Max-Leaves Spanning Tree is APX-hard for Cubic Graphs

Max-Lea v es Spanning T ree is APX-hard for Cubic Gra phs P aul Bonsma Hum b o ldt Univ ersit¨ at zu Berlin, Computer Science Department, Un ter den Linden 6, 10099 Berlin. Ma y 31, 2018 Abstract W e consider the pr oblem o f findin g a spanning tree with maximum n umber o f leav es (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3 / 2-approximation algorithm when restricted to graphs where every vertex has degree 3 ( cubic g r aphs). MaxLeaf is known to be APX-hard in genera l, and NP-ha rd for cubic graphs. W e show that the problem is also AP X-hard for cubic gr aphs. The AP X-hardness of the rela ted problem Minim um Connected Dominating Set for cubic gra phs follo ws. 1 In tro duction W e study the problem Maxim um Leaf Spann ing T ree or MaxLeaf, for whic h the ob jectiv e is to fin d in a give n connected graph a s p anning tree with m axim um num b er of lea v es. An α -appr oximation algorithm for a m aximizatio n (minimization) problem is a p olynomial time algorithm that ret urn s a so lution with ob j ectiv e v alue at least (at most) α · OPT, where O PT is the ob jecti v e v alue of an optimal s olution for th e giv en in stance 1 . MaxLeaf is k n o wn to b e APX-hard [12], which implies that there exists an ǫ > 0 such that no p olynomial time (1 − ǫ )- appro ximation algorithm is p ossible for this problem, unless P=NP [2]. Ho w ev er, constan t factor app ro ximation algorithms are known: Lu and Ravi [20] ga v e a 1 / 3-appro ximation, and this was later impr o v ed b y Solis-Oba who ga v e a linear time 1 / 2-appro ximation [23]. S o the problem is in APX – the class of optimization p roblems with constan t factor appro ximation algorithms – and therefore APX-complete. MaxLeaf is closely related to M inimum Conne cte d Dominating Set (MinCDS). This prob- lem asks, giv en a graph G , for a set S ⊂ V ( G ) of minimum size suc h that G [ S ] is connected and ev ery v ertex v 6∈ S is adjacent to a vertex in S (a c onne cte d dominating set ). The rela- tion b et ween these pr oblems is as follo ws: since the non-lea v es of a sp an n ing tree of G f orm a connected dominating set (unless G = K 2 ), G has a spanning tree with at least k leav es if and only if G has a connected dominating set of size at most | V ( G ) | − k . These problems differ from an app ro ximabilit y viewp oin t: Guha and Khuller [14] sho w ed that MinCDS admits n o constan t factor approximat ion algorithm u nder established complexity- theoretic assumptions. Ruan et al [22] giv e a 2 + ln ∆( G )-appro ximation algorithm, where ∆( G ) is the maxim um degree of G . In cubic graphs , ev ery v ertex has degree 3. The restriction of MaxLeaf to cubic graphs has receiv ed m uc h atten tion. One r eason is that these are easier to analyze algorithmically , 1 In the literature on MaxLeaf, approximation algorithms are u sually stated with α > 1 approximation ratios. F or our p roofs it is more convenien t to defi n e th ese as 1 /α -approximation algorithms. 1 y et from an appr o ximation viewp o int , this is where th e main hard ness lies. F or instance, for 5-regular graphs a 2 / 3-appro ximatio n follo w s easily f rom kn o wn b ounds [13], s ee b elo w. F or cubic graph s, more wo rk is r equired to obtain this ratio: Lory ´ s an d Zwo ´ znia k [18] ga v e a 4 / 7-a ppr oximati on for MaxLeaf on cubic graphs . This ratio was later improv ed to 3 / 5 b y Correa et al [6], and finally by Bonsma and Z ic kfeld [4] to 2 / 3. A natural question is ho w far this can b e imp ro v ed. How ev er, even the question wh ether the problem is APX-hard for cubic graph s remained op en. This qu estion was ask ed in [6] and [4]. The only kn o wn hardness result for cubic graphs is that the pr oblem is NP-hard, as w as sho wn by Lemk e in an unpu b lished tec hnical rep ort [17]. In this pap er w e settle the q u estion by showing that also for cubic graphs, the pr oblem is APX-hard. This is strictly stronger than the kn o wn hard n ess results [17, 12]. F rom this the APX-hardness of MinCDS for cubic graphs will also follo w. The pro of is int eresting by itself, since it sh o ws ho w APX-hardn ess r esults can b e pro v ed u sing extremal argumen ts. In formally sp eaking, th e pr oblem with proving APX-hardness for cubic graph s is that it seems imp ossible to fi nd ‘well -b ehav ed’ gadgets, that allo w for an easy analysis of the graph constructed in the reduction. Instead w e hav e a s imple construction, bu t need an elab orate global analysis of the constructed graph, inv olving v arious (fr actional) b oun d s and roun ding arguments. As a con trast, we giv e a n ew v ery simple and more traditional APX-hardn ess pro of for MaxLeaf in general graphs in at the end of this in tro duction. APX-hardness r esu lts for basic problems in restricted graph classes, in particular cu- bic graphs, are useful since they allo w for simple h ardness pro ofs of many other problems. The f our h ardness resu lts b y Alimonti and Kan n [1 ] ha v e often b een used for this purp ose: they sho w that the problems Minim um V ertex Cov er, Maxim um Indep enden t Set, Minim um Dominating Set and Maxim um Cut are APX-hard wh en restricted to cubic graphs. Their APX-hardness results for Maxim um In dep endent Set and