New Branching Rules: Improvements on Independent Set and Vertex Cover in Sparse Graphs

New Branc hing Rule s: Impro v emen ts on Indep enden t Set and V ertex Co v er in Sp arse Graphs ⋆ Mingyu Xiao School of Computer Science and Engineering Universit y of Electronic Science and T ec hnology of China Chengdu 610054, CHINA Email: myxiao@gmail.co m Abstract. W e present an O ∗ (1 . 0919 n )-time algorithm for find ing a max- im um ind ep endent set in an n -vertex graph with degree b oun ded by 3, whic h imp roves the previously known algorithm of running time O ∗ (1 . 0977 n ) by Bourgeois, Escoffier and P asc hos [IWPEC 2008]. W e also p resent an O ∗ (1 . 1923 k )-time algorithm to decide if a graph with degree b ounded by 3 has a vertex co ver of size k , which impro ves the previously kno wn algorithm of runn ing time O ∗ (1 . 1939 k ) by Chen, Kanj and X ia [ISA AC 2003]. Tw o new branching tec hniqu es, br anching on a b ottle and br anching on a 4 -cycle , are introduced, which help us to d esign simple and fast al- gorithms for the maxim um ind ep endent set and minim um vertex cov er problems and av oid tedious branching rules. Key words. Graph Algorithm, Indep end ent Set, V ertex Cov er, S parse Graph 1 In tro duction The maximum indep endent set problem (MIS), to find a ma ximum set of ver- tices in a gra ph such that there is no edge b etw een any t wo v ertices in the se t, is o ne of the basic NP-hard o ptimization problems a nd has b een well studied in the literature, in pa r ticular in the line of rese a rch on worst-ca s e analysis of algo - rithms for NP-hard optimization problems. In 1977, T arjan and T r o janowski [1] published the fir s t algor ithm for this problem, which runs in O ∗ (2 n/ 3 ) time and po ly nomial space. Later, the running time was impro ved to O ∗ (2 0 . 304 n ) by Jian [2]. Robson [3] obtained a n O ∗ (2 0 . 296 n )-time po lynomial-spa c e algo- rithm a nd an O ∗ (2 0 . 276 n )-time exp o nent ial-s pa ce algor ithm. In a technical re- po rt [4], Robson also claimed be tter running times. Recently , F omin et a l. [5] got a simple O ∗ (2 0 . 288 n )-time p oly nomial-space a lgorithm b y using the “Measure and Co nquer” metho d. There is also a cons ide r able amount of contributions to ⋆ The pap er was presented at t h e 2nd annual meeting of asian association for algorithms and computation ( AAAC 2009), April 11-12, 2009, Hangzhou, China. 2 Mingyu X iao the maximum independent set problem in sparse graphs, esp ecially in degree- 3 graphs [6 ],[7],[8],[9]. W e summarize the results on low-degree g raphs as well a s general graphs in T able 1. Authors Running times References Notes T arjan & T ro janowski O ∗ (1 . 2600 n ) for MIS 1977 [1] n : num b er of v ertices Jian O ∗ (1 . 2346 n ) for MIS 1986 [2] Robson O ∗ (1 . 2109 n ) for MIS 1986 [3] Exp onential sp ace Beigel O ∗ (1 . 0823 m ) for MIS 1999 [6] m : num b er of ed ges O ∗ (1 . 1259 n ) for 3-MIS 3-MIS: MIS in degree-3 graphs Robson O ∗ (1 . 1893 n ) for MIS 2001 [4] P artially computer-generated Chen et al. O ∗ (1 . 1254 n ) for 3-MIS 2003 [7 ] Xiao et al. O ∗ (1 . 1034 n ) for 3-MIS 2005 [8 ] Published in Chinese F omin et al. O ∗ (1 . 2210 n ) for MIS 2006 [5] F omin & Høie O ∗ (1 . 1225 n ) for 3-MIS 2006 [10 ] F¨ urer O ∗ (1 . 1120 n ) for 3-MIS 2006 [11 ] Razgon O ∗ (1 . 1034 n ) for 3-MIS 2006 [12 ] Bourgeois et al. O ∗ (1 . 0977 n ) for 3-MIS 2008 [9 ] Xiao O ∗ (1 . 0919 n ) for 3-MIS This paper T able 1. Exact algorithms for the maximum independent set problem In the literature, there ar e sev eral metho ds of des igning algor ithms for find- ing ma ximum indep endent sets in gra phs. One metho d is to find a minimum vertex cover (a set of vertices such that ea ch edge in the graph has at leas t one endpo int in the set), and then to get a maximum indep endent set by taking all the remaining vertices, s uch a s the algo rithms presented in [7],[13]. In this kind of algor ithms, the domina ting part of the r unning time is the r unning time for finding a minimum v ertex cov er. Another metho d is based on the sea rch tr ee metho d. W e will use a branch-and-reduce pa r adigm. W e choose a para meter, such as the num b er o f v ertices or edg es or others, as a measure of the size of the problem. When the parameter is zero o r a nega tive num ber, the pr oblem can be solved in p olynomial time. W e br a nch on the cur rent graph G into serv al gr aphs G 1 , G 2 , · · · , G l such that the parameter r i of graph G i is le ss than the parameter r of gra ph G ( i = 1 , 2 , · · · , l ), and a maximum independent set in G can b e found in poly no mial time if a maxim um independent set in each of the l graphs G 1 , G 2 , · · · , G l is known. By this metho d, we can build up a search tree, and the exp onential part of the running time of the algo