Question

Alice wants to throw a party and is deciding whom to call. She has n people to choose from, and she has made up a list of which pairs of these people know each other. She wants to pick as many people as possible, subject to two constraints: at the party, each person should have at least five other people whom they know and five other people whom they don’t know. Give an efficient algorithm that takes as input the list of n people and the list of pairs who know each other and outputs the best choice of party invitees. Give the running time in terms of n

Short answer

Each vertex with each member inside the vertex set, such as$uandv$ , indicates that the person v knows the person u . So, find the subset v where another vertex has a value greater than 5 , then repeat the process with both the modified degrees for every node until a network is formed in which neither vertex may be destroyed after the procedure is finished. As a result, the running time's total complexity has increased.

Step-by-step solution

Individual String ‘Fi’

Assuming that each individual attends the party with at least 5 different other things they trust and five others they don't. Throughout a few places in the question, every string 'fi' has been substituted with '?'. If you have any doubts about my answer, please leave a remark.

Vertex for Each Set

Assume the graph $G\left(V,E\right),$ which has a vertex for each person in the vertex set. If an edge e connects the vertices $uandv,$ , it means that the person v is acquainted with the person u . So, in the induced network [where V ' is the new set of vertices], we may restrict the task to identifying a subset V of V whereby each vertex has a value more than 5 and less than$|V\text{'}|-5$ .

The continuous strategy can be used, in which all nodes are examined at first, and any node with a degree more than role="math" localid="1658920458569" $|V\text{'}|-5orlessthan5$ is eliminated. This graph is changed to be the induced graph, with the remaining vertices being assigned to the vertex set. This technique is repeated with the updated degrees for each node until a graph is formed in which no vertex may be deleted once the operation is completed. $G\text{'}$ is the symbol for this graph. There are n iterations in the algorithm, and each iteration takes $O\left(n\right)$ time. As a result, the overall complexity of the running time is role="math" localid="1658920588246" $O\left(n2\right).$

Conclusion  

A vertex for each person in the vertex set like u and v means the person v is acquainted with the person u. So, identifying of sub set v where each vertex has value more than 5. same way this technique is repeated with the updated degrees for each node until a graph is form in which no vertex may deleted once the operation is complete. As a result overall complexity of the running time.