ar ( y i , x ) , and for ev ery j ∈ { 1 , . . . , l } , for ev ery ar ( y j , z ) ∈ T with z ∈ N + ( x ) , delete the ar ( y j , z ) and add the ar ( x, z ) . As y i is not rea hable in T ′ from an y v ertex y ∈ N − ( x ) − y i , there is no yle in T . Hene T is an outbran hing of D ro oted at r with at least k lea v es. Moreo v er, deiding the existene of a ut v ertex and nding one if su h exists an b e done in p olynomial time. 8 (2) Let P b e a bipath of length 4. Let u , x , y , z and t b e the v erties of P in this onseutiv e order. Let D ′ b e the ro oted digraph obtained from D b y on trating x and y . Let T b e an outbran hing of D . Let T ′ b e the ro oted digraph obtained from T b y on trating y with its father in T . T ′ is an outbran hing of D ′ with as man y lea v es as T . Let T ′ b e an outbran hing of D ′ . If the father of xy in T ′ is z , then T ′ − ( z , xy ) ∪ ( z , y ) ∪ ( y , x ) is an outbran hing of D with at least as man y lea v es as T ′ . If the father of xy in T ′ is u , then T ′ − ( u , xy ) ∪ ( u, x ) ∪ ( x, y ) is an outbran hing of D with at least as man y lea v es as T ′ . (3) Let x b e a v ertex of D . Let y ∈ N − ( x ) b e a v ertex su h that N − ( x ) − y uts y from r . Let D ′ b e the ro oted digraph obtained from T b y deleting the ar ( y , x ) . Ev ery outbran hing of D ′ is an outbran hing of D . Let T b e an outbran hing of D on taining ( y , x ) . There exists a v ertex z ∈ N − ( x ) − y whi h is an anestor of x . Th us T − ( y , x ) ∪ ( z , x ) is an outbran hing of D ′ with at least as man y lea v es as T . W e apply these rules iterativ ely un til rea hing a r e du e d instan e , on whi h none an b e applied. Lemma 5 L et D b e a r e du e d r o ote d digr aph with a vertex of inde gr e e at le ast k . Then D is a TR UE instan e. Pr o of : Assume x is a v ertex of D with in-neigh b ourho o d N − ( x ) = { u 1 , . . . , u l } , with l ≥ k . F or ev ery i ∈ { 1 , . . . , l } , N − ( x ) − u i do es not ut u i from r . Th us there exists a path P i from r to u i outside N − ( x ) − u i . The ro oted digraph D ′ = ∪ i ∈{ 1 ,...,l } P i is onneted, and for ev ery i ∈ { 1 , . . . , l } , u i has outdegree 0 in D ′ . Th us D ′ has an outbran hing with at least k lea v es, and su h an outbran hing an b e extended in to an outbran hing of D with at least as man y lea v es. 4 Quadrati k ernel In this setion and the follo wing, a v ertex of a 2-onneted ro oted digraph D is said to b e sp e ial if it has indegree at least 3 or if one of its inoming ars is simple. A non sp eial v ertex is a v ertex u whi h has exatly t w o in-neigh b ours, whi h are also outneigh b ours of u . A we ak bip ath is a maximal onneted set of non sp eial v erties. If P = { x 1 , . . . , x l } is a w eak bipath, then the in-neigh b ours of x i , for i = 2 , . . . , l − 1 in D are exatly x i − 1 and x i +1 . Moreo v er, x 1 and x l are ea h outneigh b our of a sp eial v ertex. Denote b y s ( P ) the in-neigh b our of x 1 whi h is a sp eial v ertex. This setion is dediated to the pro of of the follo wing statemen t: Theorem 3 A digr aph D of size at le ast (3 k − 2 )(30 k − 2) r e du e d under the r e dution rules of pr evious se tion has an outbr anhing with at le ast k le aves. 9 Pr o of : By Theorem 1 and Theorem 2 , if there are at least 6 k + 24 k − 1 sp eial v erties, then D has an outbran hing with at least k lea v es. Assume that there are at most 30 k − 2 sp eial v erties in D . As D is redued under Rule (2), there is no bipath of length 4. W e an asso iate to ev ery w eak bipath B of D of length t a set A B of ⌈ t/ 3 ⌉ out- ars to w ard sp eial v erties. Indeed, let P = ( x 1 , . . . , x l ) b e a w eak bipath of D . F or ev ery three onseutiv e v erties x i , x i +1 , x i +2 of P , 2 ≤ i ≤ l − 3 , ( x i − 1 , x i , x i +1 , x i +2 , x i +3 ) is not a bipath b y Rule (2), hene there exists an ar ( x j , z ) with j = i, i + 1 or i + 2 and z / ∈ P . Moreo v er z m ust b e a sp eial v ertex as ars b et w een non-sp eial v erties lie within their o wn w eak bipath. The set of these ars ( x j , z ) has the presrib ed size. By Lemma 5, an y v ertex in D has indegree at most k − 1 as D is redued under Rule (3), hene there are at most 3( k − 1)(30 k − 2) non sp eial v erties in D . T o sum up, the k ernelization algorithm is as follo ws: starting from a ro oted digraph D , apply the redution rules. Let D ′ b e the obtained redued ro oted digraph. If D has size more than (3 k − 2)(30 k − 2) , then redue to a trivially TR UE instane. Otherwise, D ′ is an instane equiv alen t to D of size quadrati in k . Our analysis for this quadrati k ernel for R OOTED M AXIMUM L EAF O UTBRANCHING is atually tigh t up to a onstan t