rithm is corresp o nding to the size of the search tree. The r unning time analysis lea ds to a linear r ecurrence for each no de in the sear ch tr ee that c an b e solved by using standard techniques. Let C ( r ) deno te the worst-case size of the sea rch tree when the para meter of graph G is r , then we ge t r ecurrence relation C ( r ) ≤ P l i =1 C ( r i ). Solving the recur - rence, we get C ( r ) = [ α ( r , r 1 , r 2 , · · · , r l )] r , wher e α ( r, r 1 , r 2 , · · · , r l ) is the large s t ro ot of the function f ( x ) = 1 − P l i =1 x r i − r . As for the measur e (the parameter Indep endent Set and V ertex Co ver in Sp arse graphs 3 r ), a natural one is the n umber of v ertices or edges in the gra ph. Most previous algorithms for the maximum indep endent set pro blem are analyzed by using the nu mber o f vertices as a mea sure [1],[2],[3],[5]. The num b er of edg es is consider ed in Beige l’s algor ithm [6]. There a re also some other measures. Xia o et a l. [8] used the num ber of degr ee-3 vertices as a measure to a nalyze alg o rithms and g o t an O ∗ (1 . 1034 n )-time algor ithm for MIS in degree-3 graphs. Unfortunately , that pap er was published in Chines e. Rece nt ly , Razg on [12] a lso go t an O ∗ (1 . 1034 n )- time algor ithm for MIS in deg ree-3 gr aphs by mea suring the num ber of degr ee-3 vertices. But the tw o algorithms are totally different . F ¨ urer [11] designed an al- gorithm for MIS in degree-3 gra phs by tackling m − n , where m is the n umber of edges and n the num b er of vertices. Ba sed up o n a refined branching with resp ect to F ¨ urer’s alg orithm, Bourge o is et a l. [9] g o t the current bes t alg orithm for MIS in degr ee-3 gr aphs with r unning time O ∗ (1 . 0977 n ). In this pape r, we still use the n umber o f degree-3 v ertices as a measure to analy z e our algo rithm. Ba sed on tw o new branching rules, br anching on a b ottle and br anching on a 4 -cycle , we desig n an even faster algorithm for MIS in deg r ee-3 graphs, which r uns in O ∗ (1 . 0919 n ) time. Our a lgorithm is s imple a nd do es not contain ma ny branching rules. F urther more, it can b e used to solve the k -vertex c over pr oblem (to de c ide if the g raph has a vertex cov er of size k ) in deg ree-3 g raphs in O ∗ (1 . 1923 k ) time, which improves the previously known result of O ∗ (1 . 1939 k ) by Chen et a l. [7]. 2 Preliminaries W e sha ll try to b e co nsistent in using the following notation. The n umber of ver- tices in a g raph will b e denoted b y n and the n umber of degr ee-3 v ertices (vertices of deg ree ≥ 4 will also b e counted with a weight) by r . F or a vertex v in a graph, d ( v ) is the degree o f v , N ( v ) the s et of all neighbo rs of v , N [ v ] = N ( v ) ∪ { v } the set of vertices with distance a t mo st 1 fr om v , and N 2 ( v ) the set o f vertices with distance exa ctly 2 fro m v . W e say edge e is incident on a vertex set V ′ , if at least one endp oint o f e is in V ′ . In our alg orithm, when we remov e a s e t o f vertices, we also r emov e all the edg es that a re incident on it. Througho ut the pa p e r we use a mo dified O notation that suppresses all p olyno mially b ounded facto r s. F or t wo functions f and g , w e wr ite f ( n ) = O ∗ ( g ( n )) if f ( n ) = O ( g ( n ) poly ( n )), where pol y ( n ) is a p olyno mial. Our alg orithms are based on the branch-and-reduce paradigm. W e will fir s t apply some reduction r ules to r educe the size of instances of the problem. Then we apply some branching rules to branch on the gr a ph by including some vertices in the indepe ndent set o r excluding some vertices from the indep endent s e t. In each bra nch, we will get a maxim um indep endent set problem in a g raph with a smaller measure. Next, w e intro duce the reduction rules a nd branching rules that will b e used in our a lgorithms. 2.1 Reduction Rules There ar e several standard prepr o cesses to reduce the size o f instances of the problem. F olding a de gr e e- 1 or de gr e e- 2 vertex and r emoving a dominate d vertex 4 Mingyu X iao are frequently used rules. Besides these r eduction r ules, we still need to reduce some other lo c al structure s called 2 - 3 st ructur e , 3 - 3 struct ur e and 3 - 4 structur e . F oldi ng a degree-1 v ertex F olding a de gr e e-1 vertex v me ans r emoving v and u fr om t he gr aph, wher e u is the unique n eighb or of v . F oldi ng a degree-2 v ertex F olding a de gr e e- 2 vertex v (with two neighb ors u and w ) me ans ( a ) r emoving v , u and w fr om t he gr aph, when u and w ar e adjac ent. ( b ) r emoving v , u and w fr om the gr aph and intr o ducing a n ew vertex s that is adjac ent to al l neighb ors of u and w in G ( exc ept the r emove d vertex v ) , when u and w ar e nonadjac ent. Please refer to Figure 1 for an illustration of the op