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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

The Maximum Independent Set Problem

Sarah Bleiler

DIMACS REU 2005

Advisor: Dr. Vadim Lozin, RUTCOR

definitions
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.
slide3
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
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
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
Equivalent Problems
  • Maximum Clique Problem in .

G= =

  • Minimum Vertex Cover Problem in G.

G= G=

applications
Applications
  • Computer Vision/Pattern Recognition
  • Information/Coding Theory
  • Map Labeling
  • Molecular Biology
  • Scheduling
np hard
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
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.
definitions10
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
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
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
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
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
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>