Treewidth computation and extremal combinatorics

T reewidth computation and extremal com binatorics ⋆ F edor V. F omin and Yngve Villanger Department of Informatics, Universit y of Bergen, N-5020 Bergen,Norw ay { fedor.fomin,yngv e.villanger } @ii.uib.no Abstract. F or a given graph G and in tegers b, f ≥ 0, let S b e a subset of vertices o f G of size b + 1 suc h that the sub graph of G induced by S is connected and S can b e separated from oth er v ertices of G by remo ving f vertices . W e prov e that every g raph on n vertices con tains at most n ` b + f b ´ such vertex subsets. This result from extremal combinatorics app ears to b e very useful in the design of severa l enumeration and ex act algorithms. In p articular, w e use it to provide algorithms that for a give n n -vertex graph G – compute th e treewidth of G in time O (1 . 7549 n ) by making use of exp onen tial space and in time O (2 . 6151 n ) and p olynomial space; – decide in time O (( 2 n + k +1 3 ) k +1 · kn 6 ) if th e treewidth of G is at most k ; – list all minimal separators of G in time O (1 . 6181 n ) and all p oten tial maximal cliques of G in time O (1 . 7549 n ). This significantly imp roves previous algorithms for these p roblems. 1 In tro duction The aim of exact algorithms is to optimally solve hard pr o blems exp onentially faster than brute-force sear c h. The first pap ers in the area date back to the sixties and seven ties [18, 26]. F or the last tw o decades the amount of literature devoted to this topic has b een tremendous and it is imp ossible to give here a list of representativ e r eferences without missing significant results. Recent surveys [14, 20 , 25, 28] pr ovide a compr e hensiv e information on ex act algor ithms. It is very natura l to a ssume the existence of str ong links betw een the area of ex act algorithms and some ar eas of extrema l combinatorics, esp ecially the part o f extremal co m binatorics which studies the max imum (minimum) cardinalities of a system of s ubsets of so me set satisfying certain pr o perties. Strangely enough, there are not so many examples o f suc h links in the literature, a nd the ma jority of exact algorithms are ba sed o n the so-called branching (backtrac k ing ) technique which traces back to the w o rks of Da vis , Putnam, Logemann, and Loveland [11, 12]. In this pa per, we pr o ve a combinatorial lemma which app ears to be very useful in the analysis of certain en umer a tion and e x act algorithms. F or a vertex ⋆ This researc h w as partially supp orted by the Researc h Council of Norw ay . 2 F edor V. F omin and Yngve Villanger v of a gr aph G and integers b, f ≥ 0, let t ( b, f ) be the maximum num b er of connected induced subgr aphs of G of size b + 1 such that the intersection of all these subgra phs is no nempt y and ea c h such a subgraph has exactly f neighbor s (a neighbor of a subgraph H is a vertex of G \ H which is adjacent to a vertex of H ). Then the combinatorial lemma states that t ( b, f ) ≤  b + f b  (and it is easy to chec k that this b ound is tight ). This can b e s een as a v ariation of Bollob´ ass Theorem [7], which is one of the corner -stones in extremal set theory . (See Section 9.2.2 o f [21] for detailed discussions on B ollob´ ass Theorem and its v aria n ts.) W e use this combinatorial res ult to obtain fas ter algor ithm for a num b er of problems related to the treewidth o f a graph. The treewidth is a fundamental graph pa rameter from Graph Minors Theory by Rob ertson and Seymour [2 4] and it ha s numerous a lg orithmic applications, see the surveys [4, 6 ]. The problems to compute the treewidth is known to b e NP -hard [1] and the b est k no wn appr o x- imation a lg orithm for treewidth ha s a factor √ log OP T [13]. It is an old op en question whether the treewidth can b e a ppro xima ted within a co nstan t factor. T reewidth is known to b e fixed parameter tractable. Mor eo ver, for any fixed k , there is a linear time algo rithm due to Bo dlaender [3] computing the treewidth of gr aphs of tree width at most k . Unfortunately , h ug e hidden constants in the running time of Bo dlaender’s alg orithm is a serio us obstac le to its implemen- tation. F or small v alues of k , the classic a l algorithm of Arnbo r g, Corneil and Prosk ur o wski [1] from 19 87 w hich runs in time O ( n k +2 ) can be used to decide if the treewidth of a gra ph is at most k . The first exact algo rithm computing the tr e e width of an n - v ertex g raph is due to F omin et al. [15] and has running time O (1 . 