LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

18 Artiom Alhazov, Carlos Martín-Vide, and Linqiang Pan
7 Solving the Clique Problem
Given a graph G with n vertices and m edges, a clique of G is a subset of the
vertices such that every vertex is connected to every other vertex by an edge; the
clique problem (denoted by CP) asks whether or not there exists a clique for G
of size k, wherekis a given integer less than or equal to n. The clique problem
is an NP-complete problem [1].
For a graph G =(V,E), we define the complementary graph G ′ =(V,E ′ ),
where E ′ = {(vi,vj) �∈ E | vi,vj ∈ V }. In figure 2, an example of a graph with
five vertices and five edges is given.
In this section we will solve the clique problem in linear time (3n +2k + 12)
by recognizing P systems with restricted elementary active membranes.
Theorem 3. The clique problem can be solved in linear time with respect to the
number of vertices of the graph by recognizing P systems with restricted elementary
active membranes.
2
1
5
3 4
3 4
(a) (b)
Figure 2. The original graph (a) and its complementary graph (b).
Proof. Let G =(V,E) be a graph with n vertices and m edges, V = {v1,v2,
··· ,vn}, E = {e1,e2, ··· ,em}. The instance of the problem is encoded by a
(multi)set in the alphabet Σ(〈n, k〉) ={pi,j,n+2k+5,ei,j,n+2k+5 | 1 ≤ i, j ≤ n, i �=
j}. The object pi,j,n+2k+5 stands for (vi,vj) ∈ E, while ei,j,n+2k+5 represents
(vi,vj) /∈ E. For given values of n and k we construct a recognizing P system
(Π(〈n, k〉),Σ(〈n, k〉),i(〈n, k〉)) with i(〈n, k〉) =2
Π(〈n, k〉) =(O(〈n, k〉),H,µ,w1,w2,R),
where
O(〈n, k〉) ={ai,di,ti,fi,yi,zi | 1 ≤ i ≤ n}
∪{hi,h ′ i | 0 ≤ i ≤ k}∪{gi | 0 ≤ i ≤ n +3}
∪{ei,j,l | 1 ≤ i ≤ n +2, 1 ≤ j ≤ n +2, 0 ≤ l ≤ n +2k +5}
∪{ci | 0 ≤ i ≤ 3n +2k +11}∪{a, b, g, p, q, u, yes, no},
H = {1, 2},
µ =[ 1 [ 2 ] 0 0
2 ] 1 ,
w1 = c3n+2k+11,
w2 = a1a2 ···and1,
2
1
5