eration in case ( b ) of folding a degree- 2 vertex. Let α ( G ) denote the size of a maximum indep endent set of g raph G and G ⋆ ( v ) the gr aph after fo lding a degree-1 or degree -2 vertex v . Then we ha ve the following lemma. Lemma 1. F or any de gr e e- 1 or de gr e e- 2 vertex v in gr aph G , α ( G ) = 1 + α ( G ⋆ ( v )) . v 3 2 1 b a 3 2 1 s Folding a degree-2 vertex c b a u v 3 2 1 4 3 2 1 4 s Folding a 2-3 structure c b a w v 3 2 1 4 u 3 2 1 4 s Folding a 3-3 structure Fig. 1. Illustrations of folding o p erations The correctness of folding a degree-1 or degr ee-2 vertex ha s b e e n discussed in many pe r vious pa p e rs. In fact, general folding rules a re known in the literature, Indep endent Set and V ertex Co ver in Sp arse graphs 5 which can deal with a v ertex of degree ≥ 3 or a set of independent v ertices [13],[5]. In this pap er, we still need to fo ld the following three loc a l s tructures ca lled 2 -3 structure, 3-3 structur e and 3-4 structure. Let u and v b e tw o indep endent degree- 3 v ertices, if they hav e three co mmon neighbors a, b and c , then we say that the five v ertices comp ose a 2 - 3 struct ur e (see Figure 1), a nd denote it by { u , v } - { a, b, c } . Let v b e a degr ee-3 vertex, and u and w t w o adjacent vertices of degr ee ≥ 3. If N ( u ) ∪ N ( w ) − { u, w } = N ( v ), then we say that the six vertices { u, v , w } ∪ N ( v ) co mp ose a 3 - 3 stru ctur e (see Figure 1), and deno te it b y { v , u , w } - { a, b , c } . Let u, v and w b e thr ee independent vertices of degr ee ≥ 3 , if they ha ve exact four neigh b o rs a, b , c and d , then we say that the s even v ertices comp os e a 3 - 4 st ructur e , and deno te it by { u, v , w } - { a, b, c, d } . F oldi ng a 2 - 3 structure, 3 - 3 structure or 3 - 4 structure L et A - B b e a 2 - 3 structur e or 3 - 3 st ructur e or 3 - 4 stru ct ur e. F oldi ng A - B me ans ( a ) r emoving A ∪ B fr om the gr aph, when B is not an indep endent set . ( b ) r emoving A ∪ B fr om the gr aph and intr o ducing a new vertex s that is adjac ent to al l neighb ors of vertic es in B ( exc ept t he r emove d vertic es ) , when B is an indep endent set. Lemma 2. If gr aph G has a 2 - 3 stru ctur e or 3 - 3 structur e, t hen α ( G ) = 2 + α ( G ⋆ 2 ) , wher e G ⋆ 2 is the gr aph after folding a 2 - 3 structu r e or 3 - 3 structu r e in G . If gr aph G has a 3 - 4 struct u r e, then α ( G ) = 3 + α ( G ⋆ 3 ) , wher e G ⋆ 3 is the gr aph after folding a 3 - 4 structu r e in G . A deg ree-2 v ertex c an b e reg arded as a 1-2 structure according to our def- initions. In fact, a degr ee-2 v ertex, 2-3 structure and 3- 4 structure a re specia l cases describ ed in Lemma 2 . 4 in [1 3]. The 3 - 3 structure is for the fir st time b e- ing in tro duced. The correctnes s o f folding an A - B str uc tur e (a 1-2 s tructure, 2 -3 structure, 3 - 3 structure o r 3-4 structure) follows from this obser v ation: When B is not an independent set, there is a maximum indep endent s e t that contains A (or tw o independent v ertices in A , when A - B is a 3-3 structure). When B is a n independent set, there is a maximum indep endent set that cont ains either B or A (o r tw o indep endent vertices in A , when A - B is a 3-3 structure). W e ig nore the detailed pro o f here. Dominance If ther e ar e two vertic es v and u such that N [ u ] ⊆ N [ v ] , we say u dominates v . Lemma 3. If vertex v is dominate d by any other vertex in gr aph G , then α ( G ) = α ( G − { v } ) . Definition 1. A gr aph is c al le d a r educed gr aph , if it has no de gr e e- 1 vertex, de gr e e- 2 vertex, dominate d vertex, 2 - 3 structu re , 3 - 3 structu r e or 3 - 4 structure. 6 Mingyu X iao 2.2 Branc hing Rules Next we intro duce tw o bra nching techniques, br anching on a b ottle and br anching on a 4 -cycle , which are simple and obvious, but ca n av oid tedious bra nching rules in the description of the alg orithms. Let a b e a deg ree-3 v ertex, and b, c, d the three neighbors of a . If t wo neighbo r s of a , say c and d , a re adjac e nt , then we say that the four vertices comp ose a b ottle and denote it by b - a - { c, d } . Lemma 4. L et b - a - { c, d } b e a b ottle in gr aph G , then ther e is a maximum in- dep endent set S in G such that either a ∈ S or b ∈ S . Pr o of. If b is no t in a maximum indep endent s et, we can directly r emov e b from the g raph. In the remaining gra ph a b ecomes a degree -2 vertex and the tw o neighbors o f it are adjacent. In this case, there is a maxim um indep endent set that contains a . Based on Lemma 4, we get the following branching rule. Branc hing on a b ottle Br anchi ng on a b ottle b - a - { c, d } me ans bra nching by either including a in the indep endent set or including b in the indep endent set. Note. In fact, w e can fold a bo ttle by using the general folding