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# rec8 - ORIE 321/521 RECITATION 8 Spring 2007 The purpose of...

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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 2-matching 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 non-negative 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 2-matching problem. Is this an optimal solution to the traveling salesman problem? Be sure to explain your answer. Are any of the cut-set 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 “2-matching” 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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