An Efficient Algorithm for Enumerating Pseudo Cliques

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An Efficient Algorithm for Enumerating Pseudo
Cliques

• Takeaki Uno
• National Institute of Informatics
• The Graduate University for Advanced Studies

Dec/18/2007 ISAAC, Sendai
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Introducing Pseudo Cliques
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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
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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)
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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
ave. ratio of vertices adjacent to a vertex
For given ?, K is a pseudo clique ? (density of
K) ? ?
We want to solve the problem of
enumerating all pseudo cliqus of the given graph
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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 ? 1
• Approximation algorithms for maximizing the
  density for size k
• - O(V1/3-e) approaximation algorithm
• - O((n/k)e) approx. if optimal solution is dense
  Tokuyama el al.
• - PTAS if O(n2) edges Arora et al.
• Many heuristic algorithms in data mining, data
  engineering, natural sciences
• However, no algorithm for "complete"
  enumeration

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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

The existence of pseudo clique is NP-comp.
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Proof of the Hardness
Theorem 1

• For given graph G, threshold ?, and vertex
  set U, the problem of checking the existence of a
  pseudo clique including U is NP-complete

Proof reducing the problem of clique of k
vertices
input graph G(V,E)
only (U clique) is pseudo clique density
increases by increase of pseudo clique
size setting es.t. clique of size at least k
induces a pseudo clique
density
V2 -1 V2
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Is This Really Hard?

• We proved NP-hardness for "very dense graphs"
• ? unclear for middle dense graph
• ? possibility for polynomial time enumeration

hard
easy
?????
easy
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Polynomial Time Enumeration
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Reverse Search Approach

• Introduce an acyclic parent-child relation on
  all pseudo cliques

objects
Enumeration by traversing the tree induced by the
relation
Need an algorithm for listing up all children
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Parent of Pseudo Clique

• v(K) min. deg. min. index vertex in GK
• The parent of pseudo clique K ? K\v(K)

The parent of K
K
Density of K ave. degree GK /
(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
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Ex. Enumeration Tree

• threshold .7

1
2
4
5
3
7
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Finding Children

• A child is obtained by adding a vertex to the
  parent
• degK(v) vertices in K adjacent to v
• (can be maintained in O(?) time for vertex
  addition)
• K?v is a child of K ?
• ? K?v is a pseudo clique ? lower bound for
  degK(v)
• ? v(K?v) v ? upper bound for degK(v)
• - degK(v) lt min. deg. of K ? K?v is
  always a child
• - degK(v) gt min. deg. of K 1 ? K?v never be
  a child
• degK(v) min. deg. of K or 1 ? next slide

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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(?2) time

top of S(K) is v(K)
Computation time for one iteration is O(?2 log
V) ( O(?k log V) if k-degenerate)
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Computational Experiments
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Implementation

• Code is a simple version
• - update degK(vi) at each addition
• ? adding u to K takes O(deg(u)) time
• - to find children, vi satisfying
• ?K(K1) - (edges in K) ? degK(vi)
  ? d(K)1
• ? O( C d(K)) O(E) time O(1) time
  for each
• C vertices vi, degK(vi) d(K),
  d(K)1

Seems to be not large for children
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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
  ik with probability 1/2
• - graphs generated from real-world data
• (co-author graph)

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Random Graphs

• p 0.1, vertices 200 to 2000, threshold 0.8,
  0.9

Computation time linearly increase as ave. degree
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Locally Dense Random Graph

• make edge from a vertex to its neighbors with
  p0.5
• vertices 100 to 25600, threshold 0.8, 0.9

10 times slower than clique enumeration
computation time per one clique does not change
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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
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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
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Bottom-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!

Long time
Short time
When pseudo cliques are sufficiently large (over
5?) min. degree is small on average ? computation
time is short on average at lower levels
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Conclusion

• 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