9601 n ). Later these r esults were improv ed in [1 6, 27] to O (1 . 8899 n ). Both a lgorithms use exp onen tial space. The fas test po lynomial spa c e a lgorithm for treewidth prior to this work is due to Bo dlaender et a l. [5] and runs in time O (2 . 9512 n ). Our resu l ts. W e in tro duce a new (expo nen tial space) a lgorithm computing the treewidth o f a graph G on n vertices in time O (1 . 7 549 n ) and a p olyno- mial spa ce a lgorithm computing the treewidth in time O (2 . 61 51 n ). W e also show that if the treewidth of G is at most k , then it can be computed in time O (( 2 n + k +1 3 ) k +1 · k n 6 ). This is a r efinemen t of the clas sical result of Arn b org et al. Running times of all these algo rithms strongly dep end on p ossibilities of fast enum er ation of sp ecific structures in a gra ph, namely , p oten tial maximal cliques, and minimal sepa r ators [5, 8, 9, 15 , 27]. The new combinatorial lemma is cr ucial in obtaining new combinatorial bo unds and en umer a tion algorithms for mini- mal separa to rs a nd p oten tial ma ximal cliques, which, in tur n, pr o vides faster algorithms for tr eewidth. Similar improvemen ts in r unning times from O (1 . 8899 n ) to O (1 . 7549 n ) can be obtained for a num b er of re s ults in the liter ature o n problems r elated to treewidth (we skip definitions her e). F or example, by co m bining the ideas from [15] it is p ossible to c o mpute the fill-in o f a gr aph in time O (1 . 7549 n ). Another example are the treelength and the Chordal Sandwich problem [23] which also can b e solved in time O (1 . 7549 n ) b y ma k ing use of our technique. T reewidth computation and ext remal combinatorics 3 The remaining part of the pa per is organized as follows. In the next s ection we provide definitions and pr eliminary results. In Section 3, we pr o ve our main combinatorial to ol. By ma k ing use o f this to ol, in Section 4 , we prov e combi- natorial b ounds o n the num b er of minimal separato rs and p oten tial maxima l cliques a nd obta in algorithm enumerating these structure s . Thes e results form the ba sis for all our algorithms computing the tr eewidth of a graph pr esen ted in Sections 5, 6 , and 7. 2 Preliminaries W e denote by G = ( V , E ) a finite, undirected and simple graph with | V | = n vertices and | E | = m edges. F or any non-empty subset W ⊆ V , the subgr aph of G induced b y W is denoted by G [ W ]. W e s a y that a vertex se t S ⊆ V is c onne cte d if G [ S ] is connected. The neighb orho o d of a vertex v is N ( v ) = { u ∈ V : { u, v } ∈ E } a nd for a vertex set S ⊆ V we set N ( S ) = S v ∈ S N ( v ) \ S . A clique C of a graph G is a subset of V such that all the vertices of C are pairwise adjacent. Minimal separators. Let u and v b e tw o non a djacen t vertices of a gr aph G = ( V , E ). A set o f vertices S ⊆ V is a n u , v -sep ar ator if u a nd v are in different connected comp onen ts of the graph G [ V \ S ]. A connected compo nen t C of G [ V \ S ] is a ful l compo nen t a s socia ted to S if N ( C ) = S . S is a minimal u, v -sep ar ator of G if no pro per subset of S is an u, v -separ ator. W e say that S is a minimal s ep ar ator of G if there are tw o vertices u and v such that S is a minimal u, v -separator. Notice that a minimal separato r ca n b e str ictly included in another one. W e deno te by ∆ G the set of all minimal s eparators of G . W e need the following result due to Ber ry et al. [2] (se e also Klo ks et a l. [22 ]) Prop osition 1 ([2]). Ther e is an algorithm listing al l minimal sep ar ators of an input gr aph G in O ( n 