The Maximum Clique Problem

Transcription

1 November, 2012

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

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

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

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

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

7 Outline s 4

8 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.

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10 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.

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12 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?

13 Equivalent Problems Maximum Independent Set Problem in G

14 Equivalent Problems Minimum Vertex Cover Problem in G

15 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity 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.

16 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound 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.

17 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound 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

18 Bron Kerbosch Algorithm Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-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.

19 Without Pivoting Strategy Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound 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}

20 With Pivoting Strategy Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound 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}

21 Example Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound

22 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound R = X =, P = (1, 2, 3, 4, 5, 6) Choosing the pivot element u as 4. 4 in P\N(v) = (1, 2, 3, 4, 5, 6) \ (1, 2, 3, 5, 6) = 4 in 4 Finds the values of R new, P new, X new P new = P N (v); R new = R v; X new = X N (v) R new = 4; P new = (1, 2, 3, 5, 6) ; X new = 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

23 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound BronKerbosch(4,(1,2,3,5,6), ) BronKerbosch((4,3),(1), ) BronKerbosch((4,3,1),, ) Report (4,3,1) as one of the other Maximal Clique. BronKerbosch(4,(1,2,3,5,6), ) BronKerbosch((4,2),(1,5), ) BronKerbosch((4,2,5),, ) Report (4,2,5) as an other Maximal Clique. BronKerbosch(4,(1,2,3,5,6), ) BronKerbosch((4,6),, ) Report (4,6) as the Maximal Clique.

24 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound 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.

25 Example Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound

26 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound

27 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound

28 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound 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.

29 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound Any n-vertex graph has at most 3 n/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(3 n/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)

30 Bron Kerbosch Algorithm Backtracking Algorithm Time Complexity Brand-and-Bound f (x) = n x i min i=1 x i + x j 1, (i, j) E x {0, 1} n f (x) = x T Ax min x {0, 1} n, wherea = A G I

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

32 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

33 References 1. D.-Z. Du and P.M.Pardalos Handbook of Combinatorial Optimization. Kluwer Academic Publishers, http : //en.wikipedia.org/wiki/clique - problem.

34 Thank you for your attention!