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Introduction
Algorithms
Application
Conclusion

The Maximum Clique Problem

Dam Thanh Phuong, Ngo Manh Tuong

November, 2012

Dam Thanh Phuong, Ngo Manh Tuong The Maximum Clique Problem

Introduction
Algorithms
Application
Conclusion

Motivation

How to put as much left-over stuﬀ as possible in a tasty meal
before everything will go oﬀ?


Introduction
Algorithms
Application
Conclusion

Motivation

Find the largest collection of food where everything goes
together! Here, we have the choice:


Introduction
Algorithms
Application
Conclusion

Outline

1 Introduction
2 Algorithms
3 Applications
4 Conclusion


Introduction
Algorithms
Application
Conclusion

Graph (G): a network of vertices (V(G)) and edges (E(G)).
Graph Complement (G ): the graph with the same vertex set
of G but whose edge set consists of the edges not present in G.
Complete Graph: every pair of vertices is connected by an
edge.
A Clique in an undirected graph G=(V,E) is a subset of the
vertex set C ⊆ V ,such that for every two vertices in C, there
exists an edge connecting the two.


Introduction
Algorithms
Application
Conclusion


Introduction
Algorithms
Application
Conclusion

Maximum Clique: A Clique of the largest possible size in a
given graph. The clique number, ω (G ), is the cardinality of
the maximum clique.
Maximal Clique: A Clique that cannot be extended by
including one more adjacent vertex.
Independent Set: a subset of the vertices such that no two
vertices in the subset are connected by an edge of G.
Vertex cover: a subset of the vertices of G which contains at
least one of the two endpoints of each edge.


Introduction
Algorithms
Application
Conclusion

Maximum Clique Problem
Does there exist an integer k such that G contains an clique
of cardinality k?
What is the clique in G with maximum cardinality?
What is the clique number of G?


Introduction
Algorithms
Application
Conclusion

Equivalent Problems
Maximum Independent Set Problem in G


Introduction
Algorithms
Application
Conclusion

Equivalent Problems
Minimum Vertex Cover Problem in G


Introduction Bron Kerbosch Algorithm
Algorithms Backtracking Algorithm
Application Time Complexity
Conclusion Brand-and-Bound

NP-hard

A problem is NP-hard if solving it in polynomial time would
make it possible to solve all problems in the class of NP
problems in polynomial time.
All 3 versions of the Maximum Clique problem are known to
be NP-hard for general graphs.



NP : the class of decision problem which can be solved by a
non-deterministic polynomial algorithm.
P: the class of problems which can be solved by a
deterministic polynomial algorithm.
NP-hard: the class of problems to which every NP problem
reduces.
NP-complete (NPC): the class of problems which are NP-hard
and belong to NP.



Method to Solve Maximum Clique Problem

Non polynomial-time algorithms.
Polynomial-time algorithms providing approximate solutions.
Polynomial-time algorithms providing exact solutions to
graphs of special classes.
Two eﬀective algorithms for dealing with NP-complete
Problems: backtracking, branch and bound




The Algorithm was Designed and Published in 1973 by the
Dutch scientists Joep Kerbosch and Coenradd Bron.
Bron Kerbosch Algorithm is for Finding the Maximal Cliques
in undirected graph.
It is known to be one of the most efficient algorithm which
uses recursive backtracking to find Cliques is practically
proven.
The Bron Kerbosch Algorithm uses the vertex in graph and its
neighbour with few functions to generate some effective
results.



Without Pivoting Strategy

BronKerbosch(R, P, X )
if {P = X = ∅}
Report R as the Maximal Clique
for each vertex v in P
BronKerbosch(R ∪ {v } , P ∩ N {v } , X ∩ N {v })
P := P {v }
X := X ∪ {v }



With Pivoting Strategy

BronKerbosch(R, P, X )
if {P = X = ∅}
Report R as the Maximal Clique
Choose Pivot Vertex u in P ∪ X
for each vertex v in P
BronKerbosch(R ∪ {v } , P ∩ N {v } , X ∩ N {v })
P := P {v }
X := X ∪ {v }



Example



R = X = ∅, P = (1, 2, 3, 4, 5, 6)
Choosing the pivot element u as 4.
4 in PN(v ) = (1, 2, 3, 4, 5, 6) (1, 2, 3, 5, 6) = 4 in 4
Finds the values of Rnew , Pnew , Xnew
Pnew = P ∩ N (v ); Rnew = R ∪ v ; Xnew = X ∩ N (v )
Rnew = 4; Pnew = (1, 2, 3, 5, 6) ; Xnew = ∅
BronKerbosch(4,(1,2,3,5,6),∅)
BronKerbosch((4,1),(2,3),∅)
BronKerbosch((4,1,2),∅,∅)
Report (4,1,2) as one of the Maximal Clique



BronKerbosch((4,3),(1),∅)
Report (4,3,1) as one of the other Maximal Clique.
BronKerbosch((4,2),(1,5),∅)
Report (4,2,5) as an other Maximal Clique.
BronKerbosch((4,6),∅,∅)
Report (4,6) as the Maximal Clique.



This backtracking algorithm is a method for finding all the
sub sets in an undirected graph G.
Given a graph G with V vertices and E edges, G=(V,E)
Let us take an integer variable k.
This algorithm is used in scientific and engineering
applications.
This algorithm is a Depth First Search algorithm.
The algorithm for finding k-clique in an undirected graph is a
NP-complete problem.



List out all the possibilities in the subgraph and check for each
and every edge.
Check for a subgraph in which every node is connected to
every other node.
Check for all possible Cliques in the graphs.
Check the size of clique whether it is equal to k or not.



Any n-vertex graph has at most 3n/3 Maximal Cliques
The worst-case running time of the BronKerbosch algorithm
(with a pivot strategy that minimizes the number of recursive
calls made at each step) is O(3n/3 )
This Backtracking algorithm runs in polynomial time if size of
k is ﬁxed. If k is varying then it is in exponencial time
Running time of the algorithm is O (nk), where k = O(log n)



n
f (x) = − xi → min
i=1
xi + xj ≤ 1, ∀ (i, j) ∈ E
x ∈ {0, 1}n
f (x) = x T Ax → min
x ∈ {0, 1}n , whereA = AG − I


Introduction
Algorithms
Application
Conclusion

Scheduling
Coding Theory: Hamming and Johnson Graphs
Map Labeling
Computer Vision and Pattern Recognition


Introduction
Algorithms
Application
Conclusion

Problem: finding maximum cliques of a graph efficiently
Hard task (in terms of memory and runtime)
Bron-Kerbosch algorithm is one efficient solution
Several applications


Introduction
Algorithms
Application
Conclusion

References

1. D.-Z. Du and P.M.Pardalos ”Handbook of Combinatorial
Optimization”. Kluwer Academic Publishers, 1999
2. http : //en.wikipedia.org/wiki/Clique - problem.


Introduction
Algorithms
Application
Conclusion

Thank you for your attention!


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