3 | ∆ G | ) t ime. The following prop osition is an exercis e in [1 7]. Prop osition 2 (F olklore). A set S of vertic es of G is a m inimal a, b -sep ar ator if and only if a and b ar e in differ ent ful l c omp onents asso ciate d to S . In p artic- ular, S is a minimal sep ar ator if and only if ther e ar e at le ast two distinct ful l c omp onents asso ciate d to S . P otential maximal cliques . A graph H is chor dal (or triangulate d ) if every cycle of length at least four ha s a chord, i.e. an edge b et ween tw o non-consecutive vertices of the cycle. A triangulation of a graph G = ( V , E ) is a chordal gra ph H = ( V , E ′ ) such that E ⊆ E ′ . H is a m inimal triangulation if for any se t E ′′ with E ⊆ E ′′ ⊂ E ′ , the graph F = ( V , E ′′ ) is not chordal. A set of vertices Ω ⊆ V of a gr aph G is ca lled a p otential maximal clique if there is a minimal triang ulation H of G such that Ω is a maximal clique o f H . W e denote by Π G the set of all p otential maximal cliques of G . The following result on the structure of p oten tial maximal cliques is due to Bouchitt ´ e and T o dinca. 4 F edor V. F omin and Yngve Villanger Prop osition 3 ([8]). L et K ⊆ V b e a set of vertic es of the gr aph G = ( V , E ) . L et C ( K ) = { C 1 ( K ) , . . . , C p ( K ) } b e the set of the c onne cte d c omp onents of G [ V \ K ] and let S ( K ) = { S 1 ( K ) , S 2 ( K ) , . . . , S p ( K ) } wher e S i ( K ) , i ∈ { 1 , 2 , . . . , p } , is the set of those vertic es of K which ar e adjac ent to at le ast one vertex of the c omp onent C i ( K ) . Then K is a p otential maximal clique of G if and only if: 1. G [ V \ K ] has no ful l c omp onent asso ciate d to K , and 2. the gr aph on the vertex set K obtaine d fr om G [ K ] by c ompleting e ach S i ∈ S ( K ) into a clique, is a c omplete gr aph . The following result is a lso due to Bouchitt ´ e and T o dinca. Prop osition 4 ([8]). Ther e is an algorithm that, given a gr aph G = ( V , E ) and a set of vertic es K ⊆ V , verifies if K is a p otential maximal clique of G . The time c omplexity of the algorithm is O ( nm ) . T ree width. A tr e e de c omp osition of a gr aph G = ( V , E ) is a pair ( χ, T ) in which T = ( V T , E T ) is a tree a nd χ = { χ i | i ∈ V T } is a family of subsets of V such that: (1) S i ∈ V T χ i = V ; (2) for ea c h edge e = { u , v } ∈ E there ex ists a n i ∈ V T such that b oth u and v belo ng to χ i ; and (3) for all v ∈ V , the set of no des { i ∈ V T | v ∈ χ i } forms a connected subtree of T . T o distinguish betw een vertices of the o riginal g r aph G and vertices o f T , we call vertices of T no des and their corresp onding χ i ’s b ags . The maximum size of a bag minus one is called the width o f the tree dec o mposition. The tr e ewidth of a gra ph G , tw ( G ), is the minim um width over all p ossible tree decomp ositions of G . An alterna tiv e definition of treewidth is via minimal tria ngulations. The tr e ewidth of a g raph G is the minimum of ω ( H ) − 1 taken over all tria ngula- tions H of G . (By ω ( H ) we denote the maximum clique-size of a graph H .) Our alg orithm for treewidth is based o n the following result. Prop osition 5 ([15] ). Ther e is an algorithm t hat, given a gr aph G to gether with the list of its minimal sep ar ators ∆ G and the list of its p otential maximal cliques Π G , c omputes t he tr e ewidth of G in O ( n 3 ( | Π G | + | ∆ G | ) time. Mor e over, the algorithm c onstructs an optimal triangulation for the tr e ewidth. 