64

# Extract maximum independent set using eigenvalue relation

The maximum clique problem is an NP-hard problem. If there is an algorithm that extracts the maximum clique in polynomial time, the two complexity classes P and NP are the same. We try to delete the vertices that do not belong to the maximum clique from the input graph G with vertex set V using eigenvalue relation. In this paper, we identify the vertices that do not belong to the maximum clique by controlling the change of the eigenvalue distribution of the adjacency matrix due to the vertex deletion. First, we create the graph G_s, which does not have the integer eigenvalue from G. Next, we prove the following; (1) Let k_m be the number of the eigenvalues of A_s smaller than -1. We keep k_m by adding edges with the weights of -1 between the vertices in V before and after deletion to keep the size of the maximum clique. (2) Suppose the input graph is not a complete graph and if k_m decreases for any vertex deletion. When we can add an edge with the weight -1 that increases k_m after deleting vertex v, it does not belong to the maximum clique. (3) Let k_q be the integer series that does not exceed the positive eigenvalues of the adjacent matrix of G_s. If k_q keeps before and after deletion, the vertex does not belong to the maximum clique. And then, we demonstrate an algorithm to extract the maximum clique in polynomial time using this result.

March 27, 2020

## Transcript

2. ### About me Yasunori Ohto – Trying to solve P vs

NP problem 2 / 20
3. ### Background Maximum Clique problem is an NP-hard. If it can

solve in polynomial time, computation complexity class P and NP is equal 3 / 20
4. ### Extract maximum clique 1.Make graph G s such that the

adjacency matrix has no positive and negative eigenvalue before and after vertex removal 2.Find a vertex that does NOT belong to the maximum clique using Eigenvalue Interlacing Theorem and delete it 4 / 20
5. ### Definition -Graph Graph is an ordered pair of vertices and

edges 5 / 20 1 2 3 4 5 G
6. ### Definition -Adjacency matrix Adjacency matrix A of graph G with

size n is a symmetric matrix size n such that its element a i,j is the weight of the edge between the vertices v i and v j 6 / 20 1 2 3 4 5 G 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 A
7. ### Spectral graph theory Studies the relation between the properties of

a graph and its characteristic polynomial, eigenvalues, eigenvectors, adjacency matrix, and Laplacian matrix 7 / 20 1 2 3 4 5 G 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 A ⇒ 2:1, (1±√5)/2:2
8. ### Spikes Add a edge to each vertex of input graph

8 / 20 Make graph G s such that the adjacency matrix has no positive and negative eigenvalue before and after vertex removal Weight: 0.5 vertex set V independent set
9. ### 0 1 0 0 1 1 0 1 0 0

0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 Principal submatrix is a adjacency matrix of induced subgraph Adjacency matrix of the maximum clique is the maximum size of it Symmetric matrix Maximum clique 9 / 20 Principal submatrix clique
10. ### 0 1 0 0 1 1 0 1 0 0

0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 The relation of the eigenvalues of symmetric matrix and its principal submatrix λ i μ ≧ i λ ≧ n-m+i Eigenvalue Interlacing Theorem 10 / 20 clique Principal submatrix μ 1 … μ ≧ ≧ n Symmetric matrix λ 1 … λ ≧ ≧ n
11. ### It from the interlacing theorem is . min(2n-k p -1,

2n-k m, λ max +1) . k p : number of eigenvalues over -1 k m : number of eigenvalues under -1 -1 Upper bound of size of maximum clique of G s 11 / 20 k p k m no eigenvalue
12. ### Let k q be the integer series that does not

exceed the positive eigenvalues of the adjacent matrix A s of G s . If k q keeps before and after deletion a vertex v, v does not belong to the maximum clique. Keeping k q 12 / 20
13. ### keep k m by adding edges with the weights of

-1 between the vertices in V before and after deletion to keep the size of the maximum clique. -1 Decrease k m before and after deletion 13 / 20 k p k m Add edge with weight -1 Decrease K m
14. ### Suppose k m decreases for any vertex deletion. If k

m can increase by adding an edge the weight -1 after delete a vertex v, v does not belong to the maximum clique. Expose deletable vertex that keeps k m 14 / 20 vertex deletion Add edge with weight -1 to recover decreased k m
15. ### Algorithm 1.Create G s from input graph G and decrease

k m 2.Delete vertices that keeps k q or k m 3.While V is not the complete graph of vertex set 1.If we find an edge with weight -1 increases k m , after deleting v i , then delete v i 2.Delete vertices that keeps k q or k m 15 / 20
16. ### Count the number of eigenvalues If the degree of a

polynomial equation is greater than four, we cannot solve analytically in general. However, we need only to count the number of positive eigenvalues. 16 / 20
17. ### Count the number of eigenvalues Count the number of eigenvalues.

1.Get Frobenius normal form from adjacent matrix 2.Get characteristic equation from each Frobenius block 3.Count the number of positive eigenvalues using Strum's method from each characteristic equation 17 / 20
18. ### Computational complexity Total computational complexity is O(n7) except the cost

of the rational operation 18 / 20
19. ### Conclusion Extraction of the maximum independent set can be executed

in polynomial time It concludes P = NP. 19 / 20