Mathematics-the Computational complexity theory, the Parameterized complexity, perfect code (in the Graph Theory), split graphs

 

I need for more explanation and rewriting in simple, understandable language
The writer must have a good knowledge about the problems that was reviewed in the article and can rewrite this for general public

SRBPC parameterized by the number of parts in R is W[1]-hard.

This paper gives the proof for The W[1]-hardness proof of SRBPC (Special Red-Blue Perfect Code) which is variant of Perfect Code. This paper provides us with a theorem which is accompanied by a series of lemmas which completes the proof as discussed below.
Theorem
The theorem statement can be stated as: «SRBPC parameterized by the number of parts in R is W[1]-hard».
Definition: W hierarchy
The W hierarchy is a collection of computational complexity classes. A parameterized problem is in the class W[i], if every instance (x, k) can be transformed (in FPT-time) to a combinatorial circuit that has weft at most i, such that (x, k) ∈ L if and only if there is a satisfying assignment to the inputs, which assigns 1 to at most k inputs. The weft is the largest number of logical units with unbounded fan-in on any path from an input to the output. The total number of logical units on the paths (known as depth) must be limited by a constant that holds for all instances of the problem.
Many natural computational problems occupy the lower levels, W[1] and W[2].
Definition: Perfect Code
A perfect code in a graph G = (V , E) is a subset of vertices V’ such that for each vertex v ∈ V, the subset V’ includes exactly one element of the closed neighborhood N[v] of v, that is, exactly one element among v and all vertices adjacent to v.
Definition: SPLIT CONTRACTION and SRBPC
Formally, given a graph G and an integer k, Split Contraction asks whether there exists X ⊆ E(G) such that G/X is a split graph (a graph in which the vertices can be partitioned into a clique and an independent set) and |X| ≤ k. Split Contraction is fixed-parameter tractable. Split Contraction, despite its deceptive simplicity, is W[1]-hard. Under the Exponential Time Hypothesis, Split Contraction cannot be solved in time 2^o(l^2 ) n^(O(1)) where n is the vertex cover number of the input graph. I also verify that this lower bound is essentially tight. To the best of my knowledge, this is the first tight lower bound of the form 2^o(l^2 ) n^(O(1)) for problems parameterized by the vertex cover number of the input graph. SRBPC is a variant of Perfect Code which is known to be W[1]-hard.
Now let (G,R= R_1⊎, R_2⊎ … ⊎R_k,β) be an instance of SRBPC so that I assume the magnitude of ,β to be equal to dk i.e., |β| = dk. Otherwise, the occasion is an inconsequential NO example of SRBPC. For specialized reasons, I accept that |β|=e ˃4k which implies that d>4 absolutely if β|=e≤4k then for each partition P_1… P_K of β into k parts with the end goal that each part is non-exhaust, I first figure a stage π on k elements and then for every i∈[k], I check whether there exists a vertex r_(π(i)) ∈ R_(π(i)) that commands precisely all the vertices in P_i. Plainly, this can be done in time 2^O(klogk) n^(O(1)) here, n denotes the number of vertices in the input graph. Moreover, I additionally expect that k≥2 else the issue is resolvable in polynomial time. Presently I give the coveted reduction. I build an occurrence (G^’,K^’) of SPLIT CONTRACTION as follows. Originally, V(G^’ )=R ⋃ B and E(G^’ )=E(G) for all b,b^’∊ B b≠b^’ I increment the edge b,b^’ to E(G^’ ). That is, I change B into a coterie let t = 2k + 2 for each b_i ∈B add a set of y_1^i,… y_t^i each adjacent to b_i in G^’. I add a vertex s adjacent to every vertex r∈R in G^’. Also, I add a set of t vertices q_1,…q_(t ) each adjacent to s in G^’. For each i∈[k], add a vertex x_i adjacent to each vertex r∈ R_i. Lastly, for all i∈[k] add a set of t vertices w_1^i,…w_t^i adjacent to x_i in G^’. Now set the new parameter k^’ to be 2k this finishes the depiction of reduction. The following lemmata are used to prove certain structural properties of the instance. G^’,K^’) which will be used later to show that (G,R= R_1⊎, R_2⊎ … ⊎R_k,β) is a YES instance of SRBPC if and only if (G^’,K^’) is a YES instance of SPLIT CONTRACTION.
Definition. (Lemma 1): Let G^’,K^’ be a YES instance of SPLIT CONTRACTION then for all v∈({s}⋃B⋃{x_i} | i∈[k] I have φv∈ć.
