New Algorithms for Enumerating All Maximal Cliques

1. New Algorithms for Enumerating All Maximal Cliques Kazuhisa Makino Takeaki Uno Osaka University National Institute of JAPAN Informatics, JAPAN 9/Jul/2004 SWAT 2004

2. Background Recently, Enumeration algorithms are interesting ・There are still many unsolved nice problems (unlike to ordinal discrete algorithms) ・Recent increase of computer power makes many enumeration problems practically solvable  many applications have been appearing, such as, genome, data mining, clustering, so on ・Some (theoretical) algorithms use enumeration as subroutines (recognition of perfect graph)

3. Background (cont.) ・My institute has 100 researchers of informatics ・ At least 5 researchers (independently) use implementations of enumeration algorithms ・Suppose that there are 100,000 researchers of informatics in the world 5000 researchers use enumeration algorithms ?????

4. Problems and Results Problem1 : for a given graph G=(V, E), enumerate all maximal cliques in G Problem2 : for a given bipartite graph G=(V1∪V2, E), enumerate all maximal bipartite cliques in G ( Problem2 is a special case of Problem1 ) ・ We propose algorithms for solving these problems, reduce the time complexity in dense cases and sparse cases. ・ Computational experiments for random graphs and real-world data

5. Difficulty ・ Consider branch-and-bound type enumeration: divide maximal cliques into two groups maximal cliques includingv / not includingv ・ If a group includes no maximal clique,  cut off the branch  Finding a maximal clique not including given vertices of S is NP-Complete  Can not cut off subproblems(branches) including no maximal clique v1∈K v1∈K v2∈K v2∈K

6. Existing Studies and Ours O(|V||E|): Tsukiyama, Ide, Ariyoshi & Shirakawa, O(|V||E|),lexicographic order: Johnson, Yanakakis & Papadimitriou O(a(G)|E|): Chiba & Nishizeki ( a(G): arboricity of Gwith m/(n-1)≦a(G) ≦m1/2 ) ・ many heuristic algorithms in data mining, for bipartite case Ours: O(|V|2.376) (dense case) O(Δ4) (sparse case) O((Δ*)4 + θ3 ) (θ vertices have degree >Δ* ) O(Δ3) (bipartite case) O(Δ2) (bipartite case with using much memory)

7. 9 4 10 7 3 6 8 Enumeration of Maximal Cliques ・Improved version of algorithm of Tsukiyama et. al. Idea: Construct a route on all maximal cliques to be traversed ・ For a maximal clique K of G = ( V, E ): C (K) : lexicographically maximum maximal clique including K K≦i: vertices of K with indices ≦i i(K) :minimum index s.t. C(K≦i) =C(K≦i+1) parent of a maximal clique K : C(K≦i(K)-1) ・parent is lexicographically larger than K Lexicographically larger 9 4 1 11 7 1,2,3>1,2,4 3 10 1,3,6>1,4,5 2 K 6 8 i(K) 5

8. Graph Representation of Relation ・Parent-child relation is acyclic graph representation forms atree (enumeration tree) Visit all maximal cliques by depth-first search ・need to find children of a maximal clique

9. 10 9 4 K[8] 8 Child of Maximal Clique Γ(vi): vertices adjacent to vi K[i] = C ( K≦i∩Γ(vi)∪ {vi} ) ・ H is a child of K only if H = K[i] for some i>i(K) (H is a child of K if the parent of K[i] is K ) ・ i(K[i]) = i ・construct K[i] in O(|E|) time ・construct parent in O(|E|) time ( O(Δ2 ) time) ・for i=i(K)+1,…,|V| in O(|V||E|) time enumerate O(|V||E|) time per maximal clique K,i(K)=6 9 4 1 11 7 3 10 2 6 8 5

10. 5 1 4 K≦5∪ Characterization of Child The parent of K[i]=K⇔ (1) no vj , j<i is adjacent to all vertices in K≦i∩Γ(vi) ∪ {vi} (2) no vj , j<i is adjacent to all vertices in K≦i∩Γ(vi) ∪ K≦j (1) is not satisfied ⇔K[i] and parent of K[i] includes vj∈K (2) is not satisfied ⇔ parent of K[i] includes vj∈K K = {3,4,7,9} K[10] = {3,7,10} K≦5= {3,4} K ≦7∩Γ(v10) = {3,7} 7 4 9 3 10 K ≦10∩Γ(v10) ∪ {v10}