rule mentioned in [5] (also in [6 ]), but this folding rule is helpless for our analysis, esp ecially when the three neighbor s o f the degree - 3 v ertex are high-degr ee vertices. Let a, b , c and d be four vertices in graph G , if G has four edg es ab , b c , cd and da , then we say tha t abcd is a 4 -cycle in G . Lemma 5. L et ab c d b e a 4 -cycle in gr aph G , then for any indep endent set S in G , either a, c / ∈ S or b , d / ∈ S . Pr o of. Since any independent set contains at most 2 vertices in a 4-c ycle and the tw o v ertices can not b e adjace nt, w e know the lemma holds. Based on Lemma 5, we get the following branching rule. Branc hing on a 4 -cycle Br anchi ng on a 4 -cycle abcd me ans br anching by either ex cluding a and c fr om the indep endent set or excluding b and d fr om the indep en dent set. 3 A Simple Algor ithm Our algorithm for the ma ximum indep endent set problem is described in Fig- ure 2. It works as follows. If the graph has a comp onent of at most 15 vertices, we find a maximum indep endent set in this comp o nent dir ectly ( Step 1). If the graph has a degr ee-1 o r deg ree-2 vertex, we fold it in Step 2. If the gr aph has a do minated vertex, we re mov e it in Step 3. If the gra ph has a 2-3 str ucture or 3-3 structure or 3-4 structure, we fold it in Step 4 and Step 5. When the g r aph can not b e reduced, we apply our branching rules. If there is a b o ttle, we branch Indep endent Set and V ertex Co ver in Sp arse graphs 7 on a b o ttle ( Step 6). Els e if ther e is a 4-cycle, we br anch on a 4- cycle ( Step 7). Else in Step 8, w e gr eedily select a vertex of max imu m degr ee and br anch on it by including it in the independent set or excluding it from the indep endent set. M I S ( G ) Input : A graph G . Output : The size of a maxim um indep endent set in G . 1. If { G has a component P of at most 15 vertice s } , return t + M I S ( G − P ), where t is the size of a minim um indep endent set in P . 2. Else if {∃ v ∈ V : d ( v ) = 1 or 2 } , return 1 + M I S ( G ⋆ ( v )). 3. Else if {∃ v , u ∈ V : N [ u ] ⊆ N [ v ] } , return M I S ( G − { v } ). 4. Else if { there is a 2-3 structure or 3-3 stru cture } , return 2 + M I S ( G ⋆ 2 ). 5. Else if { there is a 3-4 structure } , return 3 + M I S ( G ⋆ 3 ). 6. Else if { there is a b ottle b - a - { c, d }} , return max { 1 + M I S ( G − N [ a ]) , 1 + M I S ( G − N [ b ]) } . 7. Else if { there is a 4-cycle abcd } , return max { M I S ( G − { a, c } ) , M I S ( G − { b, d } ) } . 8. Else, pick up a vertex v of max imum degree, and return max { M I S ( G − { v } ) , 1 + M I S ( G − N [ v ]) } . Note : With a few mo difi cations, th e algorithm can p ro vide a maximum inde- p endent set. Fig. 2. The Algorithm M I S ( G ) 4 The A nalysis T o a nalyze the time complexity of our algor ithm, we will consider recur rence relations r elated to parameter r , the num b er of degree- 3 vertices (vertices of degree ≥ 4 will a ls o b e counted with a w eight) in the cor resp onding gra ph. When r = 0, the gra ph has o nly degree-0 , degree- 1 a nd degree- 2 v ertices and the ma ximum indep endent set problem can b e so lved in linea r time. W e use C ( r ) to denote the worst-case size of the search tree in our algo r ithm when the parameter of the graph is r . In our alg orithm, it is p o ssible to crea te a vertex of degree ≥ 4 when folding. W e will rega rd a de g ree- d ( d ≥ 3) vertex as a combination of d − 2 degr e e-3 vertices and count d − 2 in parameter r . Then when a degree- d vertex is r emov ed, parameter r will b e reduced by d − 2. When an edge incident on a degree- d vertex is r emov ed, pa rameter r will b e reduced by 1. In the remaining of the pap er, when we say a graph has x deg r ee-3 vertices, it do es not mean that the g r aph rea lly has exactly x vertices of degree 3. In fact, 8 Mingyu X iao all the vertices of degree ≥ 3 a re coun ted. Next, we ana lyze ho w muc h r ca n b e reduced in each s tep of our algor ithm. Lemma 6. Aft er folding a de gr e e- 1 or de gr e e- 2 vertex, p ar ameter r wil l not incr e ase. Lemma 7. L et G b e a gr aph having no de gr e e- 1 or de gr e e- 2 vertex, then after folding a 2 - 3 structu r e or 3 - 3 stru ctur e or 3 - 4 stru ct ur e, or r emoving a dominate d vertex in G , p ar ameter r wil l b e r e duc e d by at le ast 4 . Pr o of. In each case, a degree-3 vertex is remov ed (or a n e ven better case o ccurs), and then w e ca n further r educe r by 3 from 3 neighbor s o f the vertex. T otally r will b e reduced by a t least 4. Lemma 8. L et G b e a c onne cte d gr aph. If G has at le ast x de gr e e- 1 vertic es and x vertic es of de gr e e ≥ 3 (a de gr e e- d ( d ≥ 3 ) vertex wil l b e r e gar de d as d − 2 de gr e e- 3 vert ic es), then after iter atively foldi ng de gr e e- 1 vertic es until the gr aph has no de gr