On the cubicity of bipartite graphs

On the Cubi cit y of Bipartite Graphs L. Sunil Chandran ∗ , Anita Das † , Na v een Siv adasan ‡ ∗ ∗ , † Computer Science and Automation departmen t, Indian I nstitute of Science Bangalore- 560012, India. { sunil, anita } @csa.iisc.ernet.in ‡ Adv anced T ec hnolo gy Cen ter, T ata Consultancy Services 1, Soft w a re Units La y out, Madhapur Hyderabad - 500081, India s.na v een@atc.tcs.com Abstract A unit cub e in k -dimension (or a k -cub e) is define d as the c artesian pr o duct R 1 × R 2 × · · ·× R k , wher e e ach R i is a close d interval on t he r e al line of the form [ a i , a i + 1] . The cubicity of G , denote d as cub ( G ) , is t he minimum k such that G is the interse ction gr aph of a c ol le ction of k -cub es. Many NP-c omplete gr aph pr oblems c an b e solve d effic iently or h ave go o d appr oximation r atios in gr aphs of low cubicity. In most of these c ases the first step is to get a low dimensional cub e r epr esent ation of the given gr aph. It is known that for a gr aph G , cub ( G ) ≤  2 n 3  . Re c ently it has b e en shown that for a gr aph G , cub ( G ) ≤ 4 (∆ + 1) ln n , whe r e n and ∆ ar e the numb er of vertic es and max imu m de gr e e of G , r esp e ctively. In this p ap er, we show that for a bip artite gr aph G = ( A ∪ B , E ) with | A | = n 1 , | B | = n 2 , n 1 ≤ n 2 , and ∆ ′ = min { ∆ A , ∆ B } , wher e ∆ A = max a ∈ A d ( a ) and ∆ B = max b ∈ B d ( b ) , d ( a ) and d ( b ) b eing the de gr e e of a and b in G r esp e ctively, cu b ( G ) ≤ 2(∆ ′ + 2) ⌈ ln n 2 ⌉ . We also give an efficient r andomize d algori thm to c onstruct the cub e r epr esentation of G in 3(∆ ′ + 2) ⌈ ln n 2 ⌉ dimensions. The r e ader may n ote that in gener al ∆ ′ c an b e much smal ler than ∆ . Keyw ords: Cubicity , a lgorithms, intersection gra phs. 1 In tro duction Let F b e a family of non-empty sets. An und irected graph G is an in tersection graph for F if th ere exists a one-one corresp ond en ce b et ween the v ertices of G and th e sets in F su c h that t wo vertic es in G are adjacen t if and only if the corresp onding sets h a v e non-empt y in tersection. If F is a family of in terv als on real line, then G is calle d an interval gr aph . If F is a family of in terv als on real line suc h that all the in terv als are of equal length, then G is called a unit interval gr aph . A u nit cub e in k -dimens ional sp ace or a k -cub e is defined as th e cartesian pro duct R 1 × R 2 × · · · × R k , where eac h R i is a close d interv al on the real line of the form [ a i , a i + 1]. A k -cub e represent ation of a graph is a mapp ing of the v ertices of G to k -cub es suc h that t wo v ertices in G ∗ This research was funded by the DS T grant SR/S3/EECE/62/20 06 1 are adjacen t if and only if their corresp onding k -cub es hav e a non-empt y inte rsection. The cub i city of G is the minimum