Minim um V ertex Co v er will b e used for the t w o redu ctions in this pap er. W e n o w review some algorithmic results on MaxLeaf. Recen tly , the generalization of MaxLeaf to directed graphs or digr aph s h as receiv ed a lot of atten tion. V ery recen tly Daligault and Thomasse [7] ga v e a constan t factor approximat ion algorithm for this p roblem (more precisely , a 1 / 92-appro ximatio n algorithm), impr oving on th e Ω(1 / √ OPT)-appro ximation of Drescher and V etta [9]. The pap er of Daligault and Thomasse [7] also deals with the parameterized v arian t of the decision v ersion of Directed MaxLeaf. See [10, 16] for other parameterized r esults on (u n)directed MaxLeaf. Undirected MaxLeaf has also b ee n studied in th e area of fast exact algorithms. F omin et al [11] ga v e an algorithm for fin ding a minimum connected dominating set, and th erefore a maximum leaf spanning tree, that run s in time O (1 . 9407 n ) w here n is the num b er of v ertices. Com binatorial b oun ds form an imp ortant ingredient for many of the ab o v e results. F or instance, it is kn o wn that connected graphs with m in im um d egree δ ≥ 3 on n vertices admit a s panning tree with at least n/ 4 + 2 lea v es [15]. A str onger version of this b oun d app e ars in [5]. F or cubic graph s, see [4] for an impro v ed b ou n d. When δ ≥ 4, 2 n/ 5 + 8 / 5 lea v es are p ossible [15 , 13], and for δ ≥ 5, n/ 2 + 2 lea v es are p ossible [13]. In [3] and [7] b ound s for the directed case can b e found. One may w onder why it is muc h hard er to prov e APX-hardness f or cubic graphs than it is to pro v e NP-h ardness for cub ic graphs [17] or APX-hard n ess for general graphs [12]. Indeed, for general graph s a very s im p le APX-hardn ess pro of can b e giv en, usin g a red u ction from the APX-hard problem Cub ic Minimum V ertex Cov er: let G b e a cubic graph on n v ertices and m = 3 2 n edges for whic h w e search a minimum vertex c over , i.e. a minimum 2 set S ⊆ V ( G ) suc h that ev ery ed ge of G is inciden t with some vertex of S . Let k b e the size of a minim um v ertex co v er. Constru ct a MaxLeaf ins tance G ′ as follo ws: introduce a new vertex x , and add edges from x to every other vertex. Next, sub d ivide every edge n ot inciden t with x with a single ve rtex. It can b e c hec k ed that an y spanning tree in G ′ can b e transformed into a spanning tree with at least as man y lea v es, where all the degree 2 v ertices are lea ves. F rom th is it follo w s that G has a vertex co v er w ith at most y vertice s if and only if G ′ has a sp anning tree with at least n − y + m lea v es. S ince G is cub ic, k ≥ m/ 3. A (1 − ǫ )-approximati on algorithm f or MaxLeaf now yields a solution with at least (1 − ǫ )( n − k + m ) = n − k + m − ǫ (5 / 3 m − k ) ≥ n − k + m − ǫ (5 k − k ) = n − (1 + 4 ǫ ) k + m lea v es, and therefore a ve rtex cov er of size at most (1 + 4 ǫ ) k . This concludes the APX-hardn ess pro o f. It seems ho w ev er imp ossible to giv e a similar simple p ro of for cubic graphs. Considering the NP-hardn ess pr o of for cu bic graphs, Lemk e [17] ga v e a reduction from Exact Cover by 3-Sets . Here a 3-uniform hypergraph G on n vertice s is give n (i.e. all edges conta in three v ertices). The question is w h ether there is a su bset of the ed ges Q su c h that ev ery v ertex is con tained in exactly one edge of Q . F or every instance G , in [17] a graph is constructed that has a sp anning tree without v ertices of d egree 2 if and only if G is a ‘yes’-instance. It is easily seen that suc h a tree is optimal. Ho w ev er, an app r o ximation preserving redu ction from an APX-hard pr oblem needs also to take in to accoun t cases where the tree is not optimal, that is, it conta ins some d egree 2 vertice s. I n th is case the b ehavior of the subgraphs in Lemke’s construction, or eve n any cubic construction, b ecomes muc h hard er to analyze. In Section 2 we giv e defin itions and n otations, an d in Section 3 the constr u ction of our APX-hardness pro of, which uses an app ro ximation pr eserving redu ction from Cubic Maxim um Indep en den t Set. Sections 4 and 5 show ho w leafy sp anning trees yield large indep enden t sets and vice v ersa, and in S ection 6 these b oun ds are com bined to conclude the pro of. 2 Preliminaries F or basic graph theoretic d efinitions, we follo w [8]. By d G ( v ) w e denote the degree of v in graph G . Th e subscript is omitted when the grap h in question is clear. By δ ( G ) and ∆( G ) w e denote the minimum and maximum degree of G , r esp ectiv ely . By G − S w e denote the graph obtained from G by d eleting th e v ertex or edge set S . A directed grap h or digr aph D consists of a verte x set V ( D ) and arc set A ( D ), whic h is a set of ordered 2-tuples of v ertices. F or an arc ( u, v ) ∈ A ( D ), u is called the tail and v the he ad of ( u, v ). The in- de gr e e d − ( v ) ( out-de g r e e d + ( v )) of a v ertex v is th e num b er of arcs of wh ic h v is th e h ead (tail). A directed graph ( digr aph ) D is an orientation of an undirected graph G if V ( D ) = V ( G ) an d there exists a bijection f : A ( D ) → E ( G ) with f (( u, v )) = { u, v } for all ( u, v ) ∈ A ( D ). An out-tr e e orientation of a tree T is an orien tation T ′ of the (giv en undirected) tree T such that T ′ is an out-tr e e , that is, there is exactly one vertex with in -degree 0, whic h is called the r o ot . Note that ev ery other ve rtex then h as in-degree 1. A verte x sequence v 0 , . . . , v k is called a path or cycle in a d igraph D if it is a p ath or cycle in th e und erlying u ndirected graph (i.e. ( v i , v i +1 ) ∈ A ( D ) or ( v i +1 , v i ) ∈ A ( D ) holds for all i ). Directed paths and cycles, w here ( v i , v i +1 ) ∈ A ( D ) h olds for all i are called dip aths and dicycles . A path fr om u to v is also called a ( u, v ) -p ath . In an undirected graph G , v is s aid to b e r e acha ble from u if a ( u, v )-path exists in G . In a digraph D , v is reac hable from u if a ( u, v )-dipath exists. An ind uced su bgraph H of an und ir ected graph G is called a k - terminal sub gr aph if H 3 (a) b h f i c a d g e (b) Figure 1: Gadgets u sed in the construction. G : c 1 c 2 c 3 c 4 c 5 c 0 G ′ : G : Figure 2: Constructing a W eigh ted MaxLeaf ins tance f rom a C u bic MIS Instance. con tains exactly k vertice s that ha v e n eigh b ors outside of H , these are called its terminals . 3 The Constru ction of a W eigh ted M axLeaf In stance W e n o w pro v e that Cub ic MaxLeaf is APX-hard (and th us APX-complete), u sing a reduction from Cubic Maximum Indep endent Set (Cubic MIS) . Th is p r oblem has as inp ut a cubic graph G , and asks for a m axim um size set S ⊆ V ( G ) such that no tw o v ertices in S are adjacent. T o improv e the pr esentati on, we w ill pr o v e that the follo wing p roblem v arian t is APX-hard, from whic h APX-hardness of cub ic MaxLeaf easily follo ws. The pr oblem Wei ghte d MaxL e af has as inpu t a graph G w ith ∆( G ) ≤ 3 and δ ( G ) ≥ 2, and the ob jectiv e is to find a sp anning tree T that maximizes the num b er of vertice s v with d T ( v ) = 1 and d G ( v ) = 3. W e will also call v ertices of G with degree 3 weig hte d vertic es and the other vertices unweighte d . S o the ob jectiv e is to m aximize the num b er of weighte d le aves . By ℓ ( T ) w e w ill denote the num b er ℓ ( T ) of we igh ted lea v es of T . F rom instances G of W eigh ted MaxLeaf, it is easy to constru ct equiv ale nt Cu b ic MaxLeaf instances: replace ev ery verte x of degree 2 by the 1-terminal subgraph as shown in Figure 1(a) (the t w o half edges in dicate th e terminal). The next lemma is easily observed. Lemma 1 L et G ′ b e the cubic gr aph obtaine d fr om a gr ap h G with δ ( G ) = 2 , ∆( G ) = 3 b y r eplacing al l x vertic es of de gr e e 2 as shown in Figur e 1(a). Then G ′ has a sp anning tr e e with at le ast l + 3 x le aves if and only if G has a sp anning tr e e with at le ast l weighte d le aves. The construction of W eigh ted MaxLeaf instances uses the follo wing gadgets. A vertex gadget of G is an induced 4-terminal subgraph of G as sho wn in Figure 1(b), where the four vertex gadget terminals are ind icated by half edges. Note th at one v ertex h as degree 2, and therefore do es not count to w ards the num b er weigh ted lea v es. 4 Construction Let G b e a Cub ic MIS instance on n vertice s. W e use this to construct G n in p olynomial time a weig h ted MaxLeaf in stance as follo ws. First, w e assu m e w .l.o.g. that G 6 = K 4 , and thus we can construct a prop er 3-coloring of G , using colors red , green and blue. (By Bro o ks’ Theorem su c h a coloring exists, and in addition it can b e fou n d in p olynomial time, see also [19].) Let r and g b e th e n umber of red and green v ertices resp ect iv ely , and r g w.l.o.g. assume r ≥ 1 and g ≥ 1. Number the ve rtices of v 0 , . . . , v n − 1 suc h that v 0 , . . . , v r − 1 v 0 , . . . , v n − 1 are red, v r , . . . , v r + g − 1 are green, and v r + g , . . . , v n − 1 are blu e. W e constru ct a graph G as follo ws. The construction is illus tr ated in Figure 2. 1. Start with G . Add a cycle consisting of n c onne ction vertic es c 0 , . . . , c n − 1 and ed ges c onne ction vertic es c 0 , . . . , c n − 1 c i c ( i +1) mo d n for i ∈ { 0 , . . . , n − 1 } . 2. Add edges v i c i for all i ∈ { 0 , . . . , n − 1 } . 3. Sub divide every edge with one new v ertex (of degree 2). 4. Replace ev ery v ertex v i of degree four with a v ertex gadget H i , such that ev ery terminal H i of H i b ecomes adj acen t to a different neigh b or of v i . (Cho ose arbitrarily whic h terminals b ecome adjacen t to w hic h neigh b ors.) Let G b e the r esulting graph , and let G ′ b e the graph obtained after Step 2 in th is constr u ction. G G ′ Recall that by our defi n ition of W eigh ted MaxLeaf, the ve rtices in tro duced in Step 3 d o not coun t to w ards the num b er of weigh ted lea v es. F or th e pro ofs b elo w it will b e u s eful to denote ho w end vertice s of edges of G ′ corresp ond to vertice s of G . In Step 3, edges uv of G ′ are sub d ivided with a new v ertex w to yield t w o ed ges uw and v w . In S tep 4, the edge uw ma y b e replaced by an edge tw , where t is a terminal of a ve rtex gadget. If this is the case, t uv ( u ) t uv ( u ) will denote this terminal t , otherw ise t uv ( u ) will denote u . W e will pr o ce ed to sho w that for ev ery x ∈ R , if G has a spanning tree with at least 3 . 75 n + 1 . 5 x weigh ted leav es, then G h as an indep end en t set of size at least x − 1 3 (Section 4), whic h can b e constructed in p ol ynomial time. In add ition, if G has an indep enden t set of size x , G has a span n ing tree with at least ⌊ 3 . 