e e- 1 vertex, p ar ameter r wil l b e r e du c e d by at le ast x . Pr o of. Let V ′ 6 = ∅ b e the set of vertices o f degree ≥ 2 in the remaining gra ph after iteratively folding degree-1 v ertices (The lemma obviously ho lds, w he n V ′ = ∅ ). Assume there are y edges b etw een V ′′ = V − V ′ and V ′ . After removing V ′′ , we c a n r educe y deg ree-3 v ertices from V ′ . W e will prov e that there are at least x − y degree-3 vertices in V ′′ . T o prove that, we first construct a new graph G ′ from G b y contracting V ′ int o a single vertex v and remov e all self-lo o ps incident on it (keeping parallel edges). Then we only need to prov e that e xcept vertex v , G ′ has at least x − y degree-3 vertices. Since all the x degree -1 vertices of G are in V ′′ , G ′ has at leas t x ′ degree-1 vertices, where x ′ = x + 1 when v is a degree-1 vertex and x ′ = x when v is not a deg ree-1 vertex. Note that a tree with x ′ degree-1 vertices has at le ast x ′ − 2 degree-3 vertices. W e know that G ′ has least x ′ − 2 degree-3 vertices ( G ′ is a connected graph). W e c o nsider the following thr ee cases . Case 1: y = 1. F o r this case, v is a degree-1 vertex and x ′ = x + 1. Then G ′ has at least x ′ − 2 = x − 1 degree-3 vertices. Case 2: y = 2. F or this case, v is a degree - 2 vertex and x ′ = x , and G ′ still has at lea st x − 2 degree-3 vertices. Case 3: y ≥ 3. F or this case, v is a degree- y vertex and x ′ = x . Ex c epting y − 2 degr ee-3 vertices count ed from v , there ar e sill x − 2 − ( y − 2 ) = x − y degr ee-3 vertices. Therefore, after removing V ′′ , r will b e reduced by a t least x . Corollary 1. L et G b e a gr aph having not any c omp onent of a p ath. If G has any de gr e e- 1 vertex, then we c an r e du c e r by at le ast 1 by iter atively folding de gr e e- 1 vertic es. If G has exactly 2 de gr e e- 1 vertic es, then we c an r e duc e r by at le ast 2 by iter atively folding de gr e e- 1 vertic es. Lemma 9. L et G b e a r e duc e d gr aph and v a de gr e e- 3 vertex in G . Then not any de gr e e- 0 vertex or c omp onent of a 1 - p ath or c omp onent of a 2 -p ath is cr e ate d after re moving N [ v ] . Indep endent Set and V ertex Co ver in Sp arse graphs 9 Pr o of. If a degree-0 vertex u is created, then G has a 2 - 3 structur e { v , u } - N ( v ). If a 1-pa th ab is created, then there is a 3-3 s tr ucture { v , a, b } - N ( v ). If a 2-path abc is crea ted, then there is a 3-4 structure { a, c, v } - N ( v ) ∪ { b } . Lemma 10. Le t G b e a c onne cte d r e duc e d gr aph of mor e than 7 vertic es and v a de gr e e- 3 vertex in G . Then after r emovi ng N [ v ] , p ar ameter r wil l b e r e duc e d by at le ast 8 . F u rthermor e, if e ach 3 -cycle in G c ontains at le ast one vertex of de gr e e ≥ 4 , then after r emoving N [ v ] , p ar ameter r wil l b e r e duc e d by at le ast 10 . Pr o of. There is a t mos t one edg e with bo th endpoints in N ( v ), otherwise v will dominate a neighbor of it. Therefore, there are at least four edg es b e t ween N ( v ) and N 2 ( v ). If | N 2 ( v ) | ≥ 4, r will b e reduced by 4 + 4 = 8 directly after removing N [ v ]. If | N 2 ( v ) | ≤ 3 , it is imp os s ible to cr eate a co mp onent of a l - path ( l ≥ 3) after removing N [ v ]. By Lemma 8 and Corolla ry 1 and Lemma 9 we know tha t even tually r will b e reduced by at lea st 8. Next, we assume that in each 3 -cycle in G ther e is a vertex of deg r ee ≥ 4. W e distinguish the following t wo ca s es. Case 1: All vertices in N ( v ) ar e degree-3 vertices. In this case, no ne pair of vertices in N ( v ) ar e a djacent and there are exactly six edges b etw een N ( v ) a nd N 2 ( v ), which means at most 3 degree-1 vertices w ill be crea ted a fter removing N [ v ]. It is imp oss ible to cr eate a comp onent of a path after r emoving N [ v ] (Obviously , no path o f length ≥ 4 will be cr eated. Lemma 9 shows no path of length ≤ 2 will be created. If a 3-path is created, then the gr a ph G has only 7 vertices). So b y Coro llary 1, if a comp onent with 1 or 2 degree- 1 vertices is created a fter re moving N [ v ], we ca n further reduce r by 1 or 2 b y further reducing deg ree-1 vertices in the comp onent. If a comp onent with 3 degree-1 vertices is c r eated, then the co mpo nent a lso contains at least 3 degree -3 v ertices, otherwise the only p ossibility o f the comp o nent is that it has 4 vertices: a degree-3 vertex adjacent with thre e degree-1 vertices, which also implies a c ontradiction–the gra ph G has only 7 v ertices. By Lemma 8 , we still can fur ther r educe r by leas t 3. In any cas e, totally we ca n reduce r b y at least 4 + 6 = 1 0. Case 2: There is a vertex of degree ≥ 4 in N ( v ). Then there are at least five edges betw een N ( v ) and N 2 ( v ) (Note that there is at mo s t one edge with b oth endp oints in N ( v )). By