k such that G h as a k -cub e r epresen tation. Note that a k -cub e r epresen tation of G usin g cub es with u nit sid e length is equiv alen t to a k -cub e repr esentati on wh ere the cub es ha v e side length c for some fixed p ositiv e n u m b er c . The graph s of cubicit y 1 are exactly the class of un it int erv al graphs. The concept of cubicit y was in tro duced b y F. S. Rob erts [9] in 19 69. T his concept generaliz es the concept of u n it in terv al graph s. If w e requir e that eac h verte x of G corresp ond to a k -dimen s ional axis-parallel b o x R 1 × R 2 × · · · × R k , where eac h R i , 1 ≤ i ≤ k , is a closed in terv al of the form [ a i , b i ] on the real line, then the minimum dimension r equired to r epresen t G is calle d its b oxicity denoted as box ( G ). Clearly box ( G ) ≤ cub ( G ), for a graph G . It h as b een s ho wn that deciding whether the cubicit y of a giv en graph is at least three is NP-co mplete [11]. Compu ting the b o xicit y of a graph w as sh o w n to b e NP-hard b y C ozzens in [5]. This w as later stren gthened b y Y annak akis [11], and finally by Krato ch vil [6] w h o sho we d th at d eciding wh ether b o xicit y of a graph is at most t wo itself is NP-complete. Th us, it is interesting to design efficien t algorithms to r epresen t small cubicit y graphs in low dimension. There hav e b een man y attempts to b oun d the cub icit y of graph classes with sp ecial structure. The cub e and b ox represen tations of sp ecial classes of graphs lik e hyp ercub es and complete multipartite graphs w ere inv estigated in [1, 2, 3, 7, 8, 9, 10]. 1.1 Our results Recen tly Chandr an et al. [4] h a v e sho wn that for a graph G , cub ( G ) ≤ 4(∆ + 1) ln n , where n and ∆ are the num b er of v ertices and maximum degree of G , r esp ectiv ely . In this pap er, we p r esen t an efficien t randomized algorithm to construct a cub e r epresen tation of bip artite gr aphs in lo w dimension. In particular, we sho w that for a b ipartite graph G = ( A ∪ B , E ), cub ( G ) ≤ 2(∆ ′ + 2) ⌈ ln n 2 ⌉ , wh ere | A | = n 1 , | B | = n 2 , n 1 ≤ n 2 , and ∆ ′ = min { ∆ A , ∆ B } , where ∆ A = max a ∈ A d ( a ) and ∆ B = max b ∈ B d ( b ), d ( a ) and d ( b ) b eing th e degree of a and b in G , r esp ectiv ely . T h e algorithm present ed in this p ap er is not v ery different from that of [4] bu t this h as the adv an tage that it giv es a b etter result in th e case of b ipartite graph s. Note that, ∆ ′ can b e m uc h smaller than ∆ in general, wh er e ∆ is the maximum degree of G . In p articular, when | A | ≪ | B | , then the b ound for cubicit y giv en in this pap er can b e muc h b etter than that giv en in [4]. Also, the complexit y of our algorithm is comparable with the complexit y of the algorithm prop osed in [4]. 