The Maximum Independent Set Problem

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The Maximum Independent Set Problem. Sarah Bleiler DIMACS REU 2005 Advisor: Dr. Vadim Lozin, RUTCOR. Definitions. Graph (G) : a network of vertices (V(G)) and edges (E(G)).

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### The Maximum Independent Set Problem

Sarah Bleiler

DIMACS REU 2005

Definitions
• Graph (G): a network of vertices (V(G)) and edges (E(G)).
• Graph Complement ( ): the graph with the same vertex set of G but whose edge set consists of the edges not present in G.
• Complete Graph (Kn): every pair of vertices is connected by an edge.
Clique: a complete subgraph of G.
• Vertex cover: a subset of the vertices of G which contains at least one of the two endpoints of each edge:
Independent Sets
• An independent set of a graph G is a subset of the vertices such that no two vertices in the subset are connected by an edge of G.

α(G)=3

• The independence number, α(G), is the cardinality of the maximum independent set.
Maximum Independent Set (MIS) Problem
• Does there exist an integer k such that G contains an independent set of cardinality k?
• What is the independent set in G with maximum cardinality?
• What is the independence number of G?
Equivalent Problems
• Maximum Clique Problem in .

G= =

• Minimum Vertex Cover Problem in G.

G= G=

Applications
• Computer Vision/Pattern Recognition
• Information/Coding Theory
• Map Labeling
• Molecular Biology
• Scheduling
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 MIS problem are known to be NP-hard for general graphs.
Methods to Solve MIS Problem
• Non polynomial-time algorithms
• Polynomial-time algorithms providing approximate solutions
• Polynomial-time algorithms providing exact solutions to graphs of special classes.
Definitions
• Bipartite graph: a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent.
• Line graph L(G): associate a vertex with each edge of G and connect two vertices with an edge iff the corresponding edges of G are adjacent.

G= L(G)=

a

1

a

2

e

c

b

b

c

e

4

d

3

d

Maximum Matching Problem
• Matching: a set of edges of G such that no two of them share a vertex in common.

• The Maximum Matching Problem is solvable in polynomial time and is applied to find a solution to the MIS problem for bipartite and line graphs.
• Line graphs: Matching Algorithm (Edmonds 1965)
• Bipartite graphs: (König’s Minimax Theorem) α(G) + |E(max. matching)| = n
Augmenting Graphs
• Let S be any independent set in G.
• Label V(S) as black and V(G-S) as white.
• A bipartite graph H=(P,Y,E) is said to be augmenting for S if:
Theorem of Augmenting Graphs
• An independent set S in a graph G is maximum if and only if there are no augmenting graphs for S.
• The process of finding augmenting graphs is also NP-hard but is a useful option to:
• Develop approximate solutions
• Bound α(G)
• Solve in polynomial time for special classes
My Research Problem
• Alekseev (1983) proved that if a graph H has a connected component which is not of the form Si,j,k, then the MIS is NP-hard in the class of H-free graphs.
• The solution for line graphs has been extended to claw-free graphs.
• We are looking to extend these results to larger classes of Si,j,k-free graphs.

i

Claw, K1,3, S1,1,1

k

j

Si,j,k

References

[1] A. Hertz, V.V. Lozin, The Maximum Independent Set Problem and Augmenting Graphs. Graph Theory and Combinatorial Optimization, 1:1-32, 2005.

[2] Eric W. Weisstein. "Maximum Independent Set." From Mathworld--A Wolfram Web Resource.<http://mathworld.wolfram.com/Ma ximumIndependentSet.html>