3 Com binatorial Lemma The following lemma is our main co mbinatorial to ol. Lemma 1 (M ain Lem m a). L et G = ( V , E ) b e a gr aph. F or every v ∈ V , and b, f ≥ 0 , the numb er of c onne cte d vertex subsets B ⊆ V such that ( i ) v ∈ B , ( ii ) | B | = b + 1 , and ( iii ) | N ( B ) | = f is at most  b + f b  . T reewidth computation and ext remal combinatorics 5 Pr o of. Let v b e a vertex o f a graph G = ( V , E ). F or b + f = 0 Lemma trivially holds. W e pro ceed by inductio n ass uming that for some k > 0 and ev er y b and f suc h that b + f ≤ k − 1, Lemma holds. F or b and f such that b + f = k w e define B as the set of sets B sa tisfying ( i ) , ( ii ) , ( iii ). W e claim that |B | ≤  b + f b  . Since the claim a lways holds for b = 0, let us assume that b > 0. Let N ( v ) = { v 1 , v 2 , . . . , v p } . F o r 1 ≤ i ≤ p , we define B i as the s et of a ll connected s ubsets B such that – V ertices v , v i ∈ B , – F or every j < i , v j 6∈ B , – | B | = b + 1, – | N ( B ) | = f . Let us no te, that every set B satisfying the conditions of the lemma is in some set B i for some i , and that for i 6 = j , B i ∩ B j = ∅ . Therefore, |B | = p X i =1 |B i | . (1) F or every i > f + 1, |B i | = 0 (this is beca use for every B ∈ B i , the se t N ( B ) contains v er tices v 1 , . . . , v i − 1 and thus is of size at lea st f + 1.) Thus (1) can be rewritten as fo llo ws |B | = f +1 X i =1 |B i | . ( 2 ) Let G i be the graph o btained from G by contracting edge { v , v i } (r emo ving the lo op, r educe do uble edges to single edg es, and calling the new v er tex by v ) and r emo ving vertices v 1 , . . . , v i − 1 . Then the car dinalit y o f B i is eq ua l to the nu mber of the connected vertex subsets B o f G i such that – v ∈ B , – | B | = b , – | N ( B ) | = f − i + 1. By the induction assumption, this n umber is at most  f + b − i b − 1  and (2) yields that |B | = f +1 X i =1 |B i | ≤ f +1 X i =1  f + b − i b − 1  =  b + f b  . ⊓ ⊔ The inductive pro of of the Main L e mma can b e eas ily turned int o a recursive po lynomial spac e enumeration algo rithm (we skip the pro of here). Lemma 2. Al l vertex sets of size b + 1 with f neighb ors in a gr aph G c an b e enumer ate d in time O ( n  b + f b  ) by making us e of p olynomial sp ac e. 6 F edor V. F omin and Yngve Villanger 4 Com binatorial b ounds In this section we provide combinatorial b ounds on the num b er of minimal sep- arator s and p otential maximal cliq ue s in a gra ph. B o th b o unds are o btained b y applying the Main Lemma o n the resp ectice problems . 4.1 Minimal separators Theorem 1. L et ∆ G b e the set of al l minimal sep ar ators in a gr aph G on n vertic es. Then | ∆ G | = O (1 . 6181 n ) . Pr o of. F or 1 ≤ i ≤ n , le t f ( i ) be the num ber of all minimal sepa r ators in G of size i . Then | ∆ G | = n X 1 f ( i ) . (3) Let S be a minimal s e parator of size αn , where 0 < α < 1 . By Prop osition 2 , there exists tw o full comp onents C 1 and C 2 asso ciated to S . Let us a ssume that | C 1 | ≤ | C 2 | . Then | C 1 | ≤ (1 − α ) n/ 2. F rom the definition of a full comp onent C 1 asso ciated to S , we have tha t N ( C 1 ) = S . Thus, f ( αn ) is a t most the n umber of connected vertex sets C o f size at most (1 − α ) n/ 2 with neig h b orho ods o f s iz e | N ( C ) | = αn . Hence, to b ound f ( αn ) we c a n use the Main Lemma for every vertex of G . By Le mma 1, we hav e that for every vertex v , the num b er o f full comp onen ts of size b + 1 = (1 − α ) n/ 2 cont a ining v and with neig h b orho ods of size αn is at most  b + αn b  ≤  (1 + α ) n/ 2 b  . Therefore f ( αn ) ≤ n · (1 − α ) n/ 2 X i =1  i + αn i  < n · (1 − α ) n/ 2 X i =1  (1 + α ) n/ 2 i  . (4) F or α ≤ 1 / 3, we hav e (1 − α ) n/ 2 X i =1  (1 + α ) n/ 2 i  < 2 (1+ α ) n/ 2 < 2 2 n/ 3 < 1 . 59 n , and thus n/ 3 X i =1 f ( i ) = O (1 . 59 n ) . (5) F or α ≥ 1 / 3, one can use the well known fact tha t the sum P ⌊ j / 2 ⌋ k =1  j − k k  is equal to the ( j + 1 )-st Fib o nacci n umber to show that T reewidth computation and ext remal combinatorics 7 (1 − α ) n/ 2 X i =1  (1 + α ) n/ 2 i  < n · ϕ n , where ϕ = (1 + √ 5) / 2 < 1 . 6181 n is the go lden ratio. Therefore, n X i = n/ 3 f ( i ) = O (1 . 