MO412 - Thiago Laitz

A post office must plan a route to visit five cities (A, B, C, D, E) and return to the start. Distances (cost units) are given in the graph below. The graph satisfies the triangle inequality and is undirected. Apply the cheapest insertion heuristic (from the traveling salesperson problem), starting with triangle ABC (initial cycle: A → B → C → A). At each step, insert the vertex that yields the smallest cost increase according to the cheapest insertion heuristic. Which final cycle is obtained and what is its total cost? A) A → B → D → C → E → A, total cost 25 B) A → B → E → D → C → A, total cost 27 C) A → C → B → D → E → A, total cost 29 D) A → D → B → E → C → A, total cost 27 E) None of the above Original idea by: Thiago Soares Laitz

Considering an undirected graph network with no degree correlation. Let (e_jk) be the probability to find a node with degree j and degree k at the two ends of a randomly selected edge and let (q_k) be the probability to have a degree k node at the end of a link. You are given the average degree: <k> = 10 and the following degree-distribution probability masses (all other (p_k) can be anything consistent and are not needed): p_20 = 0.04 p_30 = 0.03 p_50 = 0.02 Using only the information above, compute the 3×3 submatrix of e_jk for degrees 20, 30, and 50. Which option matches this block? Consider the theoretical e_jk for the network with no degree correlation and that the matrix rows and columns correspond to degrees 20, 30, and 50 in that order. A) {0.0064  0.0072  0.0080 } {0.0072  0.0081  0.0090 } {0.0080  0.0090  0.0100 } B) {0.0032   0.0045   0.0060 } {0.0045   0.0075   0.0080 } {0.0060   0.0080...