Definition. (Lemma 2): Let G^’,K^’ be a YES instance of SPLIT CONTRACTION then for all i∈[k] there exist r_i ∈ R_i such that (x_i,r_i )∈S. For each i∈[k] I arbitrary chose the vertex r_i^* ∈R_i such that e_(i )^*=(x_i,r_i^* ) this existence of a vertex is guaranteed by this lemma. From this lemma, I know that each i∈[k] I have r_i^* ∈R_i such that (x_i,r_i )∈S.
Definition. (Lemma 3): Let G^’,K^’ be a YES instance of SPLIT CONTRACTION then for all ∈[k] and h_i= φ(r_i^* ), I have |W(h_i ) |≥3 (for all elements of k and h_i is equal to φ(r_i^* ), the magnitude of W(h_i ) is greater or equal to 3. Addition, there is an edge e_i≠e_i^* in S incident to exactly one of x_i,r_i^* and not incident to the vertices in {w_1^i,… w_t^i }.
Definition. (Lemma 4): Let G^’,K^’ be a YES instance of SPLIT CONTRACTION then for all i∈[k], r_i^* ∈W(φ(s)) meaning I is an element of k and , r_i^*is an element of W(φ(s)).
Assume that there is i ∈ [t] such that for all sY ∈ Si ,(bi ,sY )∈Si T. Recall that NG_0. There exists v ∈ A such that |W(h0)|= 1, where h0 = ϕ(v). This follows from the fact that at most 2k0 is equal to 4t vertices in A can be incident to an edge in T, as t can be assumed to be larger than 6, else the graph has constantly many edges and I can solve the problem in polynomial time. From Lemma 3 it follows that (h,h0)∈ E(H), but W(h),W(h 0) are not adjacent in G_0, contradicting that W is an H witness structure of G0 (bi) = Si ∪ {b0i j | j ∈ [4t + 2]} meaning bi is equal to Si which is a subset of b0ij such that j is an element of 4t + 2.
Proof for this can be done as follows: Let h_s= φ(s) and Rˆ={r_i^* | ∈[k]∈W(h_s )}, Rˆ is equal to r_i^* such that I is an element of k and k is an element of W(h_s ) . Assuming that the magnitude of Rˆ is less than k, |Rˆ| < k. Thus every edge in S is incident to either x_i or r_i^*. Observe that the only vertices in W(h_s ) that can be adjacent to a vertex in B are in Rˆ. However, every vertex in Rˆ has exactly d neighbors in B. This together with the fact that the magnitude of B is equal to e which is also equal to dk but greater than d|Rˆ| ,|B|= e=dk>d|Rˆ| this implies that there exist a subset B^’ of size d(k-|Rˆ| such that none of these vertices are adjacent to any vertex in Rˆ. However, at most k-|Rˆ| Vertices in B^’ can be incident to an edge in S. This implies that there exists a vertex b which is an element of B prime with h being equal to ϕ(b) i.e. b ∈ B^’ with h = ϕ(b) such that it is not incident to any edge in S and thus the magnitude of |W(h)| = 1. But then I can conclude that W(h) and W(h_s ) are not adjacent in G^’.
Example: Consider a unique fixed point L_ϕ=ϕL_ϕof the function ϕ: a^N͢ a^N where N represents set of natural numbers. dϕ(f),ϕ(g)≤1/2 d(f,g). If g is an sequence then L_ϕ=lim┬(i͢ α)⁡〖ϕ^i (g) 〗, this means that the concept L_ϕ which I have defined recursively via ϕ is the limit of a Cauchy sequence.
Definition. (Lemma 5): (G,R= R_1⊎, R_2⊎ … ⊎R_k,β) is a YES instance of SRBPC if and only if (G^’,K^’) is a YES instance of SPLIT CONTRACTION This lemma can be proved in two ways namely forward and reverse direction as follows.
For an arbitrary solution F of (G^’,K^’), if there are at least two vertices in KH that are finally in the independent set IF , they must be contained in a same witness set W0 because they are adjacent originally. Thus the number of such vertices is bounded by k + 1, implying that there is a vertex u ∈ KH that is in the clique KF (i.e., not in IF) since |KH| > 2k. By the definition of witness sets, I see that the induced subgraph G^’ [W0] has a spanning tree whose edges are all contained in F. Let x be an arbitrary leaf in this spanning tree. I can remove one edge from F to separate the vertex x from the witness set W0, and add an edge ux into F to make x adjacent to all vertices in KF after contracting F. It is easy to see that the resulting set obtained from F is also a solution of(G^’,K^’), and x is no longer in the witness set W0. I repeat this operation on F until there is exactly one vertex in the witness set W0. Then I obtain a solution set F satisfying the requirement.
Definition: Fixed Parameter Tractable (FTP). A parametrized problem Q is a subset of Σ (N) for some finite alphabet Σ. The second component is called the parameter. The problem Q is fixed-parameter tractable (FPT) if it admits an algorithm deciding whether (I,k) ∈ Q in time f(k)|I| O(1), where |I| is the size of I and f is a computable function depending only on k.