75 n + 1 . 5 x ⌋ w eigh ted lea v es (Section 5). In Section 6 it is then shown that this yields a (1 − 141 ǫ )-appro ximation algorithm for Cu bic MIS, when a (1 − ǫ )-appr o ximation algorithm for Cubic MaxLeaf is giv en. This pro v es APX-h ard ness for Cubic MaxLeaf. 4 Constructing an In dep enden t Set fr om a Spann ing T ree W e fir st tak e a closer lo ok at the b eha vior of vertex gadgets, by b ound ing the num b er of w eigh ted lea v es a sp anning tree ma y cont ain within one giv en v ertex gadget. Prop osition 2 L et G b e a weighte d MaxL e af instanc e, T b e a sp anning tr e e of G and H b e a vertex gadget of G . L et T ′ b e an out-tr e e orientation of T with r o ot r ∗ ∈ V ( G ) \ V ( H ) . Then the fol lo wing b ounds hold: (i) H c ontains at most si x weighte d le aves of T . (ii) If T ′ c ontains at le ast one ar c le aving V ( H ) , then H c ontains at most f our weighte d le aves of T . 5 4 3 3 4 6 3 1 G : r ∗ T ree T of G : G ′ : Figure 3: A spannin g tree with 24 = ⌈ 3 . 75 · 6 + 1 . 5 ⌉ we igh ted lea v es yields a s ize 1 ind ep endent set. (iii) If T ′ c ontains at le ast two ar cs le aving V ( H ) , then H c ontains at most thr e e weighte d le aves of T . Pr o of: In the pro o f we w ill refer to the vertex lab els of H as sho wn in Figure 1(b). (i) { a, d, f } and { b, g , i } are v ertex cuts of G , so b oth conta in at least one non-leaf of a sp anning tree. They are disjoint, so H con tains at least t w o weig ht ed n on -lea v es of T . (ii) Since eve ry arc of T ′ that lea v es V ( H ) is part of a dipath in T ′ that starts at the ro ot, T con tains a path P in H from one terminal of H to another, where all v ertices of P are non-lea v es. Su pp ose b is one of the ends of P . Then either P con tains at least four w eigh ted v ertices, or P con tains the v ertices b , c , f and i . In the second case th e vertex cut { a, e, g } shows there is at least one m ore non-leaf, so in b oth cases we ha v e foun d f our w eigh ted n on -lea v es. Now supp ose g is one of the end s of P . If h is the other end this ensures that g and h are non-lea v es, an d the tw o disj oin t v ertex cuts { a, d, f } and { b, e, i } show there are at least tw o more w eigh ted n on-lea v es. If i is the other end, P either has length at least four (in w hic h case we are done), or it con tains g , h and i . Then the v ertex cut { a, d, f } sho ws there must b e at least one more w eigh ted n on-leaf. Finally , if P go es from h to i , the t w o ve rtex cuts { b, f } and { a, e, g } sh o w that there are at least four weigh ted non-lea v es. (iii) Because there are at least t w o arcs lea ving V ( H ), in this case T − L ( T ) con tains a subgraph of H of one of th e follo wing tw o forms: it con tains a tree T H that conta ins at least three terminals of H , or it con tains tw o paths b et ween disjoint terminal p airs of H . (Note that all vertice s of these subgraphs are n on-lea v es.) In the latter case fiv e weig ht ed non-lea v es are easily foun d by considerin g s h ortest path lengths. Similarly , fi v e n on-lea v es are also easily found when { b, g , h } ⊆ V ( T H ) or { b, g , i } ⊆ V ( T H ). If { b, h, i } ⊆ V ( T H ), four we igh ted leav es are only p ossible when a , d , e and g are lea v es, but this is not p ossible since { a, e, g } is a v ertex cut. Finally , wh en { g , h, i } ⊆ V ( T H ), the three v ertex cuts { b, f } , { b, d, e } and { a, d, f } sho w there are at least tw o additional w eigh ted non-lea v es.  In the r emainder of this section, w e will p ro v e the next lemma, which shows th at an indep end en t set I of G of sufficient size can b e constructed when a sp anning tree T of G is giv en. The construction is illustrated in Figure 3. The constru cted indep enden t set consists of the single encircled vertex. Num b ers indicate num b ers of weigh ted lea v es. The c hoice of the orien tations is explained b elo w. The intuiti v e idea b ehind the next pro o f is as follo ws. Not to o man y ve rtex gadgets in G can conta in six w eigh ted lea v es of a spanning tree T , since edges in ve rtex gadgets are 6 needed to connect T . In p articular, su c h v ertex gadgets cann ot b e adjacent and th us form our in dep endent set. With a similar more d elicate argu m en t we will also show that not all v ertex gadgets can con tain four lea v es of T . Ho w muc h ev ery v ertex gadget con tributes to ‘connecting T ’ is enco ded by the out-degrees of ve rtices of G ′ in the pro of b elo w. Th e pro of of the lemma consists of a num b er of claims. Lemma 3 L et G b e c onstructe d fr om a cub ic gr aph G on n vertic es as shown in Se ction 3. If G has a sp anning tr e e T with ℓ ( T ) ≥ 3 . 75 n + 1 . 5 x , then an indep endent se t I of G with | I | ≥ x − 1 3 c an b e c onstructe d i n p olyno mial time. Let T b e a s panning tr ee of G with ℓ ( T ) ≥ 3 . 75 + 1 . 