6181 n ) . (6) Finally , the theorem follows from the formulas (3 ),(5) and (6). ⊓ ⊔ 4.2 P otential maximal cliques Definition 1 ([ 8]). L et Ω b e a p otential m ax imal clique of a gr aph G and let S ⊂ Ω b e a minimal sep ar ator of G . We say that S is an active separa to r for Ω , if Ω is not a clique in the gr aph obtaine d fr om G by c ompleting al l the minimal sep ar ators c ontaine d in Ω , exc ept S . A p otential maximal clique Ω c ontaining an active sep ar ator (for Ω ) is c al le d a nice p otential maximal clique . W e need the following r esult by Bouchitt ´ e and T o dinca. Prop osition 6 ([9]). L et Ω b e a p otential maximal clique of G = ( V , E ) , let u b e a vertex of G , and let G ′ = G [ V \ { u } ] . Then one of the fol lowing holds: 1. Either Ω , or Ω \ { u } is a p otential maximal clique of G ′ ; 2. Ω = S ∪ { u } , wher e S is a minimal sep ar ator of G ; 3. Ω is a nic e p otential maximal clique. Let Π n be the maximum num b er of nice p oten tial ma x imal cliques that can be contained in a gra ph on n vertices. Pr opositio n 6 is useful to b ound the num b er of p oten tial ma ximal cliques in a gr aph by the num ber of minimal separato rs ∆ G and Π n . Lemma 3. F or any gr aph G = ( V , E ) , | Π G | ≤ n ( n | ∆ G | + Π n ) . Pr o of. Let v 1 , v 2 , ..., v n be an or dering of V and let V i = S i j =1 v j . The pro of of the lemma follows fro m the following claim Π G [ V i +1 ] ≤ Π G [ V i ] + n | ∆ G | + Π n which can be pr o ved by making inductive use o f Pro position 6. ⊓ ⊔ Definition 2. L et Ω ∈ Π G , v ∈ Ω , and C v b e the c onne cte d c omp onent of G [ V \ ( Ω \ { v } )] c ontaining v . We c al l the p air ( C v , v ) by vertex re pr esen tation of Ω . Lemma 4. L et ( C v , v ) b e a vertex r epr esentation of Ω . Then Ω = N ( C v ) ∪ { v } . 8 F edor V. F omin and Yngve Villanger Pr o of. By Prop osition 3, ev er y v er tex u ∈ Ω \ { v } , is either adjacent to v , or there exists a connected comp onen t C of G [ V \ Ω ] such that u, v ∈ N ( C ). Since C ⊂ C v , we hav e that Ω \ { v } ⊆ N ( C v ). E v ery connected comp onent C of G [ V \ Ω ] that contains v ∈ N ( C ) is co n tained in C v and N ( C ) ⊂ Ω for every C , therefor e Ω \ { v } = N ( C v ). ⊓ ⊔ W e need als o the following result from [27]. Prop osition 7 ([27] ). L et Ω b e a nic e p otential maximal clique of size αn in a gr aph G . Ther e ex ists a vertex r epr esentation ( C v , v ) of Ω such that | C v | ≤ ⌈ 2(1 − α ) n 3 ⌉ . Now everything is settled to apply Main Lemma. Lemma 5. The numb er of nic e p otential maximal cliques in a gr aph G = ( V , E ) is O (1 . 7549 n ) . Pr o of. By Pr o position 7, for every nice p otential ma ximal clique Ω o f car dinalit y αn , there ex ists a vertex representation ( C v , v ) of Ω such tha t | C v | ≤ ⌈ 2 n (1 − α ) / 3 ⌉ . Let b + 1 b e the num b er of vertices in C v . By Lemma 1, for every vertex v , the num b er o f such pairs ( C v , v ) is at most 2(1 − α ) n/ 3 X i =1  (2 + α ) n/ 3 i  . As in the pr oof of Theo rem 1, for α ≤ 2 / 5 the a bov e sum is O (1 . 7 549 n ). F or α ≥ 2 / 5, by making use of the fact that P ⌊ j / 2 ⌋ k =1  j − k 2 k  is equal to the ( j + 1)-s t nu mber of the sequence { a i } ∞ i =0 such that a i = 2 a i − 1 − a i − 2 + a i − 3 , w ith a 0 = 0 , a 1 = 1, and a 2 = 2, it is pos s ible to show that the v a lue of the ab o ve sum, and th us the num b er o f nice po ten tial maximal cliques, is O (1 . 7549 n ). ⊓ ⊔ By combining Lemma 3, 5 a nd The o rem 1 we arr iv e at the main result of this subsection. Theorem 2. F or any gr aph G , | Π G | = O (1 . 754 9 n ) . 5 Exp onen tial space exact algorithm for treewidth Our a lgorithm computing the treewidth of a graph is ba sed on Prop osition 5. By making us e o f Pr opositio n 5 we need to know how to list minimal se pa rators and p oten tial maximal c liques. By Pro position 1 a nd Theorem 1, all minimal separato r s ca n b e listed in time O (1 . 6 181 n ). The pro of o f the following lemma is p o stponed till the full version of this pa per. Lemma 6. F or any gr aph G on n vertic es, the set of p otential m ax imal cliques c an b e liste d in O (1 . 7549 n ) t ime. As a n immediate corolla r y o f Prop osition 1 a nd Lemma 6, w e hav e the fol- lowing result. Theorem 3. The tr e ewidth of a gr aph on n vertic es c an b e c ompute d in time O (1 . 