11. Use of Matrix Multiplication ・ Check the conditions (1) and (2) by matrix multiplication (1) no vj , j<i is adjacent to all vertices in K ≦i∩Γ(vi) ∪ {vi} ith row of left ⇒K≦i∩Γ(vi)∪{vi} jth column of right ⇒Γ(vj) ij cell of product ⇒ |K≦i∩Γ(vi)∪{vi} ∩Γ(vj) | = |K≦i∩Γ(vi)∪{vi}| ? Γ(vj) ∩ K ≦i∩Γ(vi) ∪ {vi} K≦i∩Γ(vi)∪{vi} Γ(vj) Condition (2) can be checked in the same way Checked in O(|V|2.368 ) time ⇒ time complexity is O(|V|2.368 ) for each

12. O((Δ*)4 + |Θ|3 ) if partially dense Δ*: max. degree in V＼Θ Sparse Cases ・If vi is adjacent to no vertex in K K[i] = C ( K≦i∩Γ(vi)∪ {vi} ) = C ({vi}) parent of K[i] = C ( C ({vi}) ≦i ) If C ({vi}) ≦i＝φ,parent of K[i] is K0 If C ({vi}) ≦i≠φ,(1) is not satisfied If K≠ K0,K[i] is not a child of K ・ Since |K|≦Δ+1 , at most Δ(Δ+1) vertices are adjacent to K ・ Each K[i] takes O(Δ2) time to construct the parent Δ: max. degree O(Δ4 ) per maximal clique

13. Bipartite Clique ・ Enumerate maximal bipartite cliques in G =(V1 ∪V2 ,E ) ( = maximal cliques in G’ =(V1 ∪V2 , E ∪V1 ×V1∪V2×V2))  enumerated in O(|V|2.368 ) time for each ・ But a sparse bipartite graph will be dense  need some improvements for sparse cases V1 V2

14. K[i] vi Fast Construction of K[i] ・ For any maximal bipartite clique K K∩V2= ∩v∈K∩V1Γ(v) K∩V1= ∩v∈K∩V2Γ(v) ・K[i]∩V1for all i are computed in O(Δ2) time ・K[i]for all i are computed in O(Δ3) time K[v1] K[v6] V1 1 2 3 4 V2

15. K[i] vi Checking the Parent ・・・ V1 1 2 3 |V1|-1 |V1| ・ Put small indices to V1 , large indices to V2 K[i] is a child of K ⇔ K[i]≦i = K≦i checked in O(Δ)time V2 ・・・ |V1|+1 |V1|+2 V1 V2 Enumerated in O(Δ3) time for each O(Δ2) by using memory

16. Computational Experiments ・ for graphs randomly generated ・ vertex viis connected to vertices from i-rto i+rwith probability 1/2 ・ Faster than Tsukiyama’s algorithm ・ Computation time is linear in maximum degree

17. Benchmark Problems ・ Problem of finding frequent closed item sets from database  equivalent to maximal bipartite clique enumeration ・ Used on KDDcup (data mining algorithm competition ) BMS-WebView1　 (from Web-log data) 　　　 |V|=60,000, ave. degree2.5 BMS-WebView2　(from Web-log data) 　　　 |V|=80,000, ave. degree5 BMS-POS(from POS data) 　　　|V|=510,000, ave. degree 6 IBM-Artificial　 (artificial data) 　　　|V|= 100,000, ave.degree10

18. Results

19. Conclusion and Future Work ・ Proposed fast algorithms for enumerating maximal cliques: O(|V|2.376), O(Δ4 ), O((Δ*)4 + θ3 ) maximal bipartite cliques: O(|V|2.376), O(Δ3 ), O(Δ2) ・ Examined benchmark problems of data mining, and showed that our algorithm performs well. Future work: ・ Can we improve more? What is the difficulty ? ・ Can we enumerate other maximal (minimal) graph objects ? ・ Can we apply matrix multiplication to other enumeration problems ? ・ What can be enumerated efficiently in practice ?

20. Frequent Sets customer1 customer2 customer3 customer4 beer nappy milk Input graph: An item and a customer is connected iff the customer purchased the item In a maximal bipartite clique: Customers: have similar favorites Items: frequently purchased together [Agrawal et al. 96, Zaki et al. 02, Pei 00, Han 00, … ]

21. Few Large Degree Vertices ・Very few vertices (denoted by Θ) have large degrees ・Divide the maximal cliques into two groups: (a) cliques not included in Θ (b) cliques included in Θ ・(a) can be enumerated in O(Δ’4) time ・ Maximal clique K in the induced graph by Θ is a maximal clique of G⇔K is not included in any of (a)  O(|Θ|3) timefor each small degree < Δ’ large degree O(Δ’4 + |Θ|3 ) per maximal clique

22. Avoid Duplications by Using Memory ・We can avoid duplications by storing all maximal bipartite cliques ・ From K∩V1=Γ(K∩V2) ,we store all K∩V1 1. Get a K from memory (which is un-operated) 2. generate all K[i]∩V1 3. Store each K[i]∩V1 if it is not in memory 4. Go to 1 if a maximal clique is un-operated Enumerated in O(Δ2) time for each