On the cubicity of bipartite graphs
aa r X i v : . [ c s . D M ] O c t On the Cubicity of Bipartite Graphs
L. Sunil Chandran ∗ , Anita Das † , Naveen Sivadasan ‡ ∗∗ , † Computer Science and Automation department, Indian Institute of ScienceBangalore- 560012, India. { sunil, anita } @csa.iisc.ernet.in ‡ Advanced Technology Center, Tata Consultancy Services1, Software Units Layout, MadhapurHyderabad - 500081, [email protected]
Abstract
A unit cube in k -dimension (or a k -cube) is defined as the cartesian product R × R ×· · ·× R k ,where each R i is a closed interval on the real line of the form [ a i , a i + 1] . The cubicity of G ,denoted as cub ( G ) , is the minimum k such that G is the intersection graph of a collection of k -cubes. Many NP-complete graph problems can be solved efficiently or have good approximationratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensionalcube representation of the given graph.It is known that for a graph G , cub ( G ) ≤ (cid:4) n (cid:5) . Recently it has been shown that for a graph G , cub ( G ) ≤ n , where n and ∆ are the number of vertices and maximum degree of G ,respectively. In this paper, we show that for a bipartite graph G = ( A ∪ B, E ) with | A | = n , | B | = n , n ≤ n , and ∆ ′ = min { ∆ A , ∆ B } , where ∆ A = max a ∈ A d ( a ) and ∆ B = max b ∈ B d ( b ) , d ( a ) and d ( b ) being the degree of a and b in G respectively, cub ( G ) ≤ ′ + 2) ⌈ ln n ⌉ . We also givean efficient randomized algorithm to construct the cube representation of G in ′ + 2) ⌈ ln n ⌉ dimensions. The reader may note that in general ∆ ′ can be much smaller than ∆ . Keywords:
Cubicity, algorithms, intersection graphs.
Let F be a family of non-empty sets. An undirected graph G is an intersection graph for F if thereexists a one-one correspondence between the vertices of G and the sets in F such that two verticesin G are adjacent if and only if the corresponding sets have non-empty intersection. If F is a familyof intervals on real line, then G is called an interval graph . If F is a family of intervals on real linesuch that all the intervals are of equal length, then G is called a unit interval graph .A unit cube in k -dimensional space or a k -cube is defined as the cartesian product R × R ×· · · × R k , where each R i is a closed interval on the real line of the form [ a i , a i + 1]. A k -cuberepresentation of a graph is a mapping of the vertices of G to k -cubes such that two vertices in G ∗ This research was funded by the DST grant SR/S3/EECE/62/2006 k -cubes have a non-empty intersection. The cubicity of G is the minimum k such that G has a k -cube representation. Note that a k -cube representationof G using cubes with unit side length is equivalent to a k -cube representation where the cubeshave side length c for some fixed positive number c . The graphs of cubicity 1 are exactly the classof unit interval graphs.The concept of cubicity was introduced by F. S. Roberts [9] in 1969. This concept generalizes theconcept of unit interval graphs. If we require that each vertex of G correspond to a k -dimensionalaxis-parallel box R × R × · · · × R k , where each R i , 1 ≤ i ≤ k , is a closed interval of the form [ a i , b i ]on the real line, then the minimum dimension required to represent G is called its boxicity denotedas box ( G ). Clearly box ( G ) ≤ cub ( G ), for a graph G . It has been shown that deciding whether thecubicity of a given graph is at least three is NP-complete [11]. Computing the boxicity of a graphwas shown to be NP-hard by Cozzens in [5]. This was later strengthened by Yannakakis [11], andfinally by Kratochvil [6] who showed that deciding whether boxicity of a graph is at most two itselfis NP-complete.Thus, it is interesting to design efficient algorithms to represent small cubicity graphs in lowdimension. There