7549 n ) . T reewidth computation and ext remal combinatorics 9 6 Computing treewidth at most k In this sectio n we s ho w how the lemma bounding the num b er of connected comp onen ts can be use d to refine the clas sical result of Arnborg et a l. [1]. By Pro p osition 5 , the treewidth of a gra ph c a n b e computed in O ( n 3 ( | Π G | + | ∆ G | )) time if the list of all minimal separators ∆ G and the list o f all p otential maximal c liq ues Π G of G are g iven. Actually , the res ults of Pr opositio n 5 can be str e ng thened (with almost the same pro of as in [16]) as follows. Let ∆ G [ k ] be the set of minimal separa tors and let Π G [ k ] b e the s et of po tential maximal cliques of size a t most k . Lemma 7. Given a gr aph G with sets ∆ G [ k ] and Π G [ k + 1 ] , it c an b e de cide d in time O ( n 3 ( | Π G [ k + 1 ] | + | ∆ G [ k ] | )) if the tr e ewidth of G is at most k . Mor e over, if the t r e ewidth of G is at m ost k , an optimal tre e de c omp osition c an b e c onstructe d within the same time. By Le mma 2 and Equatio n (4), | ∆ G [ k ] | ≤ k n · ( n − k ) / 2 X i =1  ( n + k ) / 2 i  ≤ k n 2 ·  ( n + k ) / 2 k  , (7) and it is p ossible to list a ll vertex subsets containing all separators from ∆ G [ k ] in time O ( k n 2 ·  ( n + k ) / 2 k  )) . F or each such a subset one ca n chec k in time O ( n 2 ) if it is a minimal separa tor or not, and th us all minimal separa tors of siz e a t most k ca n b e listed in time O ( k n 4 ·  ( n + k ) / 2 k  ) . Let Π n [ k ] be the maximum num b er of nice p otential ma x imal cliques of s ize at most k that can b e in a gr aph on n vertices. By P ropo sition 7, | Π n [ k ] | ≤ k n · ( n − k )2 / 3 X i =1  (2 n + k ) / 3 i  ≤ k n 2 ·  (2 n + k ) / 3 k  , and b y making use of Prop osition 4, all nice potential maximal cliques o f size at most k ca n b e listed in time O ( k n 5 ·  (2 n + k ) / 3 k  ). Finally , we use nice p otential maximal cliq ue s and minimal separator s o f size k to g e ne r ate all p oten tial maximal cliques of size a t most k . Lemma 8. F or every gr aph G on n vertic es, | Π G [ k ] | ≤ n ( | ∆ G [ k ] | + Π n [ k ]) and al l p otential maximal cliques of G of size at most k c an b e liste d in time O ( k n 6 ·  (2 n + k ) / 3 k  ) . Pr o of. Let v 1 , v 2 , ..., v n be an ordering o f V and let V i = S i j =1 v j . By Pro po- sition 6 and Lemma 3, every po ten tial maxima l clique of G [ V i ] either is a nice po ten tial maximal clique of G [ V i ], or is a p otential maximal cliq ue of G [ V i − 1 ], or is obtained by adding v i to a minimal separ ator or a p otential ma ximal clique of G [ V i − 1 ]. This yields that | Π G [ k ] | ≤ n ( | ∆ G [ k ] | + Π n [ k ]). T o list all po ten tial maximal cliques, for ea c h i , 1 ≤ i ≤ n , we list all minimal separ ators and nice 10 F edor V. F omin and Yngve Villanger po ten tial maximal cliques in G [ V i ]. This can b e done in time O ( kn 6 ·  (2 n + k ) / 3 k  ). The total n umber of all such structures is a t most k n 3 ·  (2 n + k ) / 3 k  . B y making use of dynamic pro graming, one ca n chec k if adding v i to a minimal separ ator or po ten tial maxima l clique of G [ V i − 1 ] creates a p otent ia l ma ximal clique in G [ V i ], which by P ropo sition 4 ca n be done in time O ( n 3 ). Th us, dynamic pro gramming can b e done in O ( k n 6 ·  (2 n + k ) / 3 k  ) steps. ⊓ ⊔ Now putting Lemma 7, Lemma 8 and Equation (7) together, we obtain the main result of this sectio n. Theorem 4. Ther e exists an algorithm that for a given gr aph G and inte ger k ≥ 0 , either c omputes a tr e e de c omp osition of G of the minimum width, or c orr e ctly c oncludes that the tr e ewidth of G is at le ast k + 1 . The running time of this algorithm is O ( k n 6 ·  (2 n + k +1) / 3 k +1  ) = O ( kn 6 · ( 2 n + k +1 3 ) k +1 ) . Pr o of. By the pr evious discussions in this section we ca n list all the minimal separato r s a nd p oten tial maximal cliques o f size at most k + 1 in O ∗ (  (2 n + k ) / 3 k  ) time. These minimal separa tors and p otent ia l maximal cliques are then used as input to the dyna mic progra mming algorithm of [1 5 ]. ⊓ ⊔ 7 P olynomial space exact algorit hm for treewidth The algo rithm used in Prop osition 1 r equires exp onen tial spa c e beca use it is based on dy namic programming which keeps a table with all p otential maximal cliques. As a consequence our O (1 . 