2 Preliminaries Let G = ( A ∪ B , E ) b e a simp le, fi n ite b ipartite graph . Let | A | = n 1 , | B | = n 2 , and n 1 ≤ n 2 . Let N ( v ) = { w ∈ V ( G ) | v w ∈ E ( G ) } b e the set of n eigh b ors of v . Degree of a verte x v , denoted as d ( v ), is defin ed as the num b er of edges inciden t on v . That is, d ( v ) = | N ( v ) | . Supp ose ∆ A denote the maximum degree in A and ∆ B denote the m axim um d egree in B . Th at is, ∆ A = m ax a ∈ A d ( a ) and ∆ B = max b ∈ B d ( b ). F or a graph G , let G ′ b e a graph such th at V ( G ′ ) = V ( G ). Then, G ′ is a sup er gr aph of G if E ( G ) ⊆ E ( G ′ ). W e defin e the i nterse ction of t wo graphs as f ollo ws: if G 1 and G 2 are t wo graphs suc h that V ( G 1 ) = V ( G 2 ), then the in tersection of G 1 and G 2 denoted as G = G 1 ∩ G 2 is a graph with V ( G ) = V ( G 1 ) = V ( G 2 ) and E ( G ) = E ( G 1 ) ∩ E ( G 2 ). 2 Let I 1 , I 2 , . . . , I k b e k unit interv al graphs suc h that G = I 1 ∩ I 2 ∩ · · · ∩ I k , then I 1 , I 2 , . . . , I k is called an unit interval gr aph r epr esentation of G . The follo wing equiv alence is well kno wn . Theorem 2.1 ([9]) . The minimum k such that ther e exists a unit interval gr aph r epr esentation of G using k unit interval gr aphs I 1 , I 2 , . . . , I k is the same as cub ( G ) . 3 Construction Let G = ( A ∪ B , E ) b e a b ipartite graph. In this s ection w e describ e an algorithm to efficien tly compute a cub e represent ation of G in 2(∆ ′ + 2) ⌈ ln n 2 ⌉ dimensions, w here ∆ ′ = min { ∆ A , ∆ B } . Definition 3.1. L et π b e a p ermutation of the set { 1 , 2 , . . . , n } and X ⊆ { 1 , 2 , . . . , n } . Th e pr o- je ction of π onto X denote d as π X is define d as fol lows. L et X = { u 1 , u 2 , . . . , u r } b e such that π ( u 1 ) < π ( u 2 ) < . . . < π ( u r ) . Then π X ( u 1 ) = 1 , π X ( u 2 ) = 2 , . . . , π X ( u r ) = r . Definition 3.2. A gr aph G = ( V , E ) is a unit interval gr aph if and only if ther e exists a function f : V − → R and a c onstant c such that ( u, v ) ∈ E ( G ) if and only if | f ( u ) − f ( v ) | ≤ c . Remark: Note that the ab ov e defin ition is consisten t with the definition of the u nit inte rv al graphs giv en at the b eginning of the intro d uction. Let G = ( A ∪ B , E ) b e a bipartite graph . Giv en a p erm utation of the vertic es of A , we construct a unit in terv al graph U ( π , A, B , G ) as follo ws . Let f : A ∪ B − → R b e such that if v ∈ A , then f ( v ) = π ( v ) and if v ∈ B , then f ( v ) = n + min x ∈ N ( v ) π ( x ). Tw o v ertices u, v ∈ A ∪ B are mad e adjacen t if and only if | f ( u ) − f ( v ) | ≤ n , where n = | A | + | B | = n 1 + n 2 . Claim 1: Let G ′ = U ( π , A, B , G ). Then G ′ is a sup ergraph of G . Pr o of. Su pp ose ( a, b ) ∈ E ( G ). Without loss of generalit y supp ose a ∈ A and b ∈ B . Let s = min x ∈ N ( b ) π ( x ). So, f ( b ) = n + s . As f ( a ) = π ( a ) and a ∈ N ( b ), π ( a ) ≥ s . Therefore we h av e, | f ( b ) − f ( a ) | = n + s − π ( a ) ≤ n . Thus ( a, b ) ∈ E ( G ′ ). Hence G ′ is a sup ergraph of G . Remark: Note that if we rev erse the roles of A and B in the ab ov e constru ction, i.e., if w e start with a p erm utation of the v ertices of B rather th an that of A , then the resulting un it interv al graph will b e denoted as U ( π , B , A, G ). Clearly , U ( π , B , A, G ) will also b e a sup er graph of G . RANDUNIT Input: A bipartite graph G = ( A ∪ B , E ). Output: A unit interv al graph G ′ whic h is a sup ergraph of G . b egin if (∆ B ≤ ∆ A ) then Step 1. Generate a p erm utation π of { 1 , 2 , . . . , n 1 } (the vertice s of A ) uniformly at random. Step 2. Return G ′ = U ( π , A, B , G ). else Step 1. Generate a p erm utation π of { 1 , 2 , . . . , n 2 } (the vertice s of B ) uniformly at random. 3 Step 2. Return G ′ = U ( π , B , A, G ). end Lemma 3.1. L et a ∈ A and b ∈ B b e such that e = ( a, b ) / ∈ E ( G ) . L et G ′ b e the output of RANDUNIT( G ) . Then Pr[ e ∈ E ( G ′ )] ≤ ∆ ′ ∆ ′ +1 . Pr o of. Case I: ∆ B ≤ ∆ A . Let π b e a p ermutat ion of the v ertices in A . Let G ′ = U ( π , A, B , G ). Supp ose t w o vertice s a ∈ A and b ∈ B are non-adjacent in G . Let t = min x ∈ N ( b ) π ( x ). Claim: The v ertices a and b will b e adjacen t in G ′ if and only if π ( a ) > t . If a and b are adjacen t in G ′ , th en w e ha v e | f ( b ) − f ( a ) | = | ( n + t ) − π ( a ) | ≤ n , i.e., π ( a ) > t . Hence a is adj acen t to b in G ′ . So, Pr[ e ∈ E ( G ′ )] = Pr[ π ( a ) > t ] = 1 − Pr[ π ( a ) < t ]. (Note that π ( a ) 6 = t , since a / ∈ N ( b ).) Let X = { a } ∪ N ( b ) and π X b e the pro jection of π on X . T otal num b er of p ermutati ons of X is ( d ( b ) + 1)!. No w , it can b e easily seen that π ( a ) < t if and only if π X ( a ) = 1. Thus, Pr[( a, b ) ∈ E ( G ′ )] = 1 − d ( b )! ( d ( b ) + 1)! = d ( b ) d ( b ) + 1 ≤ ∆ ′ ∆ ′ + 1 Hence the lemma. Case I I: ∆ A ≤ ∆ B . Let π b e the p erm utation of the v ertices in B . Let G ′ = U ( π , B , A, G ). Pro of is similar to case I. Lemma 3.2. Given a bip artite gr aph G = ( A ∪ B , E ) , ther e exists a sup e r gr aph G ∗ of G with cub ( G ∗ ) ≤ 2(∆ ′ + 1) ln n 2 , such tha t if u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ) , then ( u, v ) / ∈ E ( G ∗ ) . Pr o of. Let U 1 , U 2 , . . . , U t b e the unit inte rv al graphs generated by t inv o cations of RANDUNIT( G ) . Clearly U i , for eac h i , 1 ≤ i ≤ t , is a sup er grap h of G b y Claim 1. Let G ∗ = U 1 ∩ U 2 ∩ · · · ∩ U t . No w let u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ). T hen, Pr[( u, v ) ∈ G ∗ ] = Pr   ^ 1 ≤ i ≤ t ( u, v ) ∈ E ( U i )   ≤  ∆ ′ ∆ ′ +1  t (Applying Lemma 3.1). No w, P r   _ u ∈ A,b ∈ B , ( u,v ) / ∈ E ( G ) ( u, v ) ∈ E ( G ∗ )   < n 1 n 2  ∆ ′ ∆ ′ + 1  t ≤ n 2 2  1 − 1 ∆ ′ + 1  t ≤ n 2 2 e − t ∆ ′ +1 4 If t = 2(∆ ′ + 1) ln n 2 the ab o ve probabilit y is < 1. Th us we infer that there exists a su p er