Lemma 8 a nd Lemma 9 w e know that r will b e reduced by a t least 5 + 5 = 10. Lemma 11. Le t G b e a c onne cte d r e duc e d gr aph of mor e than 8 vertic es and v a vertex of de gr e e ≥ 4 in G . Then after r emoving N [ v ] , p ar ameter r wil l b e r e duc e d by at le ast 10 . Pr o of. The lemma obviously holds when v is a vertex o f degre e ≥ 5 or a degree-4 vertex with | N 2 ( v ) | ≥ 4. Now we a ssume v is a deg ree-4 vertex and | N 2 ( v ) | ≤ 3. Case 1: | N 2 ( v ) | = 1. In this cas e, after removing N [ v ], r is re duced by at least 6 + 4 = 10, o r r is reduced by at least 6 + 3 and the only vertex in N 2 ( v ) b ecomes a degree- 1 vertex (Note tha t there are at least | N ( v ) | = 4 edges b etw een N ( v ) and N 2 ( v )). In the later case, w e can reduce r by at least 1 by folding degr ee- 1 vertices. Case 2 : | N 2 ( v ) | = 2 . The tw o vertices a, b ∈ N 2 ( v ) are adjacent, otherwise there is a 3- 4 structure { v } ∪ N 2 ( v )- N ( v ). Then after removing N [ v ], 10 Mingyu X iao at most one of a and b b eco mes a degree-1 vertex, otherwise G has only 7 vertices. Therefore, we also can reduce r by at least 4 from V − N [ v ]. Case 3: | N 2 ( v ) | = 3. If one vertex in N ( v ) is a vertex o f degree ≥ 4 , then the lemma holds. O therwise, all vertices in N ( v ) a r e degr ee-3 vertices, and then the num b er of edges b etw een N ( v ) and N 2 ( v ) is 4 or 6 o r 8 . If no degree-1 v ertex is cr eated a fter removing N [ v ], then r will b e reduced b y at lea st 6 + 4 dir ectly (Note that it is imp ossible to create tw o deg ree-0 vertices, and when one degr ee-0 vertex is cre ated, there are at lea st thr ee edg es b etw een N ( v ) and N 2 ( v ) that are incident on the other tw o vertices in N 2 ( v )). If so me degre e-1 vertices ar e crea ted but no path comp onent is cr eated, then we can further reduce r by at le a st 1 by Lemma 8. The difficult case o ccurs when a pa th c o mp o nent is created. The path can only b e a 1-path or 2 - path. If it is a 2 -path, then the graph has only 8 vertices. Therefor e the path is a 1-path. If one vertex in the pa th is a vertex of degree ≥ 4 in G , then after removing N [ v ], r is reduced by at leas t 6 + 4 = 10 directly . If the tw o vertices in the path are degr e e-3 vertices in G , then there ar e at least tw o edges betw een N ( v ) and N 2 ( v ) that are incident on the third vertex u in N 2 ( v ). So after removing N [ v ], r is re duce d b y at lea st 6 + 4 = 10, o r r is r e duced by at least 6 + 3 and u b eco mes a degree - 1 vertex, fo lding which will further reduce r by at leas t 1. W e have chec ked all the cases a nd then finished the pro of. Lemma 12. Le t G b e a c onne cte d r e du c e d gr aph of mor e than 7 vertic es. If G has a b ottle, t hen algorithm M I S ( G ) wil l br anch on a b ottle with r e curr enc e r elation C ( r ) ≤ 2 C ( r − 8) , (1) wher e C ( r ) is t he worst-c ase size of the se ar ch tr e e in our algorithm. Mor e over, if e ach 3 - cycle in G c ontains at le ast one vertex of de gr e e ≥ 4 , then M I S ( G ) wil l bra nch on a b ottle with r e curr enc e r elation C ( r ) ≤ 2 C ( r − 1 0) . (2) Pr o of. Let the bottle called by our a lgorithm b e b - a - { c, d } . Our a lgorithm will branch by either removing N [ a ] o r N [ b ]. By Le mma 10 and Lemma 11, we get (1) and (2) directly . Lemma 13. Le t G b e a c onne cte d b ottle-fr e e r e duc e d gr aph of mor e than 7 ver- tic es. If G has a 4 -cycle, t hen algori thm M I S ( G ) wil l br anch on a 4 -cycle with r e cu r re nc e r elation C ( r ) ≤ 2 C ( r − 8) . (3) Mor e over, if e ach 3 -cycle or 4 -cycle in G c ontains at le ast one vertex of de gr e e ≥ 4 , then M I S ( G ) wil l br anch on a 4 -cycle with r e cu rr enc e r elation C ( r ) ≤ 2 C ( r − 1 0) . (4) Indep endent Set and V ertex Co ver in Sp arse graphs 11 Pr o of. Let the 4- cycle called b y o ur algorithm b e ab cd . Our algor ithm will branch by removing either { a, c } or { b, d } fr o m the graph. W e lo o k at the bra nch where { a, c } is remov ed (It is the same to { b , d } ). Since none of the four vertices is dominated by o thers, ea ch of the four vertices will b e adjacent to a vertex differ- ent from the four vertices. If a fter removing { a, c } , no degree - 1 vertex is created, then we can reduce r by 8 directly in this branch. If so me degree-1 vertices are created, then our algorithm will fold one, say x , in the nex t step. Obviously , x is a degree-3 v ertex in the original graph. The ope ration of removing { a, c } and then folding x is equiv alen t to the removing of N [ x ]. W e can reduce r b y at lea st 8 b y Lemma 1 0. The r efore, we get (3). Next, we prove (4). There is at le ast o ne vertex o f degr e e ≥ 4, s ay a , in the 4-cycle. W e distinguish the following three cases. Case 1: There is only