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 (K n ): 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)= 1 2 3 4 a bc d e a b c d e

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 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 j k S i,j,k Claw, K 1,3, S 1,1,1

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.

16. bleilesa@shu.edu