The Maximum Independent Set Problem

1. The Maximum Independent Set Problem Sarah Bleiler DIMACS REU 2005 Advisor: Dr. Vadim Lozin, RUTCOR

2. 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.

3. 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:

4. 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.

5. 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?

6. Equivalent Problems • Maximum Clique Problem in . G= = • Minimum Vertex Cover Problem in G. G= G=

7. Applications • Computer Vision/Pattern Recognition • Information/Coding Theory • Map Labeling • Molecular Biology • Scheduling

8. 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.

9. 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.

10. 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

11. 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

12. 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:

13. 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

14. 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

15. 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>

16. bleilesa@shu.edu