5 x . T o construct an ind ep endent set I of T G with the desired size, w e will fi rst u se T to orient G ′ and G . O b serv e that there is some connection v ertex of G that is not a leaf of T . Cho ose r ∗ to b e suc h a ve rtex. Orient T as r ∗ out-tree with ro ot r ∗ . An orienta tion of G ′ can b e obtained from the out-tree T as follo ws: consider an edge uv ∈ E ( G ′ ), whic h wa s su b divided with a new verte x w f or constr u cting G . So uv corresp on d s to edges t 1 w and t 2 w of G , with t 1 = t uv ( u ) and t 2 = t uv ( v ). uv is n o w orien ted as follo ws: if ( t 1 , w ) ∈ A ( T ), then c ho ose the orienta tion ( u, v ). If ( t 2 , w ) ∈ A ( T ), then c ho ose th e orienta tion ( v , u ). Observe that th is u niquely d etermines th e d irection of uv in every case. Doing this for all edges of G ′ yields the orientati on of G ′ . Sin ce G is a sub graph of G ′ , this also yields th e orien tation of G that we will use. The set I no w consists of all vertices of G that h a v e out-degree 0. Clearly this is an I indep end en t set, and I can b e constructed in p olynomial time. Let n i denote the num b er of n i v ertices of G with out-degree i , so | I | = n 0 . Let n ′ i b e the num b er of vertic es of G that ha v e n ′ i out-degree i in G ′ . Observe that since r ∗ is not part of a v ertex gadget, n ′ 4 = 0. Note that v i has out-degree d in G ′ if and only if T con tains d arcs lea ving H i . So Prop o sition 2 sh o ws that if v i has out-degree 3 in G ′ , then T has at m ost three weigh ted leav es in the vertex gadget H i , etc. This yields: Claim 1 The numb er of non-c onne ction vertic es of G that ar e weighte d le aves of T i s b ounde d by 6 n ′ 0 + 4 n ′ 1 + 3 n ′ 2 + 3 n ′ 3 . Since T is an out-tree, ev ery vertex of T is reac hable from the r o o t r ∗ . Ther efore ev ery v ertex of G ′ is reac hable from r ∗ in the chosen orien tation (p ossibly by m ultiple dipaths). Observe that every connection ve rtex that is a leaf in T has out-degree 0 in G ′ . Let z b e the z n umber of connection ve rtices of G ′ that h a v e an in-neigh b or that is not a connection ve rtex. Claim 2 At most ⌈ z / 2 ⌉ c onne ction v ertic es of G ar e le aves in T . Pr o of: Let c σ 1 , . . . , c σ k b e the connection ve rtices of G th at are lea v es in T , with σ i < σ i +1 for all i . All of these v ertices h av e in-degree 3 in G ′ , which accoun ts for k connection v ertices that ha v e an in-n eigh b or that is not a connection vertex. Consider c σ i and c σ i +1 , for some i . Since these v ertices hav e in-d egree 3, they are not adjacen t in G ′ . Therefore there is at least one connection vertex c l that lies b et w een them on C (that is, σ i < l < σ i +1 ). G ′ con tains a dipath P from r ∗ to c l , whic h clearly cann ot conta in c σ i or c σ i +1 as inte rnal ve rtices. So u nless r ∗ also lies b et wee n c i and c i +1 , P must conta in a conn ection v ertex b etw een c σ i and c σ i +1 that has an in-neighbor th at is n ot a connection v ertex. 7 Since the ab ov e argum en t can b e applied for k differen t pairs of connection vertice s and r ∗ lies only b etw een one such pair, this accoun ts for k − 1 add itional suc h v ertices. It follo ws that z ≥ 2 k − 1.  A second wa y to interpret the parameter z is the follo wing: there are exactly z v ertices with different out-degrees in G and G ′ . In th is case the out-degree in G ′ is one higher. This observ ation yields the follo wing inequalit y . Claim 3 z + 3 n ′ 0 + 2 n ′ 1 + n ′ 2 = 3 n 0 + 2 n 1 + n 2 . Pr o of: Let k i denote the n umber of v ertices with out-degree i in G and out-degree i + 1 in G ′ . F rom n ′ 4 = 0, k 3 = 0 follo ws. V ertices f or whic h the out-degree increases this wa y corresp ond to in-neigh b ors of connection vertice s in G ′ , so z = k 0 + k 1 + k 2 . In addition we ha v e that n ′ i = n i − k i + k i − 1 . Sub stituting these expr essions y ields the stated equalit y .  With the ab o v e obser v ations, w e can b oun d the n um b er of w eigh ted lea v es of T . Let m = 1 . 5 n b e the num b er of arcs of G . By count ing in-degrees we h a v e m = 3 n 0 + 2 n 1 + n 2 . ℓ ( T ) ≤ 6 n ′ 0 + 4 n ′ 1 + 3 n ′ 2 + 3 n ′ 3 + ⌈ z / 2 ⌉ ≤ ⌈ 3 n + 1 . 5 | I | + 1 . 5 n ′ 0 + n ′ 1 + z / 2 ⌉ ≤ ⌈ 3 n + 1 . 5 | I | + 1 . 5 n 0 + n 1 + 0 . 5 n 2 ⌉ ≤ ⌈ 3 n + 1 . 5 | I | + 0 . 5 m ⌉ = ⌈ 3 . 75 n + 1 . 5 | I |⌉ . Here w e used Claim 1; Claim 2; n = n ′ 0 + n ′ 1 + n ′ 2 + n ′ 3 ; | I | = n 0 ≥ n ′ 0 ; z / 2 + 1 . 5 n ′ 0 + n ′ 1 + 0 . 5 n ′ 2 = 1 . 5 n 0 + n 1 + 0 . 5 n 2 (Claim 3); m = 3 n 0 + 2 n 1 + n 2 and m = 1 . 5 n , resp ectiv ely . So if ℓ ( T ) ≥ 3 . 75 n + 1 . 5 x , th en ⌈ 3 . 75 n + 1 . 5 | I |⌉ ≥ ℓ ( T ) ≥ 3 . 75 n + 1 . 5 x . Since G is a cubic graph, n is ev en. It follo ws that 3 . 75 n + 1 . 5 | I | is half inte gral, so 3 . 75 n + 1 . 5 | I | + 0 . 5 ≥ ⌈ 3 . 75 n + 1 . 5 | I |⌉ ≥ 3 . 75 n + 1 . 5 x , and thus | I | ≥ x − 1 3 . This concludes the pro of of Lemma 3 . 5 Constructing a Sp anning T re e from an Ind ep enden t Set In this section w e will p ro v e the follo wing lemma, wh ic h sho ws that a spann ing tree T with enough w eigh ted lea v es can b e constructed when an indep en den t set I of G is giv en. The pro of consists of a num b er of claims. The in tuitiv e id ea b ehind the pro of is as follo w s. When giv en an indep en d en t set I of G , w e can constru ct a spanning tree T of G that do es not u se an y vertex gadget H i with v i ∈ I for ‘connecting T ’. F or arguing that we can still m ak e T connected, w e n eed to use the 3-coloring of G . W e fi x a connection vertex as ro ot, and sho w that the red v ertices can b e r eac hed from this ro ot. This is needed to sho w that green vertic es can b e reac hed, whic h is in turn needed to s h o w th at blue v ertices can b e reac hed. Lemma 4 L et G b e c onst ructe d fr om a cu bic gr aph G on n ve rtic es as shown in Se ction 3. If G has an indep endent set I with | I | ≥ x , then G has a sp anning tr e e T with ℓ ( T ) ≥ ⌊ 3 . 