Computer Science Consider the d-Independent Set problem: Input: an undirected graph G = (V,E) such that every vertex has degree less or equal than d. Output: The largest Independent Set.     Describe a polynomial time algorithm Athat approximates the optimal solution by a factor α(d). Your must write the explicit value of α, which may depend on d. Describe your algorithm in words (no pseudocode) and prove the approximation ratio α you are obtaining. Briefly explain why your algorithm runs in polytime.

Please answer the following question in general algorithm ONLY, DO NOT answer it in ONE specific programming language. Thank you! A vertex is "disconnected" if it does not lie on any cycle. Given an undirected graph G, give an efficient algorithm to count the number of "disconnected" vertices in G. Prove that your algorithm is correct. Analyze the running time of your algorithm.   What I've done so far: I get that I'm supposed to use a depth first searching tree, so I devised an algorithm to do just that, DFS(v)     | mark v     | v.pre = clock++     | for each edge vw     |   | if w is unmarked     |        | v = parent(w)     |        | DFS(w)     | v.post = clock++   I just dont know how to find the "disconnected" vertices. My thought process is that I can perhaps find all the vertices in a cycle, and take them out of the total number of vertices to find the number of disconnected vertices. If this thought process is correct, could you help me by showing me how to make such an…

Long chain of friends: You are given a list of people, and statements of the form “x knows y”. You are asked to find, is there a chain of k distinct people, such as x1 knows x2, x2 knows x3, and xk-1 knows xk. Prove that this problem is NP-complete by using one of the known NP-complete problems (CLIQUE, 3-SAT, Hamiltonian Path, Hamiltonian Cycle, Independent Set, etc.)

10. Apply Djikstra's Algorithm to find a shortest path from A to Z. Be sure to show each step of the algoriothm on a separate picture as discussed in class. Work carefully and pay attention to details. You need to clearly show how you are applying the algorithm. Also be sure to clearly name the verticies of a final shortest path B

Write the PYTHON programming to solve with breadth-first search and build node-arcs with python code for the attached problem:

Given is a strictly increasing function, f(x). Strictly increasing meaning: f(x)< f(x+1). (Refer to the example graph of functions for a visualization.) Now, define an algorithm that finds the smallest positive integer, n, at which the function, f(n), becomes positive. The things left to do is to: Describe the algorithm you came up with and make it O(log n).

Let f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).

Algebraic Preis’ AlgorithmAlgorithm due to Preis provides a different way to solve the maximal weightedmatching problem in a weighted graph. The algorithm consists of the followingsteps.1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G3. M ← Ø4. E ← E5. V ← V6. while E = Ø7. select at random any v ∈ V8. let e ∈ E be the heaviest edge incident to v9. M ← M ∪ e10. V ← V {v}11. E ← E \ {e and all adjacent edges to e}show two ways of implementing this algorithm in Python

Algebraic Preis’ AlgorithmAlgorithm due to Preis provides a different way to solve the maximal weightedmatching problem in a weighted graph. The algorithm consists of the followingsteps.1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G3. M ← Ø4. E ← E5. V ← V6. while E = Ø7. select at random any v ∈ V8. let e ∈ E be the heaviest edge incident to v9. M ← M ∪ e10. V ← V {v}11. E ← E \ {e and all adjacent edges to e} show two ways of implementing this algorithm in Python

Python Kindly give a proper explanation and don't copy from google or any other website