have been many attempts to bound the cubicity of graph classes with specialstructure. The cube and box representations of special classes of graphs like hypercubes andcomplete multipartite graphs were investigated in [1, 2, 3, 7, 8, 9, 10]. Recently Chandran et al. [4] have shown that for a graph G , cub ( G ) ≤ n , where n and∆ are the number of vertices and maximum degree of G , respectively. In this paper, we presentan efficient randomized algorithm to construct a cube representation of bipartite graphs in lowdimension. In particular, we show that for a bipartite graph G = ( A ∪ B, E ), cub ( G ) ≤ ′ +2) ⌈ ln n ⌉ , where | A | = n , | B | = n , n ≤ n , and ∆ ′ = min { ∆ A , ∆ B } , where ∆ A = max a ∈ A d ( a )and ∆ B = max b ∈ B d ( b ), d ( a ) and d ( b ) being the degree of a and b in G , respectively. The algorithmpresented in this paper is not very different from that of [4] but this has the advantage that itgives a better result in the case of bipartite graphs. Note that, ∆ ′ can be much smaller than ∆ ingeneral, where ∆ is the maximum degree of G . In particular, when | A | ≪ | B | , then the bound forcubicity given in this paper can be much better than that given in [4]. Also, the complexity of ouralgorithm is comparable with the complexity of the algorithm proposed in [4]. Let G = ( A ∪ B, E ) be a simple, finite bipartite graph. Let | A | = n , | B | = n , and n ≤ n . Let N ( v ) = { w ∈ V ( G ) | vw ∈ E ( G ) } be the set of neighbors of v . Degree of a vertex v , denoted as d ( v ), is defined as the number of edges incident on v . That is, d ( v ) = | N ( v ) | . Suppose ∆ A denotethe maximum degree in A and ∆ B denote the maximum degree in B . That is, ∆ A = max a ∈ A d ( a )and ∆ B = max b ∈ B d ( b ).For a graph G , let G ′ be a graph such that V ( G ′ ) = V ( G ). Then, G ′ is a super graph of G if E ( G ) ⊆ E ( G ′ ). We define the intersection of two graphs as follows: if G and G are two graphssuch that V ( G ) = V ( G ), then the intersection of G and G denoted as G = G ∩ G is a graphwith V ( G ) = V ( G ) = V ( G ) and E ( G ) = E ( G ) ∩ E ( G ).2et I , I , . . . , I k be k unit interval graphs such that G = I ∩ I ∩ · · · ∩ I k , then I , I , . . . , I k iscalled an unit interval graph representation of G . The following equivalence is well known. Theorem 2.1 ([9]) . The minimum k such that there exists a unit interval graph representation of G using k unit interval graphs I , I , . . . , I k is the same as cub ( G ) . Let G = ( A ∪ B, E ) be a bipartite graph. In this section we describe an algorithm to efficientlycompute a cube representation of G in 2(∆ ′ + 2) ⌈ ln n ⌉ dimensions, where ∆ ′ = min { ∆ A , ∆ B } . Definition 3.1.
Let π be a permutation of the set { , , . . . , n } and X ⊆ { , , . . . , n } . The pro-jection of π onto X denoted as π X is defined as follows. Let X = { u , u , . . . , u r } be such that π ( u ) < π ( u ) < . . . < π ( u r ) . Then π X ( u ) = 1 , π X ( u ) = 2 , . . . , π X ( u r ) = r . Definition 3.2.
A graph G = ( V, E ) is a unit interval graph if and only if there exists a function f : V −→ R and a constant c such that ( u, v ) ∈ E ( G ) if and only if | f ( u ) − f ( v ) | ≤ c . Remark:
Note that the above definition is consistent with the definition of the unit interval graphsgiven at the beginning of the introduction.Let G = ( A ∪ B, E ) be a bipartite graph. Given a permutation of the vertices of A , we constructa unit interval graph U ( π, A, B, G ) as follows. Let f : A ∪ B −→ R be such that if v ∈ A , then f ( v ) = π ( v ) and if v ∈ B , then f ( v ) = n + min x ∈ N ( v ) π ( x ). Two vertices u, v ∈ A ∪ B are madeadjacent if and only if | f ( u ) − f ( v ) | ≤ n , where n = | A | + | B | = n + n . Claim 1:
Let G ′ = U ( π, A, B, G ). Then G ′ is a supergraph of G . Proof.