75 n + 1 . 5 x ⌋ . Throughout the pro o f we will r efer to the vertex coloring of G th at was used for the construc- tion of G . Let I b e a maximal indep enden t set of G w ith | I | ≥ x . W e use th is to construct I a sp anning tree with at least ⌊ 3 . 75 n + 1 . 5 x ⌋ lea v es as follo ws. The constru ction is illustrated 8 3 6 6 3 3 3 1 Subgraph T of G : G ′ : G : : I : C r ∗ Figure 4: A size 2 indep enden t set y ields a sp anning connected subgraph with ⌊ 3 . 75 · 6 + 1 . 5 · 2 ⌋ = 25 weig ht ed lea v es. in Figure 4, w here I is repr esented by encircled v ertices in G . First, for all v ∈ I , orien t all inciden t edges xv of G as ( x, v ), so ev ery v ∈ I h as out-degree 0. T h is is p ossible since I is an indep enden t set. F or all edges that are not incident with a ve rtex fr om I , choose the direction fr om r ed to green, f rom green to blue or from red to blu e, whic hev er applies. This yields the orien tation of G . W e extend th is to an orient ation of G ′ as f ollo ws: • If v i has out-degree 0, 1 or 3 in G , w e orient c i v i to w ards v i . • If v i has out-degree 2 in G , we orient c i v i to w ards c i . • Let C b e the set of connection v ertices c i in G ′ that no w h a v e an incoming arc ( v i , c i ). C Let g C b e the num b er of connection v ertices c i ∈ C where v i is green. F or ev ery 0 ≤ i ≤ g C n − 1, the edge c i c ( i +1) mo d n is directed to w ards c ( i +1) mo d n if |C ∩ { c 0 , . . . , c i }| m o d 2 = g C mo d 2, and to w ards c i otherwise. In Figure 4, C = { c 2 , c 4 , c 5 } . C is represent ed b y encircled vertices of G ′ . Of these v ertices, only c 2 has a green in-neigh b or, so g C = 1. Therefore c 0 c 1 is orien ted to w ards c 0 , etc. W e start with t w o simp le observ ations on these orien tations of G ′ . If a v ertex v i has out-degree 1 in G , it retains out-degree 1 in G ′ , and if it has out-degree 2 in G it receiv es out-degree 3 in G ′ . If it has out-degree 3 in G it retains out-degree 3 in G ′ . This yields: Claim 4 V ertic es v i have out-de gr e e 0,1 or 3 in G ′ . F or red vertice s v i , either d + G ( v i ) = 0 (if v i ∈ I ), or d + G ( v i ) = 3 (if v i 6∈ I ), so in either case ( c i , v i ) ∈ A ( G ′ ). Summ arizing: Claim 5 If v i is r e d, then c i 6∈ C . Let n d denote the num b er of vertic es v k with d + G ( v k ) = d . n d Claim 6 G ′ c ontains at le ast ⌊ n 2 / 2 ⌋ vertic es c i with d + ( c i ) = 0 . Pr o of: Ob serv e th at ve rtices c i ∈ C with i ≥ 1 ha v e in -degree 1 or in-degree 3 in G ′ , b ecause of the parit y based orientat ion of edges b et wee n connection ve rtices. Recall that there is at least one red ve rtex, so v 0 is red and c 0 6∈ C (Claim 5). T herefore al l v ertices in C h a v e 9 in-degree 1 or 3, in alternating order of increasing index. Since |C | = n 2 , it follo ws that there are at least ⌊ n 2 / 2 ⌋ connection v ertices with in -degree 3 (and out-degree 0).  Let r ∗ = c 0 if g C is ev en, and r ∗ = c r − 1 if it is o dd. In Figure 4, g C = 1 so r ∗ = c r − 1 = c 1 . r ∗ Claim 7 In the chosen orientation of G ′ , every vertex is r e achable fr om r ∗ . Pr o of: Out-degrees will refer to G in this pro of. First we will sho w that eve ry v ertex v i of G ′ is r eac hable from some connection v ertex. If d + G ( v i ) 6 = 2, then v i has a connection v ertex as in-n eigh b or, so the statemen t is clear. If d + G ( v i ) = 2, then v i has an in-n eigh b or v x in G ′ , with v x 6∈ I , that m ust b e red or green. If v x has a connection vertex as in-neigh b or, we ha v e pro v ed the statemen t. O therwise, v x has an in-neigh b or v y again, wh ich then m ust b e red. So v y m ust hav e a connection v ertex as in-neighbor. In an y case, w e ha v e found a dipath from some connection v ertex to v i . A connection vertex c i will b e called red , green or blue w hen its uniqu e (in- or out-) neigh b or v i is red, green or blu e resp ectiv ely . W e w ill now p ro v e that all connection vertic es c i are reac hable from r ∗ in G ′ . CASE 1: c i is red. Since there are no red v ertices c i ∈ C (Claim 5), c 0 , c 1 , . . . , c r − 1 is a dipath in G ′ if g C is ev en, and c r − 1 , c r − 2 , . . . , c 0 is a dipath if g C is o dd . So w e hav e c hosen r ∗ suc h that all red connection vertice s are r eac h able fr om r ∗ . CASE 2: c i ∈ C is green. Let v i b e the (green) in-neighbor of c i . The argu m en t w e hav e used ab ov e shows that v i is r eac hable fr om some red connection v ertex, which in tu rn is reac hable from r ∗ as sho wn in case 1. CASE 3: c i 6∈ C is green. c i has a connection v ertex as in -n eigh b or (either c i − 1 or c i +1 ). If c i − 1 is its in-neighbor, then G ′ either conta ins a dipath c r − 1 , . . . , c i , or a d ipath c j , c j +1 , . . . , c i with j < i and c j ∈ C . Both of these dipaths start at a reac hable vertex (by case 1 and 2) so c i is reac hable from r ∗ . If c i +1 is the in-neighb or of