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Problem 3: Suppose you are given an undirected graph G and a specified starting node s...
Problem 3: Bounded-Degree Spanning Trees (10 points). Recall the minimum spanning tree problem studied in class. We define a variant of the problem in which we are no longer concerned with the total cost of the spanning tree, but rather with the maximum degree of any vertex in the tree. Formally, given an undirected graph G = (V,E) and T ⊆ E, we say T is a k-degree spanning tree of G if T is a spanning tree of G,...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete. Note:...
Please answer truthfully, if you cannot answer the question, please leave it for someone else to answer. A Hamiltonian path in a graph is a simple path that visits every vertex exactly once. Prove that HAM-PATH = { (G, u, v ): there is a Hamiltonian path from u to v in G } is NP-complete. You may use the fact that HAM-CYCLE is NP-complete
Prove this is NP Complete, or it is in P. This problem is a variant of UNDIRECTED HAMILTON PATH in bounded-degree graphs. The language in question is the set of all triples (G, s, t) for which G is an undirected graph with maximum degree at most 2 containing a Hamilton path from node s to node t.
Given a graph and a node in it as the starting point, the traveling salesman problem (TSP) is about finding a least cost path that starts and ends at the starting node, and goes through each other node in the graph once and exactly once. The edges of the graph are labeled by a number representing the cost of traveling between the connecting two nodes. Formulate this problem as a search problem by specifying the states, possible initial states, goal...
3, (30 points) Given a directed graph G - N. E), each edge eEhas weight We, 3, (30 points) Given a directed graph G (V, E), each edgee which can be positive or negative. The zero weight cycle problem is that whether exists a simple cycle (each vertex passes at most once) to make the sum of the weights of each edge in G is exactly equal to 0. Prove that the problem is NP complete. 3, (30 points) Given...
Prove that the following problem is NP-complete: given an undirected graph G = (V, E) and an integer k, return a clique of size k as well as an independent set of size k, provided both exist.
Hi,. the question is below: Help if you can.. Here is some background information/ an example: 9. Let k-Color be the following problem. Input: An undirected graph G. Question: Can the vertices of G be colored using k distinct colors, so that every pair of adjacent vertices are colored differently? Suppose that you were given a polynomial time algorithm for (k + 1)-Color. Use it to give a polynomial algorithm for k-Color. This means that you need to provide a...
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?
Problem 3's picture are given below. 5. (a) Let G = (V, E) be a weighted connected undirected simple graph. For n 1, let cycles in G. Modify {e1, e2,.. . ,en} be a subset of edges (from E) that includes no Kruskal's algorithm in order to obtain a spanning tree of G that is minimal among all the spanning trees of G that include the edges e1, e2, . . . , Cn. (b) Apply your algorithm in (a)...