ORIE 321/521
RECITATION 8
Spring 2007
The purpose of this recitation exercise is to solve, by a combination of
hand computation, and with AMPL, the LP relaxation of the IP formulation
of the traveling salesman problem (TSP) (as was discussed in the lecture
notes) and then to also use this relaxation to solve the TSP itself.
Set up an AMPL formulation of the fractional 2matching problem, which
is defined as follows: given an undirected complete graph, assign a weights
to each edge, between 0 and 1, such that, for each node
i
, the total weight
assigned to edges that have
i
as one endpoint is exactly equal to 2. (Be sure
to use nonnegative variables, not binary variables.)
One simple way to do this is have two variables for each edge connecting
nodes i and j, x[i,j] and x[j.i], and to add the constraint that x[i,j]=x[j,i].
(And to set x[i,i]=0.) Your formulation should be compatible with the two
data sets posted on the web for this problem set (in terms of param names).
For the first data set, solve the fractional 2matching problem.
Is this an
optimal solution to the traveling salesman problem? Be sure to explain your
answer.
Are any of the cutset constraints
X
i
∈
S,j
6∈
S
x
[
i, j
]
≥
2
violated for some
S
for the optimal solution found for the fractional 2
matching problem?
If one of these inequalities is violated by the current
optimal LP solution
x
, we shall say that the corresponding set
S
is
bad
for
x
.
If there are any bad sets for the
x
found for the previous part, identify one,
and add it to your LP relaxation (which currently has just the “2matching”
constraints). (You may check this simply by inspection.) Use AMPL to solve
1
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this stronger LP formulation. What can you conclude about the resulting LP
optimal solution? Repeat this until you cannot identify any bad sets for the
resulting LP optimal solution. What can you say about the optimal solution
to this input for the TSP?
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 Spring '07
 SHMOYS/LEWIS
 Operations Research, Optimization, optimal solution, AMPL LP

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