The third-clique problem is about deciding whether a given graph G = (V, E) has a clique of cardinality at least |V |/3. Show that this problem is NP-complete.

Let G be a graph with n vertices. The k-coloring problem is to decide whether the vertices of G can be labeled from the set {1, 2, ..., k} such that for every edge (v,w) in the graph, the labels of v and w are different.  Is the (n-4)-coloring problem in P or in NP? Give a formal proof for your answer.  A 'Yes' or 'No' answer is not sufficient to get a non-zero mark on this question.

In the decision version of the Clique problem you are given a non-empty graph G = (V,E) and a positive integer k and asked whether G has a complete subgraph (or clique) of size k (i.e., k vertices that induce a complete subgraph).  Give the pseudocode of a recursive backtracking algorithm for the problem, and analyze your algorithm by providing a worst-case recurrence. Can you guarantee a worst-case running time in O(1.466^n)? Prove your answer

Please show written work with answer!! An independent set in a graph is a set of vertices no two of which are adjacent to each other. A clique is a complete subgraph of a given graph. This means that there is an edge between any two nodes in the subgraph. The maximal clique is the complete subgraph of a given graph which contains the maximum number of nodes. We know that independent set is a NP complete problem. Transform the independent set to max clique (in polynomial time) to show that Max Clique is also NP

please answer both of the questions.   7. The Bellman-Ford algorithm for single-source shortest paths on a graph G(V,E) as discussed in class has a running time of O|V |3
, where |V | is the number of vertices in the given graph. However, when the graph is sparse (i.e., |E| << |V |2), then this running time can be improved to O(|V ||E|). Describe how how this can be done..   8. Let G(V,E) be an undirected graph such that each vertex has an even degree. Design an O(|V |+ |E|) time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree.    Please give proper explanation and typed answer only.

Show that the decision problem version is an NP-complete problem; Exists, given a graph G and a goal cost c, a spanning tree in which the most any vertex pays is not more than c?

Prove the choice problem variant is NP-complete; Exists a spanning tree with a goal cost c for a graph G and a vertex's maximum payment?

5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…

Consider a graph G that is comprised only of non-negative weight edges such that (u, v) € E, w(u, w) > 0. Is it possible for Bellman-Ford and Dijkstra's algorithm to produce different shortest path trees despite always producing the same shortest-path weights? Justify your answer.

Can you help me solve this exercise? Please note that the greedy approach described in the advice paragraph does not work.