one vertex o f degree ≥ 4 in the 4- cycle. No matter we remov e { a, c } or { b, d } , at lea st one degree-1 w ill b e crea ted. As discuss ed ab ove, a fter further folding a degr ee-1 vertex, we c a n re duce r by at least 10 in ea ch br anch by Lemma 10. Then we get (4 ). Case 2: There are ex actly t wo vertices of degree ≥ 4 in the 4- cycle and the tw o vertices ar e adjacent to each other in the 4-cy cle (the t w o vertices ar e not { a, c } o r { b, d } ). Without loss of g enerality , we ca n assume the tw o vertices of degree ≥ 4 are a and b . In the branch where { a, c } is r emov ed, d be comes a deg ree-1 vertex. I n the bra nch wher e { b, d } is remov ed, c b ecomes a degree- 1 vertex. Then in each br anch w e will remov e N [ v ] for some degree-3 vertex v in G . W e still c an get (4 ). Case 3: The r e are ex actly tw o vertices of degre e ≥ 4 in the 4-cycle a nd the tw o vertices a re a pair of o pp o site vertices in the 4 -cycle. Then the tw o vertices o f degree ≥ 4 ar e a and c (W e hav e assumed that a is a vertex of degree ≥ 4). It is ea sy to see that after removing { b, d } , r will be reduced by at least 8. In the branch where { a, c } is removed, some degree-1 vertices ar e created (at least b and d ). Then in this branch we will remov e N [ v ] for some degree- 3 vertex v in G , where v has tw o vertices of degr ee ≥ 4 a a nd c . Since G has no bo ttle, there is not any edge with b oth endp oints in N ( v ). Therefor e, there are at lea st 8 edges b etw een N ( v ) and N 2 ( v ). If no degr ee-1 vertex is created after removing N [ v ] (Lemma 9 als o sho ws that no degree-0 vertex will be crea ted), then r is r e duced by 6 + 8 = 14 directly . If some degree-1 vertices are cr e a ted, the c ase becomes complicated. In fact, as w e do in the pr o of of Lemma 10 we can prov e that no comp onent o f less than 3 degr ee-3 vertices will b e created. B y Lemma 8, we know that r will also be reduced by at least 6 + 7 = 13 in this branch. W e g e t C ( r ) ≤ C ( r − 8) + C ( r − 13 ) . (5) Case 4: There are exactly three vertices o f degre e ≥ 4 in the 4-cycle. Without loss of generality , we assume the r emaining degree-3 v ertex is c . After r emoving { a, c } , r will b e r educed b y at least 10. After removing { b , d } , r will b e reduced by at leas t 12. W e get C ( r ) ≤ C ( r − 10) + C ( r − 1 2) . (6) 12 Mingyu X iao Case 5 : All the four v ertices in the cycle are vertices o f degree ≥ 4. It is clea r that r will be reduced by at leas t 10 in each branch. W e als o get (4 ). Since (4) cov ers (5) a nd (6), we k now that the lemma ho lds . Lemma 14. Le t G b e a c onne cte d r e du c e d gr aph of mor e than 15 vertic es that has no b ottle or 4 -cycle. If G has a vertex of de gr e e ≥ 4 , then algorithm M I S ( G ) wil l br anch on a vertex of maximum de gr e e with r e curr enc e r elation C ( r ) ≤ C ( r − 6) + C ( r − 14 ) . (7) Pr o of. Our alg o rithm will select a vertex v o f maximum degree and bra nch on it b y excluding it fro m the indep endent set or including it in the indep endent set. In the former bra nch, v is r emov ed and r decrease s b y at lea s t 2 + 4 = 6. In the latter branch, N [ v ] is removed. Since G has no bottle or 4-cycle, ther e ar e at lea s t 8 vertices in N 2 ( v ). Then in this branch, r will be reduced by at least 6 + 8 = 14. Therefo re, w e get (7). Lemma 15. Le t G b e a c onne cte d r e du c e d gr aph of mor e than 15 vertic es that has n o b ottle or 4 -cycle. If G is also a 3 - r e gu lar gr aph, t hen algorithm M I S ( G ) c an br anch with r e curr en c e r elation C ( r ) ≤ C ( r − 1 0 ) + 2 C ( r − 14) . (8) Pr o of. Our alg orithm will select a degree-3 vertex and branch on it. Since G is 3-regula r graph that has no 3-cycle or 4 -cycle, there are exactly 8 vertices in N 2 ( v ). In the branch where N [ v ] is r emov ed, 10 degree-3 vertices are reduced. So we ca n branch with recur rence relation C ( r ) ≤ C ( r − 10) + Q ( r − 4) , (9) where Q ≤ C is so me function corre s p o nding to the s ize of the branch where v is removed. Next, we fo cus on re fining a nalysis of Q . In the branch where v is remov ed, 3 nonadjacent degree-2 v ertices are cr eated. Our algor ithm will fold the three degree - 2 vertices in the next step. Let G ′ be the resulted graph. Then G ′ has exactly 3 degree- 4 vertices (Note that the original gr a ph ha s no 3-cycle or 4 -cycle. It is impo ssible to create a degree-3 vertex after folding a degree-2 vertex), and each 3-cycle or 4-cycle in the current graph contains at lea st one degree-4 vertex. If G ′ has a b ottle or 4-cy cle, we can branch with Q ( r ) ≤ 2 C ( r − 1 0) b y Lemma 12 a nd Lemma 13. If G ′ has no bo ttle or 4 -cycle, we will branch on a degree-4 v ertex v ′ . W e further distinguish three different cases. Case 1: The other tw o