ps6 - Urban OR Fall 2006 Problem Set 6(Due Wednesday...

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Urban OR Fall 2006 Problem Set 6 (Due: Wednesday, December 6, 2006) Problem 1 Problem 6.6 in Larson and Odoni Problem 2 Exercise 6.7 (page 442) in Larson and Odoni. Problem 3 Suppose we have a network G(N, A) such as the one pictured in Figure 1, which can be separated by an “isthmus edge”, (s, t) into two distinct sub-networks G(S, A s ) and G(T, A t ) such that S T = N and A s A t (s, t) = A. (Note that the set of nodes S includes node s and the set of nodes T includes node t.) Let H(T) be the sum of the weights, h j , of the nodes in the set T and H(S) be the sum of the weights, h j , of the nodes in the set S. (a) The following is known as Goldman’s majority theorem”: “If H(T) H(S) then the set of nodes T contains at least one solution to the 1-median problem on G(N, A).” Prove the theorem. To do so, assume that the solution is at some node y S and argue that J(y) J(t), a contradiction. J(.) is the objective function for the 1-median problem – see book. Note as well that t is the node on the G(T, A t ) “side” of (s, t). (b) Prove the following theorem: “If H(T) H(S) then one can find a solution to the original 1-median problem on G(N, A) by solving the 1-median problem on the sub- network G'(T, A t ) which is identical to G(T, A t ) except that the weight h t of the node t (on which the edge (s, t) is incident) is replaced by H(S) + h t .” To prove this statement argue as follows: We know from part (a) that the 1-median is in T. Show that for any node y T: J ( y ) = C + [ H ( S ) + h t ] d ( y , t ) + h j d ( y , j ) (1) j ( T t ) where C is a constant and (T-t) indicates the set of nodes, T, not including the node t. Why does (1) prove our theorem? (c) Using the theorems of parts (a) and (b) find very quickly the 1-median of the network shown in Figure 2. (For each node, an identification letter followed by the node’s weight
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is indicated; link lengths are noted next to each link.) Note that you do not have to consider the lengths of the edges in solving this problem. Problem 4 Consider the Traveling Salesman Problem with Backhauls (TSPB), a version of the TSP which is as follows: Suppose we have one “station point,” s, a set D of “delivery” points ( | D | = n) and a set P of “pick-up” points ( | P | = m). Assume the travel medium is the Euclidean plane and that all n+m+1 points in the problem are distinct, so that the Euclidean distance between any pair of points is positive. We want to design a tour of minimum length that has this description: a vehicle will begin from s, will visit first all the n points in D to deliver packages, will then (without first returning to s) visit all m points in P to pick up packages and will finally go back to s (where the tour ends). The following heuristic, based on the idea of the Christofides heuristic for the TSP, has been proposed recently for the TSPB (we shall leave Step 4 incomplete until the second question of our problem): Step 1: Construct: (a) TD, the minimum spanning tree of D, i.e., the MST that connects all n delivery points; (b) TP, the minimum spanning tree of P, i.e., the MST that connects all m pick-up points.
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