graph G ∗ of G su c h that if u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ), ( u, v ) / ∈ E ( G ∗ ) also. F rom the defin ition of G ∗ w e hav e cub ( G ∗ ) ≤ 2(∆ ′ + 1) ln n 2 . Hence the Lemma. Remark: If w e had c hosen t = 3(∆ ′ + 1) ln n 2 in the ab o v e pr o of, w e can substant ially reduce the failure pr obabilit y . More precisely we can get Pr( G ∗ do es not s atisfy th e desired p rop erty ) ≤ 1 n 2 No w w e will construct tw o sp ecial graphs H 1 and H 2 suc h that H i is a su p er graph of G for i = 1 , 2. Definition 3.3. L et A = { v 1 , v 2 , . . . , v n 1 } . F or 1 ≤ i ≤ ⌈ ln n 1 ⌉ define the fu nction f i : A ∪ B → R as fol lows: F or vertic es fr om A , f i ( v j ) = 0 if the i th bit of j i s 0 f i ( v j ) = 2 if the i th bit of j i s 1 F or vertic es in u ∈ B , f i ( u ) = 1 L et I i b e the unit interval g r aph define d on the vertex set A ∪ B such that two vertic es u and v ar e adjac ent if and only if | f i ( u ) − f i ( v ) | ≤ 1 . Now define H 1 = T ⌈ ln n 1 ⌉ i =1 I i . Thus we have cub ( H 1 ) ≤ ⌈ ln n 1 ⌉ . Definition 3.4. L et B = { u 1 , u 2 , . . . , u n 2 } . F or 1 ≤ i ≤ ⌈ ln n 2 ⌉ define the function g i : A ∪ B → R as fol lows: F or vertic es fr om B , g i ( u j ) = 0 if the i th bit of j i s 0 g i ( u j ) = 2 if the i th bit of j i s 1 F or vertic es in v ∈ A , g i ( v ) = 1 L et J i b e the unit interval gr aph define d on the v e rtex set A ∪ B suc h that two vertic es u and v ar e adjac ent if and only if | g i ( u ) − g i ( v ) | ≤ 1 . Now define H 2 = T ⌈ ln n 2 ⌉ i =1 J i . Thus cub ( H 2 ) ≤ ⌈ ln n 2 ⌉ . Lemma 3.3. H 1 is a sup er gr aph of G such that if u, v ∈ A , then ( u, v ) / ∈ E ( H 1 ) . Pr o of. It is easy to chec k that I i is a sup er graph of G for eac h i . T h us H 1 is clearly a sup er graph of G . F or u, v ∈ A , let u = v j and v = v k where k 6 = j . Then clearly there exists a t , 1 ≤ t ≤ ⌈ ln n 1 ⌉ suc h that j and k differs in the t th bit p osition. Now it is easy to v erify that u and v will not b e adjacen t in I t . It follo ws that for an y pair ( u, v ) where u, v ∈ A there exists I t suc h that ( u, v ) / ∈ E ( I t ). Then clearly ( u, v ) / ∈ E ( H 1 ) also. Hence the Lemma. Lemma 3.4. H 2 is a sup er gr aph of G such that if u, v ∈ B , then ( u, v ) / ∈ E ( H 2 ) . Pr o of. Th e pro of is similar to that of the Lemma 3.3. 5 Theorem 3.5. Given a bip artite gr aph G = ( A ∪ B , E ) , cub ( G ) ≤ 2(∆ ′ + 2) ⌈ ln n 2 ⌉ . Pr o of. By Lemma 3.2, there exists a sup er graph G ∗ of G suc h that if u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ), then ( u, v ) / ∈ E ( G ∗ ). Also let H 1 and H 2 b e the sup er graph s of G , fr om definitions 3.3 and 3.4 resp ectiv ely . No w we claim that G = G ∗ ∩ H 1 ∩ H 2 . Cleary G ∗ ∩ H 1 ∩ H 2 is a sup er graph of G , b ecause eac h of them is a sup er graph of G . Now to see that G ∗ ∩ H 1 ∩ H 2 = G w e only need to