For problems F1 to problem F3 you are to provide three things: =================

Question

 For problems F1 to problem F3 you are to provide three things:      ==================================================      a. Formulate the decision problem as a language membership problem (i.e.,     describe the language L corresponding to the decision problem). You     may assume that there is a standard way to encode graphs and their     vertices and edges as strings.      b. Sketch a proof that this language is in NP.      c. State what else would need to be proven in order to show the     language NP-complete (but do not prove).      ================================================== 

 F1. (3-COLORING) "3-COLORING" is the problem of, given three colors and a     binary relation R on a set V of "nodes", can we assign colors to nodes so     that no pair of nodes connected by R has the same color.  F2. (k-INDEPENDENT) Given a graph G, an independent set is a subset X of     the vertices of G such that no pair of vertices in X is connected by     an edge in G. The k-INDEPENDENT-SET decision problem is defined as     follows:      Given: an undirected graph G and an integer k     Question: Does G contain an independent set of size at least k?  F3. (VERTEX-COVER) Given a graph G=(V, E), and integer k <= |V|, is there     a subset of at most k vertices such that every edge in E has at least     one vertex in the subset? 

Explanation / Answer

This is typically used and applied in map coloring problems We can use the 3 primary colors – Red, Green, and Blue (RGB) The condition is that the neighboring regions cannot have the same color Least Constrained Variable and the most constrained Values are the 2 predominant entities that are followed For a given input, every decision problem will have 2 possible outputs – either Yes or No An example for decision problem can be as follows: Out of A and B whether A is greater than B? Test for prime number can be another classical example for a decision problem They can be further classified into P and NP – namely Polynomial hard (P)and non polynomial hard (NP) Emptiness condition can be tested for whether Language = {Empty Set }? We have to start from the start state and find out whether at least one lawfully acceptable state can be reached successfully from the start state Whether the language is finite? Once a language is finite, it will be regular as well – but it need not be vice versa Given a start state A, and two end states B and C, we know for sure that he start state A is distinguishable from B and C But whether one end state B can be distinguished from the other end state C? The minimum state DFA can be constructed by first selecting representative Forming a group portioning the states using the indistinguishable property

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