c i , then the num b er of C vertic es in { c 0 , . . . , c i } has d ifferen t parity than the n umber of green vertice s in C . Since all C v ertices in { c 0 , . . . , c i } are green (Claim 5), this implies that th ere is at least one more green v ertex in C . So there exists a d ipath c j , c j − 1 , . . . , c i with j > i , c j green, and c j ∈ C . c j is reac hable from r ∗ b y case 2, so c i is reac hable as well. CASE 4: c i ∈ C is blu e. By the same argument as earlier, the blue in-neighbor v i of c i is reac hable from a r ed or green connection vertex, whic h is reac hable fr om r ∗ b y case 1, 2 or 3. CASE 5: c i 6∈ C is blu e. Similar to the reasoning in case 3, we may trace a p ath back from c i consisting of connection vertic es, unt il we fi n d a d ipath starting at a ve rtex c j , where c j is either red or part of C . (Th is path ma y also b e c 0 , c n − 1 , c n − 2 , . . . , c i , so j = 0.) Case 1, 2 and 4 sho w that c j and thus c i is reac hable from r ∗ . No w we hav e considered all cases for connection ve rtices. It follo ws th at all v ertices of G ′ are reac hable from r ∗ .  10 (b) (c) (a) Figure 5: Using out-degrees to construct a spann ing tree. Whenev er we refer to the out-degree or in-d egree of v ertices b elo w, th is refers to G ′ , n ot to G , unless explicitly noted otherwise. W e use the orien tation of G ′ to construct a spann ing tree T ′ of G as follo ws. First we construct a sp anning connected subgraph T : 1. F or ev ery ve rtex gadget in G , Figure 5 sh o ws whic h subset of the edges sh ou ld b e c hosen in T , dep ending on the out-degree and out-neigh b or set of th e corresp ondin g v ertex v i in G ′ . (Note that only out-degrees 0, 1 and 3 ha v e to b e considered by Claim 4.) 2. Ev ery edge of G that is not part of a vertex gadget is added to T . 3. F or eve ry v ertex c i that has in-degree 3 in G ′ , d elete the t w o inciden t T -edges that do not corresp ond to the arc ( v i , c i ) of G ′ , making c i a leaf of T . 4. Delete edges of T until n o cycles remain, to obtain graph T ′ . T ′ T denotes the graph as it is after Step 3 ab ov e. The follo wing claim already sho ws for many T v ertices of G th at they are reac hable from r ∗ in T . Claim 8 If G ′ c ontains a dip ath P ′ = r ∗ , . . . , x, y with d + ( y ) ≥ 1 , then T c ont ains a p ath fr om r ∗ to t xy ( y ) . Pr o of: Firs t, for ev ery arc ( u, v ) of P ′ w e add th e corresp onding length 2 path in G to P . T o b e precise, this is the path t uv ( u ) , x, t uv ( v ), w h ere x is the v ertex r esulting f rom the sub d ivision of uv d uring the construction of G . O b serv e that b o th of these path edges are also part of T : in Step 3 of the constru ction of T s ome edges that are not p art of ve rtex gadgets are remo v ed fr om T , bu t on ly th ose that are incident with a v ertex c i with in-d egree 3, and thus out-degree 0. Clearly s uc h vertic es cannot b e in ternal v ertices of P ′ , and by our assu mption, the end v ertex y of P ′ also has out-degree at least 1. A t this p oint P ma y not b e a path y et; it can consist of a s equ ence of paths where one path end s at a terminal t 1 of a vertex gadget H i , and the next path starts at another terminal t 2 of H i . Joining s u c h paths together is easy wh en d + ( v i ) = 3: Figure 5(b) s ho ws the ed ges of T that are part of H i ; observ e that for eve ry terminal pair t 1 and t 2 a path from t 1 to t 2 exists in T through H i . S o it suffices to prov e that P ′ con tains no in ternal v ertices v j with d + ( v j ) 6 = 3. C learly all in ternal v ertices hav e out-degree at least 1. No v ertices v j of G ′ ha v e out-degree 2 (Claim 4), so w e only ha v e to consider the case th at d + ( v j ) = 1. Now w e w ill u se that we started with a maximal indep end en t set I : b ecause I is maximal, every vertex that is not in I has at least 11 one neighbor in I . So by choice of the orien tation of G , if v j has out-degree 1, its out-neigh b or v k is in I , and d + ( v k ) = 0. Th e dipath P ′ cannot con tain v k as in ternal v ertex, and by c hoice of P ′ , also not as end ve rtex y . Hence P ′ con tains no v ertices v j with out-degree 1. This concludes the pro of.  Using the previous tw o claims, T can b e shown to b e connected: Claim 9 Al l vertic e s u ∈ V ( G ) ar e r e achable f r om r ∗ within T . Pr o of: W e consid er four cases for u . CASE 1: u is p art of a vertex gadget H i , w ith d + ( v i ) ≥ 1. Figure 5(b) and (c) sh o w th at in ev ery case, there is an arc ( w , v i ) ∈ A ( G ′ ) such that T con tains a path from t = t w v i ( v i ) to u . So w e only need to sho w that t can b e reac hed from r ∗ within T . By Claim 7, G ′ con tains an ( r ∗ , w )-dipath, whic h then yields a dipath P ′ = r ∗ , . . . , w, v i . F r om C laim 8 it no w follo ws th at T con tains a ( r ∗ , u )-path. CASE 2: u is p art of a vertex gadget H i with d + ( v i ) = 0. W e again consider an arc ( w, v i ) ∈ A ( G ′ ) su c h that T con tains a path from t = t w v i ( v i ) to u (suc h an arc exists, see Figure 5 (a)). Let t ′ = t w v i ( w ). If t ′ is part of a ve rtex gadget, in case 1 we sh ow ed that t ′ can b e reac hed from r ∗ in T , wh ic h shows u can b e reac h ed. Otherwise, t ′ = c j with d + ( c j ) ≥ 1. C laim 7 shows that G ′ con tains