prov e that if ( u, v ) / ∈ G , th en ( u, v ) is not an edge of at least one of these three graphs. No w, if u ∈ A and v ∈ B , ( u, v ) / ∈ E ( G ∗ ) b y Lemma 3.2. If u, v ∈ A , then ( u, v ) / ∈ E ( H 1 ) by Lemma 3.3 and if u, v ∈ B , then ( u, v ) / ∈ E ( H 2 ) by L emma 3.4. No w , cub ( G ) = cub ( G ∗ ∩ H 1 ∩ H 2 ) ≤ cub ( G ∗ ) + cub ( H 1 ) + cub ( H 2 ). By Lemma 3.2 cub ( G ∗ ) ≤ 2(∆ ′ + 1) ln n 2 . Also by the definition of H 1 and H 2 w e h a ve cub ( H 1 ) ≤ ⌈ ln n 1 ⌉ and cub ( H 2 ) ≤ ⌈ ln n 2 ⌉ . Th us we ha v e, cub ( G ) ≤ 2(∆ ′ + 1) ln n 2 + ⌈ ln n 1 ⌉ + ⌈ ln n 2 ⌉ ≤ 2(∆ ′ + 1) ln n 2 + 2 ⌈ ln n 2 ⌉ as n 1 ≤ n 2 = 2(∆ ′ + 2) ⌈ ln n 2 ⌉ Hence the theorem. Remark: In view of the Remark after Lemma 3.2, we can infer that if t ≥ 3(∆ ′ + 1) ln n 2 , G = G ∗ ∩ H 1 ∩ H 2 with h igh probability . But then the cu b e repr esen tation outp u t by the algorithm will b e in 3(∆ ′ + 1) ln n 2 + ⌈ ln n 2 ⌉ + ⌈ ln n 1 ⌉ ≤ 3(∆ ′ + 2) ⌈ ln n 2 ⌉ dimensions. The follo wing T h eorem giv es the time complexit y of our rand omized algorithm to constru ct suc h a cub e representa tion. Theorem 3.6. L et G = ( A ∪ B , E ) b e a bip artite gr aph with n = n 1 + n 2 vertic es, m e dges and let ∆ ′ = m in { ∆ A , ∆ B } . Then, with high pr ob ability, the cub e r epr esentation of G in 3(∆ ′ + 2) ⌈ ln n 2 ⌉ dimensions c an b e gener ate d in O (∆ ′ ( m + n ) ln n 2 ) time. Pr o of. W e assu me that a random p ermutatio n π on n 1 v ertices can b e compu ted in O ( n 1 ) time. Recall th at we assign n inte rv als to n v ertices as follo ws. If v ∈ A , then w e assign the interv al [ π ( v ) , n + π ( v )] to v . If v ∈ B , then let t = min x ∈ N ( v ) π ( x ). No w, the inte rv al [ t + n, t + 2 n ] is giv en to the v ertex v . Since n umb er of edges in the graph m = 1 2 P u ∈ A ∪ B d ( u ), one inv o cation of RANDUNIT( G ) needs O ( m + n ) time. S ince w e need to inv oke the algorithm RANDUNIT( G ) O (∆ ′ ln n 2 ) times, the o v erall alg orithm that generates the cub e representat ion in 3(∆ ′ + 2) ⌈ ln n 2 ⌉ dimensions ru ns in O (∆ ′ ( m + n ) ln n 2 ) time References [1] S. Bellan toni, Irith Ben-Arro y o Hartman, T. M. Przytyc k a, and S. Whitesides. Grid intersec- tion graphs and b o xicit y Discr ete Mathematics , 114 (1-3):41 –49, 1993. [2] L. S. C handran, C. Mannino, and G. Oriolo. On the cubicit y of certain graphs. Inform. Pr o c ess. L ett. , 94(3):113 –118, 2005. [3] L. Sunil Chand ran and Na veen Siv adasan. The cubicit y of h yp ercub e graph s. Discr ete Math- ematics , 308(23) :5795–5800 , 2008. 6 [4] L. Sun il Chandr an, Mathew C. F r ancis, an d Nav een Siv adasan. Represen ting graphs as the in tersection of axis-parallel cub es. Man uscript, av ailable at htt p://arxiv.org/abs/cs/06 07092 . [5] M. B. Cozzens. Higher and multidimensional analo gues of interval gr aphs . 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