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)).

same vertex set of G but whose edge set consists of the edges not present in G.

• Complete Graph (K n ): 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

•

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.

a 1 a 2 G= L(G)= b e c b c 3 d 4 d e

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:

Y P

 

S

 (

S

 

S P

)  

Y



P

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 S i,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 S i,j,k -free graphs.

i

Claw, K 1,3, S 1,1,1

j

S i,j,k

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.

[email protected]