An Efficient Algorithm for Enumerating Pseudo Cliques Dec/18/2007 ISAAC, Sendai Takeaki Uno National Institute of Informatics & The Graduate University for Advanced Studies

Introducing Pseudo Cliques

Analyzing Large Scale Database By rapid growth of database size, we have to analyze databases in some computational way Finding cliques in similarity/relation graphs is a popular way to classify the data, or get characterizations of the data Group of similar or related objects Thanks to good properties such as monotonicity, (maximal) cliques can be enumerated very quickly (up to 1,000,000/sec) ・ ・ Now, we are motivated to find more rich object, dense structures, such as pseudo cliques

Finding Cliques in Graph Clique: a complete subgraph (complete bipartite subgraph  bipartite clique) Group of similar or related objects Often used for finding clusters or groups Graphs in practice are usually sparse but locally dense, scale free, and satisfy small world property Simple Backtacking (Branch-and-Bound) works well, because of the monotone property (polynomial time for each) Practically very fast even for maximal ones (up to 1,000,000/sec)

Def. Pseudo Clique For a vertex set K, the density of K is (#edges connecting vertices in K) (|K|-1)|K| /2 -  - K is a clique  density is 1 -  - K is an independent set  density is 0   if density is high, K is nearly a clique maximum #edges in S We want to solve the problem of enumerating all pseudo cliqus of the given graph pseudo clique  For given θ, K is a pseudo clique  (density of K) ≧ θ ave. ratio of vertices adjacent to a vertex

Existing Results Easy to find one pseudo clique   two connected vertices always form a pseudo clique Finding a pseudo clique of size k is NP-complete   Reducing k-clique problem by setting θ= 1 Approximation algorithms for maximizing the density for size k - - O(|V| 1/3-ε ) approaximation algorithm - - O((n/k) ε ) approx. if optimal solution is dense [Tokuyama el al.] - - PTAS if Ω(n 2 ) edges [Arora et al.] Many heuristic algorithms in data mining, data engineering, natural sciences However, no algorithm for "complete" enumeration

Hardness for Branch-and-Bound A straightforward approach is branch and bound In each iteration, divide the problem into two non-empty problems by the inclusion of a vertex v 1, v 2 v1v1v1v1 v1v1v1v1 The existence of pseudo clique is NP-comp. The existence of pseudo clique is NP-comp.

Proof of the Hardness For given graph G, threshold θ, and vertex set U, the problem of checking the existence of a pseudo clique including U is NP-complete Theorem 1 Proof: reducing the problem of clique of k vertices input graph G=(V,E) input graph G=(V,E) Add 2|V| 2 vertices as U density = |V| 2 -1 |V| 2 only (U + clique) is pseudo clique density increases by increase of pseudo clique size setting εs.t. clique of size at least k induces a pseudo clique |V| 2 -1 |V| 2 θ= +ε

Is This Really Hard? We proved NP-hardness for "very dense graphs"   unclear for middle dense graph   possibility for polynomial time enumeration θ= 1 θ= 0 easy hard ??????????

Polynomial Time Enumeration

Reverse Search Approach Introduce an acyclic parent-child relation on all pseudo cliques Need an algorithm for listing up all children objectsobjects Enumeration by traversing the tree induced by the relation

Parent of Pseudo Clique v*(K) : min. deg. min. index vertex in G[K]  The parent of pseudo clique K  K ＼ v*(K) K The parent of K = Density of K = ave. degree G[K] / (|K|-1) The parent is the removal of most "sparse" vertex from K, thus is a pseudo clique  The parent is smaller than its child  acyclic relation

Ex. Enumeration Tree threshold =

Finding Children A child is obtained by adding a vertex to the parent deg K (v): #vertices in K adjacent to v (can be maintained in O(Δ) time for vertex addition)  K ∪ v is a child of K  ①  ① K ∪ v is a pseudo clique  lower bound for deg K (v) ②  ② v*(K ∪ v) = v  upper bound for deg K (v) -  - deg K (v) < min. deg. of K  K ∪ v is always a child -  - deg K (v) > min. deg. of K +1  K ∪ v never be a child  deg K (v) ＝ min. deg. of K or +1  next slide…

Detailed Condition S(K): sequence of vertices in K in the order of (degree, index)  v is a child  v is the top of S(K ∪ v) v is child only if v is adjacent to all vertices preceding to v in S(K) For each vertex, find the first "non-adjacent vertex" in S(K) This can be done in O(Δ 2 ) time Computation time for one iteration is O(Δ 2 + log |V|) ( O(Δk + log |V|) if k-degenerate) Computation time for one iteration is O(Δ 2 + log |V|) ( O(Δk + log |V|) if k-degenerate) top of S(K) is v*(K)

Computational Experiments

ImplementationImplementation Code is a simple version - - update |deg K (v i )| at each addition   adding u to K takes O(deg(u)) time - - to find children, v i satisfying ≦≦ θ|K|(|K|+1) - (#edges in K) ≦ | deg K (v i )| ≦ d*(K)+1  =+  O( C d*(K)) = O(|E|) time + O(1) time for each :== C := #vertices v i, | deg K (v i )| = d*(K), d*(K)+1 Seems to be not large for #children

Problem Instances Pentium M 1.1GHz, 256MB memory, Cygwin, C, gcc Test instances are: - - random graphs (make edge with probability p), - - locally dense random graphs (vertex i is adjacent to vertices from i-k to i+k with probability 1/2 - - graphs generated from real-world data (co-author graph)

Random Graphs p= 0.1, #vertices = 200 to 2000, threshold 0.8, 0.9 Computation time linearly increase as ave. degree

Locally Dense Random Graph make edge from a vertex to its neighbors with p=0.5 #vertices 100 to 25600, threshold 0.8, times slower than clique enumeration computation time per one clique does not change 10 times slower than clique enumeration computation time per one clique does not change

Randomly Generated Scale Free Graph Add vertices of degree 10 iteratively, to a clique of 10 vertices Vertices to be connected are chosen according to their current degrees Computation time increases quite slowly

Real-world Instance co-author graph of academic paper database #vertices = 30,000, #edges = 125,000, scale free Computation time for one pseudo clique does not depend on threshold Computation time for one pseudo clique does not depend on threshold

Bottom-widenessBottom-wideness Why good in practice? The algorithm generates several recursive calls   recursion tree expands exponentially by going down   computation time is dominated by the lowest levels  On lower levels, small degree vertices are added  fast! When pseudo cliques are sufficiently large (over 5?) min. degree is small on average   computation time is short on average at lower levels When pseudo cliques are sufficiently large (over 5?) min. degree is small on average   computation time is short on average at lower levels ・・・ Long time Short time

ConclusionConclusion First polynomial delay polynomial space algorithm for enumerating pseudo cliques Hardness result for straight forward branch-and-bound Evaluate practical efficiency by computational experiments Future works: Explain the gap between theory and practice Introduce maximality and their enumeration Apply the technique to other structures (pseudo bla bla bla) (path, tree, bipartite clique, matching …) What is crucial for the compuation (enumeration) of structures with ambiguity