Suppose ( a, b ) ∈ E ( G ). Without loss of generality suppose 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 have, | f ( b ) − f ( a ) | = n + s − π ( a ) ≤ n . Thus ( a, b ) ∈ E ( G ′ ). Hence G ′ is a supergraph of G . Remark:
Note that if we reverse the roles of A and B in the above construction, i.e., if we startwith a permutation of the vertices of B rather than that of A , then the resulting unit interval graphwill be denoted as U ( π, B, A, G ). Clearly, U ( π, B, A, G ) will also be a super graph of G . RANDUNIT
Input: A bipartite graph G = ( A ∪ B, E ).Output: A unit interval graph G ′ which is a supergraph of G . beginif (∆ B ≤ ∆ A ) thenStep 1. Generate a permutation π of { , , . . . , n } (the vertices of A )uniformly at random.Step 2. Return G ′ = U ( π, A, B, G ). else Step 1. Generate a permutation π of { , , . . . , n } (the vertices of B )uniformly at random. 3tep 2. Return G ′ = U ( π, B, A, G ). endLemma 3.1. Let a ∈ A and b ∈ B be such that e = ( a, b ) / ∈ E ( G ) . Let G ′ be the output of RANDUNIT( G ) . Then Pr[ e ∈ E ( G ′ )] ≤ ∆ ′ ∆ ′ +1 .Proof. Case I: ∆ B ≤ ∆ A .Let π be a permutation of the vertices in A . Let G ′ = U ( π, A, B, G ). Suppose two vertices a ∈ A and b ∈ B are non-adjacent in G . Let t = min x ∈ N ( b ) π ( x ). Claim:
The vertices a and b will be adjacent in G ′ if and only if π ( a ) > t .If a and b are adjacent in G ′ , then we have | f ( b ) − f ( a ) | = | ( n + t ) − π ( a ) | ≤ n , i.e., π ( a ) > t .Hence a is adjacent to b in G ′ .So, Pr[ e ∈ E ( G ′ )] = Pr[ π ( a ) > t ] = 1 − Pr[ π ( a ) < t ]. (Note that π ( a ) = t , since a / ∈ N ( b ).)Let X = { a } ∪ N ( b ) and π X be the projection of π on X . Total number of permutations of X is( d ( b ) + 1)!. Now, it can be 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 ≤ ∆ ′ ∆ ′ + 1Hence the lemma. Case II: ∆ A ≤ ∆ B .Let π be the permutation of the vertices in B . Let G ′ = U ( π, B, A, G ). Proof is similar to caseI. Lemma 3.2.
Given a bipartite graph G = ( A ∪ B, E ) , there exists a super graph G ∗ of G with cub ( G ∗ ) ≤ ′ + 1) ln n , such that if u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ) , then ( u, v ) / ∈ E ( G ∗ ) .Proof. Let U , U , . . . , U t be the unit interval graphs generated by t invocations of RANDUNIT( G ) .Clearly U i , for each i , 1 ≤ i ≤ t , is a super graph of G by Claim 1. Let G ∗ = U ∩ U ∩ · · · ∩ U t .Now let u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ). Then,Pr[( u, v ) ∈ G ∗ ] = Pr ^ ≤ i ≤ t ( u, v ) ∈ E ( U i ) ≤ (cid:16) ∆ ′ ∆ ′ +1 (cid:17) t (Applying Lemma 3.1). Now, P r _ u ∈ A,b ∈ B, ( u,v ) / ∈ E ( G ) ( u, v ) ∈ E ( G ∗ ) < n n (cid:18) ∆ ′ ∆ ′ + 1 (cid:19) t ≤ n (cid:18) − ′ + 1 (cid:19) t ≤ n e − t ∆ ′ +1 t = 2(∆ ′ + 1) ln n the above probability is <
1. Thus we infer that there exists a super graph G ∗ of G such that if u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ), ( u, v ) / ∈ E ( G ∗ ) also. From the definition of G ∗ we have cub ( G ∗ ) ≤ ′ + 1) ln n . Hence the Lemma. Remark:
If we had chosen t = 3(∆ ′ + 1) ln n in the above proof, we can substantially reduce thefailure probability. More precisely we can getPr( G ∗ does not satisfy the desired property ) ≤ n Now we will construct two special graphs H and H such that H i is a super graph of G for i = 1 , Definition 3.3.
Let A = { v , v , . . . , v n } . For ≤ i ≤ ⌈ ln n ⌉ define the function f i : A ∪ B → R as follows: For vertices from A , f i ( v j ) = 0 if the i th bit of j is 0 f i ( v j ) = 2 if the i th bit of j is 1For vertices in u ∈ B , f i ( u ) = 1 Let I i be the unit interval graph defined on the vertex set A ∪ B such that two vertices u and v areadjacent if and only if | f i ( u ) − f i ( v ) | ≤ .Now define H = T ⌈ ln n ⌉ i =1 I i . Thus we have cub ( H ) ≤ ⌈ ln n ⌉ . Definition 3.4.
