This preview shows pages 1–3. Sign up to view the full content.
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 subnetworks 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 1median 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 1median 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 1median problem on G(N, A) by solving the 1median 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 1median 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 (Tt) 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 1median of the network
shown in Figure 2. (For each node, an identification letter followed by the node’s weight
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 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 “pickup” 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
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 11/08/2011 for the course AERO 16.72 taught by Professor Hansman during the Fall '06 term at MIT.
 Fall '06
 hansman

Click to edit the document details