degree-4 vertices are adjacen t to v ′ . In this case, w e hav e | N 2 ( v ′ ) | ≥ 8 (the three degree-4 vertices may form a triangle). In the branch where v ′ is remov ed, r is reduced by at least 6 , a nd in the branch wher e N [ v ′ ] is r emov ed, r is reduced by a t least 8 + 8 = 16. W e get Q ( r ) ≤ C ( r − 6 ) + C ( r − 1 6 ). Case 2: Ther e is o nly one deg r ee-4 vertex adjacent to v ′ . Since ther e is no b ottle and 4-cycle, we get | N 2 ( v ′ ) | ≥ 9. In the branch wher e N [ v ′ ] is remov ed, r is reduced by 7 + 9 = 16 . W e also get Q ( r ) ≤ C ( r − 6) + C ( r − 16). Case 3: Ther e is no degr ee-4 vertex adjacent to v ′ . W e will Indep endent Set and V ertex Co ver in Sp arse graphs 13 branch on v ′ with (7) directly , a nd in the br anch where v ′ is r emov ed, some o ther degree-4 vertices are left. W e can further branch with (7) at lea st. Then we get Q ( r ) ≤ C ( r − 14) + C ( r − 6 − 6) + C ( r − 6 − 14) = C ( r − 12) + C ( r − 1 4) + C ( r − 20). The worst case is that after branching with (9) we branch with (2) or (4), in which we get C ( r ) ≤ C ( r − 10) + Q ( r − 4) ≤ C ( r − 10 ) + 2 C ( r − 14), as claimed in the lemma. Among all the ca ses in our algor ithm, the worst running time co r resp onds to recurrence relation (8). Since C ( r ) = O (1 . 0919 r ) satisfies (8), we get Theorem 1. Algori thm M I S ( G ) c an find a m inimum indep endent set in a de gr e e- 3 gr aph in O ∗ (1 . 0919 n ) time. 5 Impro v emen t on k -V er tex Cov er Given a gra ph G and a par ameter k , the k - vertex c over problem is to decide if G has a vertex cov er of size at most k . The k - vertex cov er problem is one of the most extens ively s tudied pr oblems in the a rea o f Parameter ized Algo- rithms. In this section, we show that the k -vertex cov er pro blem ca n b e solv ed in O ∗ (1 . 1923 k ) time, which improv es the previous ly kno wn result of O ∗ (1 . 1939 k ) by Chen et al. [7]. Nemhauser and T rotter [14] proved the following theore m: Prop ositi on 1. F or a gr aph G = ( V , E ) with n vertic es and m e dges, we c an c ompute two disjoint vertex sets C 0 , V 0 ⊂ V in O ( √ nm ) time, such that (1) Every minimum vertex c over in induc e d sub gr aph G ( V 0 ) plus C 0 forms a minimum vertex c over of G . (2) A m inimum vertex c over of G ( V 0 ) c ontains at le ast | V 0 | / 2 vertic es. Our simple algorithm works as follows. Given an instance ( G, k ) of the k - vertex cov er problem in deg ree-3 graphs, we firs t us e Nemhauser a nd T rotter’s algorithm to construct C 0 and V 0 . If | V 0 | > 2 k , then G do es not ha ve a vertex cov er of s ize a t mos t k . Els e we use our algorithm pr esented in Section 3 to find a ma ximum indep endent set S in G ( V 0 ) in O ∗ (1 . 0919 | V 0 | ) = O ∗ (1 . 1923 k ) time. Then C 0 + V 0 − S is a minimum vertex cover of G . If k > | C 0 + V 0 − S | , then G do es no t hav e a vertex cov er of size at most k . Else C 0 + V 0 − S is sa tisfied vertex cover. Theorem 2. The k -vertex c over pr oblem in de gr e e- 3 gr aphs c an b e solve d in O ∗ (1 . 1923 k ) time. 6 Concluding Remarks In this pap er, we ha ve pre s ented a simple O ∗ (1 . 0919 n )-time algor ithm for the minim um indep endent set problem in deg ree-3 graphs and a simple O ∗ (1 . 1923 k )- time algorithm for the k - vertex cover problem in degree- 3 gr a phs. Both algo - rithms improv e previously known algo rithms. 14 Mingyu X iao Unlik e most previous algor ithms, o ur algorithms do not con tain many branch- ing rules. W e use tw o new bra nching techniques, called branching on a bottle and branching on a 4-cycle , to avoid tedio us examina tions of the lo cal structures. In fact, new branching rules catch the struc tur al prop erties of small c ycles in graphs, which make our algorithms simple and practica l. It is easy to see that many previous a lgorithms can apply these t wo new bra nching rules to simplify the description and analys is. Our algo rithm for the max imum indep endent set problem is analyzed by measuring the n umber of degree-3 vertices. W e hav e chec ked that our algorithm M I S ( G ) can a lso b e a nalyzed by measur ing para meter m − n + t to get the same running time b ound, where m is the num b er of edges, n the num ber of vertices, and t the num b er of tree co mp o ne nts in the gra ph. Reader s may note that the alg o rithms presented by F ¨ urer [11] and Bour g eois et al. [9] are analyzed by mea suring m − n . In fact, their alg o rithms also need to consider the tree comp onents cre a ted in the graphs a nd they ha ve a s eparate s e ction to a nalyze them. W e guess that considering the tree comp onents in the para meter may lead to a cleare r analy sis. References 1. T arj an, R., T ro jano wski, A.: Finding a max im um indep end ent set. 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