please answer all parts of the questions completely with all diagrams and steps
ID: 3713053 • Letter: P
Question
please answer all parts of the questions completely with all diagrams and steps of your work you did to get the answer.
21:25 pts) For the graph assigned to you, find the following using the approximation heuristics liscussed in class. (a-7) Maximal Independent Set (b-2) Minimal Vertex Cover (c-8) Maximal Clique (d-8) Minimum Connected Dominating Set Show all the work for each. 02: 20 pts) You are assigned the edge weight matrix for a complete graph. Determine an approximation to the minimum weight tour using the (i-8) Nearest neighbor heuristic (i-12) Twice around the tree heuristic. Also, show one attempt of reducing the tour weight using the 2-change heuristic for the with each of the two heuristics. i tour obtained Show all the work as well as clearly indicate the tour and its weight before and after the using the 2-change heuristic in each case. attempt of V 107 o12 3 5 V4 15 3 12 O 510 Vs 14 350 Vo lo 8 15 ioOExplanation / Answer
Solution:
Note: The first question is done as per Chegg guidelines, please repost others.
1)
a)
so the maximal independent set will be
{5, 6, 3}
There can be more than one maximal independent set.
b)
vertex cover is a set of vertices; removing which will disconnect the graph and break it into components.
Minimal vertex cover is where a vertex cannot be removed from the vertex cover set.
Minimal vertex cover {4}
c)
A maximal clique is a cycle of a maximum size which is possible in the graph.
Maximal clique is
4-7-2-8-3-1-6-4
d)
an independent set is also a dominating set if and only if it is a maximal independent set, so any maximal independent set in a graph is necessarily also a minimal dominating set.
{5, 6, 3}
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