fator. Indeed, the follo wing graph T l is re- dued under the redution rules stated on Setion 3 and has a n um b er of v er- ties quadrati in its maximal n um b er of lea v es. Let V = { v i,j | i = 1 , . . . , l, j = 1 , . . . , 3( l − 1) } . F or ev ery i = 1 , . . . , l , ( r , v i, 1 ) is an ar of T . F or ev- ery j = 1 , . . . , 3 l − 2 , i = 1 , . . . , l , ( v i,j , v i,j +1 ) is a 2-iruit of T l . F or ev ery i = 1 , . . . , l , ( v i, 3 l − 1 , v i +1[ l ] , 3 l − 1 ) is an ar of T l . F or ev ery t = 1 , . . . , l − 1 , i = 1 , . . . , l , ( v i, 3 t , v i + t [ l ] , 1 ) is an ar of T l . This digraph T l is redued under the redution rules of Setion 3, and maxl e af ( T l ) = 2( l − 1) . Finally , T l has 3 l ( l − 1 ) + 1 v erties. Note that this graph has man y 2-iruits. W e are not able to deal with them with resp et to k ernelization. F or the appro ximation on the on trary , w e are able to deal with the 2-iruits to pro due a onstan t fator appro ximation algorithm. 5 Appro ximation Let us rst p oin t out that the redution rules desrib ed in Setion 4 diretly giv e an appro ximation algorithm asymptotially as go o d as the b est kno wn appro xi- mation algorithm [10 ℄. Indeed, as these rules are indep endan t of the parameter, and as our pro of of the existene of a solution of size k when the redued graph has size more than 3( k − 1)(30 k − 2) is on trutiv e, this yields a O ( √ OP T ) appro ximation algorithm. Let us sk et h this appro ximation algorithm. Start b y applying the redution rules desrib ed in Setion 4 to the input ro oted digraph. This do es not hange the v alue of the problem. Let m b e the size of the redued 10 graph. Exhibit an outbran hing with at least p m 90 lea v es as in the pro of of Theorem 3. Finally , undo the sequene of on trations yield b y the appliation of redution rules at the start of the algorithm, repairing the tree as in the pro of of Lemma 4. The tree th us obtained has at least p m 90 lea v es, while the tree with maxim um n um b er of lea v es in the input graph has at most m − 1 lea v es. Th us this algorithm is an O ( √ OP T ) appro ximation algorithm. Let us desrib e no w our onstan t fator appro ximation algorithm for R OOTED M AXIMUM L EAF O UTBRANCHING , b eing understo o d that this also giv es an ap- pro ximation algorithm of the same fator for M AXIMUM L EAF O UTBRANCHING as w ell as for nding an out-tree (not neessarily spanning) with man y lea v es in a digraph. Giv en a ro oted digraph D ′′ , apply exhaustiv ely Rule (1) of Setion 3. The resulting ro oted digraph D is 2-onneted. By Lemma 4 , maxl e af ( D ′′ ) = maxl e af ( D ) . Let us denote b y D ns the digraph D restrited to non sp eial v erties. Reall that D ns is a dijoin t union of bipaths, whi h w e all non sp e ial omp onents . A v ertex of outdegree 1 in D ns is alled an end . Ea h end has exatly one sp eial v ertex as an in-neigh b our in D . Theorem 4 L et D b e a 2- onne te d r o ote d digr aph with l sp e ial verti es and h non sp e ial omp onents. Then max ( l 30 , h − l ) ≤ maxl eaf ( D ) ≤ l + 2 h . Pr o of : The upp er b ound is lear, as at most t w o v erties in a giv en non sp eial omp onen t an b e lea v es of a giv en outbran hing. The rst term of the lo w er b ound omes from Theorem 1 and Theorem 2. T o establish the seond term, onsider the digraph D ′ whi h v erties are the sp eial v erties of D and r . F or ev ery non sp eial omp onen t of D , add an edge in D ′ b et w een the sp eial in- neigh b ours of its t w o ends. Consider an outbran hing of D ′ ro oted at r . This outbran hing uses l − 1 edges in D ′ , and diretly orresp onds to an out-tree T in D . Extend T in to an outbran hing ˜ T of D . Ev ery non sp eial omp onen t whi h is not used in T on tributes to at least a leaf in ˜ T , whi h onludes the pro of. Consider the b est of the three outbran hings of D obtained in p olynomial time b y Theorem 1, Theorem 2 and Theorem 4 . This outbran hing has at least max ( l 30 , h − l ) lea v es. The w orst ase is when l 30 = h − l . In this ase, the upp er b ound b eomes: 92 l 30 , hene w e ha v e a fator 92 appro ximation algorithm for R OOTED M AXIMUM L EAF O UTBRANCHING . 6 Conlusion W e ha v e giv en a quadrati k ernel and a onstan t fator appro ximation algorithm for R OOTED M AXIMUM L EAF O UTBRANCHING : reduing the gap b et w een the problem of nding trees with man y lea v es in undireted and direted graphs. M AXIMUM L EAF S P ANNING T REE has a fator 2 appro ximation algorithm, and R OOTED M AXIMUM L EAF O UTBRANCHING no w has a fator 92 appro ximation 11 algorithm. Reduing this 92 fator in to a small onstan t is one hallenge. 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