Let B = { u , u , . . . , u n } . For ≤ i ≤ ⌈ ln n ⌉ define the function g i : A ∪ B → R as follows: For vertices from B , g i ( u j ) = 0 if the i th bit of j is 0 g i ( u j ) = 2 if the i th bit of j is 1For vertices in v ∈ A , g i ( v ) = 1 Let J i be the unit interval graph defined on the vertex set A ∪ B such that two vertices u and v areadjacent if and only if | g i ( u ) − g i ( v ) | ≤ .Now define H = T ⌈ ln n ⌉ i =1 J i . Thus cub ( H ) ≤ ⌈ ln n ⌉ . Lemma 3.3. H is a super graph of G such that if u, v ∈ A , then ( u, v ) / ∈ E ( H ) .Proof. It is easy to check that I i is a super graph of G for each i . Thus H is clearly a supergraph of G . For u, v ∈ A , let u = v j and v = v k where k = j . Then clearly there exists a t ,1 ≤ t ≤ ⌈ ln n ⌉ such that j and k differs in the t th bit position. Now it is easy to verify that u and v will not be adjacent in I t . It follows that for any pair ( u, v ) where u, v ∈ A there exists I t suchthat ( u, v ) / ∈ E ( I t ). Then clearly ( u, v ) / ∈ E ( H ) also. Hence the Lemma. Lemma 3.4. H is a super graph of G such that if u, v ∈ B , then ( u, v ) / ∈ E ( H ) .Proof. The proof is similar to that of the Lemma 3.3.5 heorem 3.5.
Given a bipartite graph G = ( A ∪ B, E ) , cub ( G ) ≤ ′ + 2) ⌈ ln n ⌉ .Proof. By Lemma 3.2, there exists a super graph G ∗ of G such that if u ∈ A , v ∈ B and ( u, v ) / ∈ E ( G ), then ( u, v ) / ∈ E ( G ∗ ). Also let H and H be the super graphs of G , from definitions 3.3 and3.4 respectively. Now we claim that G = G ∗ ∩ H ∩ H . Cleary G ∗ ∩ H ∩ H is a super graph of G , because each of them is a super graph of G . Now to see that G ∗ ∩ H ∩ H = G we only needto prove that if ( u, v ) / ∈ G , then ( u, v ) is not an edge of at least one of these three graphs. Now, if u ∈ A and v ∈ B , ( u, v ) / ∈ E ( G ∗ ) by Lemma 3.2. If u, v ∈ A , then ( u, v ) / ∈ E ( H ) by Lemma 3.3and if u, v ∈ B , then ( u, v ) / ∈ E ( H ) by Lemma 3.4.Now, cub ( G ) = cub ( G ∗ ∩ H ∩ H ) ≤ cub ( G ∗ ) + cub ( H ) + cub ( H ). By Lemma 3.2 cub ( G ∗ ) ≤ ′ + 1) ln n . Also by the definition of H and H we have cub ( H ) ≤ ⌈ ln n ⌉ and cub ( H ) ≤⌈ ln n ⌉ . Thus we have, cub ( G ) ≤ ′ + 1) ln n + ⌈ ln n ⌉ + ⌈ ln n ⌉≤ ′ + 1) ln n + 2 ⌈ ln n ⌉ as n ≤ n = 2(∆ ′ + 2) ⌈ ln n ⌉ Hence the theorem.
Remark:
In view of the Remark after Lemma 3.2, we can infer that if t ≥ ′ + 1) ln n , G = G ∗ ∩ H ∩ H with high probability. But then the cube representation output by the algorithmwill be in 3(∆ ′ + 1) ln n + ⌈ ln n ⌉ + ⌈ ln n ⌉ ≤ ′ + 2) ⌈ ln n ⌉ dimensions. The following Theoremgives the time complexity of our randomized algorithm to construct such a cube representation. Theorem 3.6.
Let G = ( A ∪ B, E ) be a bipartite graph with n = n + n vertices, m edges and let ∆ ′ = min { ∆ A , ∆ B } . Then, with high probability, the cube representation of G in ′ + 2) ⌈ ln n ⌉ dimensions can be generated in O (∆ ′ ( m + n ) ln n ) time.Proof. We assume that a random permutation π on n vertices can be computed in O ( n ) time.Recall that we assign n intervals to n vertices as follows. If v ∈ A , then we assign the interval[ π ( v ) , n + π ( v )] to v . If v ∈ B , then let t = min x ∈ N ( v ) π ( x ). Now, the interval [ t + n, t + 2 n ] isgiven to the vertex v . Since number of edges in the graph m = P u ∈ A ∪ B d ( u ), one invocation of RANDUNIT( G ) needs O ( m + n ) time. Since we need to invoke the algorithm RANDUNIT( G ) O (∆ ′ ln n ) times, the overall algorithm that generates the cube representation in 3(∆ ′ + 2) ⌈ ln n ⌉ dimensions runs in O (∆ ′ ( m + n ) ln n ) time References [1] S. Bellantoni, Irith Ben-Arroyo Hartman, T. M. Przytycka, and S. Whitesides. Grid intersec-tion graphs and boxicity
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