`Hey,
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OPTION C IS CORRECT
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4. The NOT-ALL-EQUAL 3SAT problem is defined as follows: Given a 3-CNF formula F, is there...
4. Approximating Clique. The Maximum Clique problem is to compute a clique (i.e., a complete subgraph) of maximum size in a given undirected graph G. Let G = (V,E) be an undirected graph. For any integer k ≥ 1, define G(k) to be the undirected graph (V (k), E(k)), where V (k) is the set of all ordered k-tuples of vertices from V , and E(k) is defined so that (v1,v2,...,vk) is adjacent to (w1,w2,...,wk) if and only if, for...
The Max Cut problem is given a undirected graph G(V, E), finding a set S so that the number of edges that go between S and V − S is maximum. This is an NPC problem. a) Show that there is always a max cut of size at least |E|/2. Hint: Decide where to put vertices according to if they have more neighbors in S or V − S.
Suppose we have 2n people, some of which are related to some of the others. We might want to split them into groups of two, so that the two people in a group are related (if this is possible). Expressing this as a graph problem, suppose we have an undirected graph G = hV;Ei. A pairing is a set P E of edges such that for all (u; v); (x; y) 2 P, the nodes u; v; x; y are...
File Edit Format View Help Graphs and trees 4. [6 marks] Using the following graph representation (G(V,E,w)): v a,b,c,d,e,f E fa,b), (a,f),fa,d), (b,e), (b,d), (c,f),(c,d),(d,e),d,f)) W(a,b) 4,W(a,f) 9,W(a,d) 10 W(b,e) 12,W(b,d) 7,W(c,d) 3 a) Draw the graph including weights. b) Given the following algorithm for Inding a minimum spanning tree for a graph: Given a graph (G(V,E)) create a new graph (F) with nodes (V) and no edges Add all the edges (E) to a set S and order them...
Explain ur working 4. [6 marks] Using the following graph representation (G(VE,w)): V a, b,c, d,e, fh E -la, b, [a, fl,la,d, (b,ej, [b,d, c,fl,fc,d],Id,el, sd, f) W(a, b) 4, W(a, f)-9, W(a, d)-10 W(b, e) 12, W (b, d)7, W(c,d) 3 a) [3 marks] Draw the graph including weights. b) [2 + 1-3 marks] Given the following algorithm for finding a minimum spanning tree for a graph: Given a graph (G(V,E)) create a new graph (F) vith nodes (V)...
Suppose we have 2n people, some of which are related to some of the others. We might want to split them into groups of two, so that the two people in a group are related (if this is possible) Expressing this as a graph problem, suppose we have an undirected graph G-(WB). A pairing is a set P C E of edges such that for all (u,v),(x,y) є P, the nodes u,v,z, y are all different. In other words, no...
Problem 5. (Lexicographical Optimisation with Paths) Provide pseudocode and an expla- nation for an algorithm that computes a path between two nodes in an undirected graph such that: . The maximum weight in the path is minimised, ie., there does not exist another path with a smaller maximum weight .Amongst all such paths, it finds the path with minimum cost. . The time complexity is no worse than 0(( and V is the set of nodes. ·IvD-log(IVD), where E is...
Show that the following problem is NP-Complete (Hint: reduce from 3-SAT or Vertex Cover). Given an undirected graph G with positive integer distances on the edges, and two integers f and d, is there a way to select f vertices on G on which to locate firehouses, so that no vertex of G is at distance more than d from a firehouse?
Have the explaination please. 4 Graph Application: Network Connectivity (Adapted from Problem 9, Chapter 3 of K&T) Think of a communications network as a connected, undi rected graph, where messages from one node s to another node t are sent along paths from s to t. Nodes can sometimes fail. If a node v fails then no messages can be sent along edges incident on v. A network is particularly vulnerable if failure of a single node v can cause...
Question 1: Given an undirected connected graph so that every edge belongs to at least one simple cycle (a cycle is simple if be vertex appears more than once). Show that we can give a direction to every edge so that the graph will be strongly connected. Question 2: Given a graph G(V, E) a set I is an independent set if for every uv el, u #v, uv & E. A Matching is a collection of edges {ei} so...