7549 n ) time alg orithm for computing the treewidth also us e s O (1 . 7 5 49 n ) space. When r estricting to p olynomial spa ce, we ca nnot store all the minimal sep- arator s and a ll the p oten tial maximal cliques . The idea used to a void this is to search for a “central” p otent ia l ma ximal clique or a minimal separa to r in the graph which can sa fely b e completed into a clique. A similar idea is us ed in [5], how ever the improv ement in the running time of our algo rithm, is due to the following lemma and the technique used for listing minimal separa tors. Bo th results are, again, ba sed on the Main Lemma. Lemma 9. F or a given gr aph G = ( V , E ) and 0 < α < 1 , one c an list in time O ( mn 2 · 2 n (1 − α ) ) and p olynomial sp ac e al l p otent ial maximal cliques of G such that for every p otential maximal clique Ω fr om this list, ther e is a c onne cte d c omp onent of G [ V \ Ω ] of size at le ast αn . Pr o of. Let Ω b e a p otent ia l max imal clique sa tis fying the co nditions o f the lemma, a nd let C b e the co nnected comp onen t o f size at lea st αn . By Pro position 3, N ( C ) is a minimal separa tor contained in Ω a nd Ω \ N ( C ) 6 = ∅ . Let ( C u , u ) be a vertex r epresentation o f Ω , wher e u ∈ Ω \ N ( C ). Since u is not adjacent to any vertex in C , we hav e that C u ∩ C = ∅ . T o find Ω , we try to find its v er tex representation by a connected vertex set such that the clos ed neig h b orho od of T reewidth computation and ext remal combinatorics 11 this set is of size a t most n (1 − α ). By the Main Lemma, the num b er of such sets is at most n · n (1 − α ) X i =1  n (1 − α ) i  = n · 2 n (1 − α ) , and by Lemma 2, all these sets can be listed in O ( n · 2 n (1 − α ) ) steps and within po lynomial s pace. Finally , for each s et we use Lemma 4 a nd P ropo sition 4 to chec k in time O ( mn ) if the set is a p oten tial maxima l clique. ⊓ ⊔ W e also use the following result which is a slight mo dification of the res ult from [5 ], where it is stated in terms of elimination o rderings. Prop osition 8 ([5]). F or a given gr aph G = ( V , E ) and a clique K ⊂ V , ther e exists a p olynomial sp ac e algorithm, that c omputes the optimum tr e e de c omp osi- tion ( χ, T ) of G , subje ct to the c ondition that t he vertic es of K form a b ag which is a le af of T . This algorithm runs in time O ∗ (4 n −| K | ) . Theorem 5. The t r e ewidth of a gr aph G = ( V , E ) c an b e c ompute d in O (2 . 6151 n ) time and p olynomial sp ac e. Pr o of. It is well known (and follows from the prop erties of cliq ue trees of chordal graphs), that there is an optimal tre e deco mposition ( χ, T ), { χ i : i ∈ V T } , T = ( V T , E T ), o f G , where every bag is a p otential maximal clique [8, 10, 19]. Among all the bags of χ , le t χ i be a bag such tha t the largest connected comp onent of G [ V \ χ i ] is o f minimum size, i.e. χ i is a bag with the minimu m v alue of max {| C | : C is a connected comp onen t of G [ V \ χ i ] } , where minimum is taken over all bags of χ . Let C i be the connected comp onent of G − χ i of max imum size. Our further s tr ategy dep ends on the s iz e of | C i | . Let us a s sume first that | C i | < 0 . 38 6 85 n . In this case , b y Lemma 9, the s et of p o ten tial maximal cliques S such tha t for every Ω ∈ S the maximum size of a comp onent of G [ V \ Ω ] is | C i | , can be listed in time O ( mn 2 · 2 n −| C i | ) and polyno mial space . Since χ i ∈ S , we hav e that there is a p oten tia l ma ximal clique Ω ∈ S such that tw ( G Ω ) = tw ( G ), where G Ω is obtained from G by turning Ω into a cliq ue . The tr eewidth of G Ω is equal to the maximum of minimum width of decomp ositions of G Ω [ C ∪ Ω ] with Ω fo r ming a leaf ba g, where C is a connected comp onent of G Ω [ V \ Ω ]. Let us remind that the size o f each such comp onen t is at most | C i | . By Pr opositio n 8, the optimu m width of G Ω [ C ∪ Ω ] for every co nnected comp onen t C o f G Ω [ C ∪ Ω ] (and with Ω fo rming a lea f bag) can b e computed in O ∗ (4 | C | ) = O ∗ (4 | C i | ), time and thus the treewidth of G can be found in time O ∗ (2 n −| C i | · 4 | C i | ) = O ∗ (2 (1 − 0 . 