an ( r ∗ , c j )-dipath, which yields an ( r ∗ , t ′ )-path in T (Claim 8 ) and th us an ( r ∗ , u )-path. CASE 3: u = c i . If d + ( c i ) ≥ 1, Claim 7 shows that G ′ con tains an ( r ∗ , c i )-dipath, wh ic h yields an ( r ∗ , c i )- path in T by Claim 8. If d + ( c i ) = 0, then the construction of T sho ws that b o th ed ges of G corresp ond in g to the arc ( v i , c i ) of G ′ are p art of T . By Case 1, ev ery v ertex of the v ertex gadget H i corresp onding to v i is reac hable from r ∗ in T , so c i is reac hable. CASE 4: d ( u ) = 2 and u is not p art of a v ertex gadget. Here u is the v ertex resu lting from the sub division of an edge xy . Let ( x, y ) b e the orien tation of this edge in G ′ . If x = c k for some k , then Case 3 s ho ws that an ( r ∗ , c k )- path exists in T . This can b e extended to the desired path; c k u ∈ E ( T ) since d + ( c k ) ≥ 1. Otherwise, x ∈ V ( H i ), where d + ( v i ) ≥ 1. Then case 1 or 2 sh ows that an ( r ∗ , x )-path exists in T , which can b e extended again.  Since Claim 9 shows that T is conn ected, clearly T ′ is connected as w ell. Since in addition T ′ con tains no cycles, T ′ is a spann ing tr ee of G . It remains to pr o v e that it has the d esired n umber of lea v es. Figur e 5 shows that a ve rtex v i con tributes six lea v es to T if d + G ′ ( v i ) = 0, four lea v es if d + G ′ ( v i ) = 1 and three lea v es if d + G ′ ( v i ) = 3. In addition, ev ery v ertex c i with in-degree 3 in G ′ is a leaf of T b y Step 3 of the construction of T . Claim 6 sho ws that there are at least ⌊ n 2 / 2 ⌋ s uc h v ertices. Recal l th at n d denotes the n umber of vertic es that h a v e out-degree d in G . In addition let n ′ d denote th e num b er of vertic es that h a v e out-degree d in G ′ . Observe that n 0 + n 1 + n 2 + n 3 = n , and let m = 1 . 5 n = 3 n 0 + 2 n 1 + n 2 b e the n umber of edges of G . T ogether this yields ℓ ( T ) ≥ 6 n ′ 0 + 4 n ′ 1 + 3 n ′ 3 + ⌊ n 2 / 2 ⌋ = 6 n 0 + 4 n 1 + 3 n 2 + 3 n 3 + ⌊ n 2 / 2 ⌋ = ⌊ 3 n + 3 n 0 + n 1 + 0 . 5 n 2 ⌋ = ⌊ 3 n + 1 . 5 n 0 + 0 . 5 m ⌋ ≥ ⌊ 3 . 75 n + 1 . 5 x ⌋ . F or the last step we used that every ve rtex u ∈ I has out-degree 0 in G and that | I | ≥ x . This concludes the p r o of of L emm a 4. 12 6 Conclusion of the Pro of Theorem 5 Cubic MaxL e af is APX -har d. Pr o of: W e sho w th at for ev ery ǫ > 0, a (1 − ǫ )-app r o ximation algorithm for cubic MaxLeaf yields a (1 − 141 ǫ )-appro ximation algorithm for Cubic MIS . Let G b e a Cu bic MIS instance on n v ertices, whic h h as a maxim um indep enden t set of size x . Observe that since G is cubic, x ≥ n/ 4. F r om G , we construct a W eigh ted MaxLeaf instance G as shown in Section 3 . G has a tree w ith at least ⌊ 3 . 75 n + 1 . 5 x ⌋ w eigh ted lea v es (Lemma 4), and it can b e c hec k ed that it has y = 4 . 5 n vertic es of degree 2. Let r = 3 . 75 n + 1 . 5 x − ⌊ 3 . 75 n + 1 . 5 x ⌋ . Note that since n is ev en, the round ed v alue is half-in tegral so r ≤ 0 . 5. F rom G , we construct a Cubic MaxLeaf instance H by r ep lacing degree 2 vertices as sh o wn in Section 3. Th en H has a tree with at least 3 . 75 n + 1 . 5 x − r + 3 y = 3 . 75 n + 1 . 5 x − r + 13 . 5 n lea v es (Lemm a 1). No w su pp ose we hav e a (1 − ǫ )-appro ximation algorithm for cubic MaxLeaf. In H , this algorithm will find a tree T with at least (1 − ǫ )(3 . 75 n + 1 . 5 x − r + 13 . 5 n ) lea v es. By Lemma 1 again, this yields tree T ′ of G with at least (1 − ǫ )(3 . 75 n + 1 . 5 x − r + 13 . 5 n ) − 13 . 5 n we igh ted lea v es. So, using x ≥ n/ 4, we obtain: ℓ ( T ′ ) ≥ 3 . 75 n + 1 . 5 x − r − ǫ (3 . 75 n + 1 . 5 x − r + 13 . 5 n ) = 3 . 75 n + 1 . 5 x − r − ǫ (17 . 25 n + 1 . 5 x − r ) ≥ 3 . 75 n + 1 . 5 x − r − ǫ (69 x + 1 . 5 x ) = 3 . 75 n + 1 . 5 x − r − γ x, where γ = 70 . 5 ǫ . No w we consider t w o cases: If γ x < 0 . 5, then ℓ ( T ′ ) > 3 . 75 n + 1 . 5 x − 0 . 5 − 0 . 5 = 3 . 75 n + 1 . 5( x − 2 3 ). (Here we used r ≤ 0 . 5.) By Lemma 3, we can construct an ind ep endent set I for G with | I | > x − 2 3 − 1 3 (note th at the inequ ality is again strict). x is intege r, so | I | ≥ x . Hence in this case we find an optimal indep enden t set. On the other h and, if γ x ≥ 0 . 5, then also γ x ≥ r , so ℓ ( T ′ ) > 3 . 75 n + 1 . 5 x − γ x − γ x = 3 . 75 n + 1 . 5( x − 4 3 γ x ). So b y Lemma 3 again, we fi nd I with | I | ≥ x − 4 3 γ x − 1 3 ≥ x − 2 γ x . In this case we h a v e an (1 − 2 γ ) = (1 − 141 ǫ ) app ro ximation. Since Cub ic MIS is APX-hard [1], the APX-hardness of Cu bic MaxLeaf follo ws.  W e remark that this reduction is an L-redu ction as introd uced in [21]. Similarly , using the fact th at cub ic graph s on n v ertices ha v e a spann ing with at least n/ 4 + 2 lea v es [15], w e fi n d th at a (1 + ǫ )-appr o ximation algorithm for MinCDS y ields a (1 − 3 ǫ )-approxima tion algorithm for Cu bic MaxLeaf on th e same graph , so: Corollary 6 Cubic MinCDS is APX -har d. Pr o of: W e consid er the trivial red uction from cubic MaxLeaf. Let G b e a cubic graph on n v ertices for whic h w e wish to find a spanning tree w ith m axim um n umber of lea v es. Let l b e the maxim um num b er of lea v es p ossible for G . Sin ce G is cubic, l ≥ n/ 4 + 2 [15]. G th en has a connected dominating set of s ize at most n − l . 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