Split Contraction can be solved in time 2^O(k_2 ) n^5. Here the algorithm starts by finding a large split subgraph H in the input graph and then considers two cases in terms of the clique size of the split subgraph H. If the clique of H is large, then almost all vertices in this clique are finally included in the clique of some target split graph. A branch-and-search algorithm is used to enumerate all edge contractions and reduce the instance to several instances of Clique Contraction that is known to be FPT. If the clique of H is small, then there will be a large independent set in the input graph.
Consider simple and undirected graphs G = (V, E), where V is the vertex set and E is the edge set. Two vertices u,v ∈ V are adjacent if and only if uv ∈ E. A vertex v is incident with an edge e iff v is an endpoint of e. The neighbor set NG(v) of a vertex v ∈ V is the set of vertices that are adjacent to v in G. The closed neighbor set of v is denoted by NG[v] = NG(v) ∪ {v}. For a set X of vertices or edges in G, G − X is used to denote the graph obtained by deleting X from G. For a set of vertices V^’ ⊆ V, I write G[V^’] to denote the subgraph of G induced by V^’ and write E[V^’] to denote the set of edges in G whose both endpoints are in V^’. A graph G is a split graph if its vertex set can be partitioned into a clique K and an independent set I, where (K; I) is called a split partition of G. The class of split graphs is hereditary and is characterized by the set {2K2, C4, C5} of forbidden induced subgraphs.
Example: Algorithm Split Contraction (G, k)
1 Find an induced split subgraph H = (KH; IH) of size (n − 2k) in G. if it does not exist return «NO». Else let Vk = V (G) − V (H).
2 If |KH| > 2k then:
2.1 Branch into instances (G^’,K^’ )by contracting edges E^’⊆ E [Vk].
2.2 Enumerate all partitions V^’k = (R, Kp, Ip).
2.3 Let T_1 = {v ∈ IH | ∃ x ∈ Ip, vx ∈ E(G^’)}. If Clique Contraction (G^’ [T_1 ∪ KH ∪ R ∪ Kp], K^’) =«YES» then return «YES».
2.4 For each w ∈ KH not adjacent to Ip do Let T_2= {v ∈ IH | ∃ x ∈ Ip ∪ {w}, vx ∈ E(G^’)}. If Clique Contraction (G^’ [T_2 ∪ (KH − {w}) ∪ R ∪ Kp], K^’) = «YES» then return «YES».
2.5 Repeat 2.1 − 2.4, return «NO» if no G^’, K^’yields «YES».
3 Else if |KH| ≤ 2k then:
3.1 Partition V (G) into disjoint sets X1, · · · ,Xd: Each Xi induces a maximal independent set, and vertices in each Xi have the same neighbors.
3.2 Reduction Rule 1: If d > 24k + 4k, then output «NO».
3.3 Reduction Rule 2: If there are more than 2k + 5 vertices in Xi for some i, then retain 2k + 5 vertices among them and remove others in Xi.
3.4 Apply brute-force search to find a solution in the reduced graph G∗.
Definition: Exponential Time Hypothesis (ETH)
This is the first tight lower bound of this form for problems parameterized by the vertex cover number of the input graph. Lately, there has been increasing scientific interest in the examination of lower bounds of forms other than 2^o(l^2 ) n^(O(1)) for some parameters s. For example, lower bounds those are «slightly super-exponential».
In this paper, I have established two important results regarding the complexity of Split Contraction. First, I have shown that under the Exponential Time Hypothesis (ETH), this problem cannot be solved in time 2^o(l^2 ) n^(O(1))where n is the vertex cover number of the input graph, and this lower bound is tight. To the best of my knowledge, this is the first tight lower bound of the form 2^o(l^2 ) n^(O(1))for issues parameterized by the vertex cover number of the information diagram. Second, I have demonstrated that Split Contraction, regardless of its misleading effortlessness, is really W[1]−hard. I might want to finish up my paper with the accompanying interesting inquiry. In the correct setting, it is anything but difficult to see that Split Contraction can be fathomed in time 2^o(l^2 ) n^(O(1)) . Can it be solved in time 2^o(l^2 ) n^(O(1))? A negative answer would suggest, for example, that it is neither one of the possibles to locate a topological inner circle minor in a given chart in time 2^o(l^2 ) n^(O(1)), which is an intriguing open issue. It may be conceivable that instruments created in my paper, for example, the use of agreeable shading, can be used to reveal insight into such issues.

References
H. Vollmer and B. Vallée. 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8–11, 2017, Hannover, Germany.
A. Agrawa, D. Lokshtanov, S. Saurabh and M. Zehavi. Split Contraction: The Untold Story. Split Conference 2017.
Gu, Q., Hell, P. and Yang, B. (2014). Algorithmic Aspects in Information and Management.

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