38685) n · 4 0 . 38685 n ) = O (2 . 6151 n ) . Thu s if | C i | < 0 . 38 685 n , we compute the tr eewidth of G , and the running time of this p olynomial space pro cedure is O (2 . 6151 n ). 12 F edor V. F omin and Yngve Villanger Let us consider the c a se | C i | ≥ 0 . 386 85 n . F or each connected comp onen t C o f G [ V \ χ i ], there exists a bag χ i ′ ⊂ N ( C ) ∪ C and a minimal sepa rator S = χ i ∩ χ i ′ in χ i that separates C from the rest of the graph. Let S = χ i ∩ χ j be the separato r in χ i that s eparates C i from the rest o f the gr aph. Let G S be the gr aph obtained fro m G by tur ning S into a clique. Then tw ( G S ) = tw ( G ). T o compute the treewidth of G S we compute the minim um width of deco mpositions of G S [ C ∪ S ] with S forming a leaf bag, where C is a connected comp onen t of G S [ V \ S ], and then take the maximum of these v a lues. By the definition of χ i , there exists a connected comp onent C j of G [ V \ χ j ], such that | C j | ≥ | C i | . By Pro position 3, χ j 6⊆ χ i . Thus χ j \ χ i 6 = ∅ , and the s ize of every connected comp onent in G [ C i \ χ j ] is at most | C i | − 1 . F urthermo re, since S = χ i ∩ χ j , we have that every connected co mponent of G [ C i \ χ j ] is a lso a connected c o mponent of G [ V \ χ j ]. This yie lds that C j ∩ C i = ∅ and that b oth C i and C j are full connected comp onents a ssosiated to S . Thus | C j | + | C i | ≤ n − | S | . Every connected comp onent of G [ V \ S ], ex c e pt C i , is a connected comp onent of G [ V \ χ j ]. Because | C i | ≤ | C j | , this implies that C j is the largest compo nen t of G [ V \ S ]. Bo th C i and C j contain at leas t 0 . 3868 5 n vertices, thus the size of S is at mo s t n (1 − 2 · 0 . 3 8685) = 0 . 226 3 n . By the a lgorithmic version o f Main Lemma, all s ets of s uc h size (and which for m the neighborho o d of a set of size | C i | ) can b e listed in po lynomial spac e and time O ( nm · 0 . 2263 n X p =1  | C i | + p p  ) . By Pr opos ition 8, we can compute the minimum width o f decomp ositions o f G S [ C ∪ S ] with S forming a leaf ba g, wher e C is a connected comp onent of G S [ V \ S ], in time O ∗ (4 | C | ) = O ∗ (4 | C j | ) and p olynomial space. Because | C j | ≤ n − | S | − | C i | , we have tha t for | S | = p , O ∗ (4 | C j | ) = O ∗ (4 n −| C i |− p ) . Thu s to compute the treewidth of G S (and the treewidth of G ), we list all sets S and for each such a set w e use Prop osition 8 for all graphs G S [ C ∪ S ]. The running time o f this pro cedure is O ∗ ( 0 . 2263 n X p =1  | C i | + p p  · 4 n −| C i |− p ) . By V a ndermonde’s identit y , we hav e tha t  | C i | + p p  = p X k =0  0 . 3868 5 n + p k  | C i | − 0 . 386 85 n k  < p X k =0  0 . 3868 5 n + p k  2 | C i |− 0 . 38685 n . T reewidth computation and ext remal combinatorics 13 Thu s 0 . 2263 n X p =1  | C i | + p p  · 4 n −| C i |− p < 0 . 2263 n X p =1 p X k =0  0 . 3868 5 n + p k  2 | C i |− 0 . 38685 n · 4 n −| C i |− p ≤ 0 . 2263 n X p =1 p  0 . 3868 5 n + p p  · 2 2((1 − 0 . 38685) n − p ) = O (2 . 6151 n ) T o conclude, if | C i | ≥ 0 . 38685 n , we compute the tre ewidth of G in p olynomial space within O (2 . 615 1 n ) steps. ⊓ ⊔ Ac kno wle dgemen t. W e are grateful to Saket Saurabh for many useful com- men ts, and to the ano n ymous referee p oin ting out that one of the b ounds matched the g olden ratio . References 1. S. Arn bor g, D. G . Corneil, and A. Pro skuro wski , Com pl exity of finding emb e ddings in a k -tr e e , SIA M J. Algebraic Discrete Metho ds, 8 ( 198 7), pp . 277– 284. 2. A. B err y, J. P. Borda t, and O. Cogis , Gener ating al l the minimal sep ar ators of a gr aph. , Int. J. F ound. Comput. Sci., 11 (2000), pp. 397–40 3. 3. H. L. 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