### Table 1: Comparison of four codes for Uncapacitated MWCP It is clear from the results that the simpler codes have di culty with the instances as the column generation technique progresses. The fully featured code, however, is robust to the changes in the node weights associated with the dual changes in the linear relaxation of CF as newer columns are added to it. The Maximum Weighted Cluster Problem can be also formulated using variables zij (corresponding to edge (i; j) in the cluster) and xi (corresponding to node i in the cluster). The problem is then to

1998

"... In PAGE 25: ...3 269 1 613.0 166 1 Table1 0: Target of 25.0, Capacity 512 method is even better as the number of subproblems solved by the combinatorial method far exceeds those solved by CPLEX in more time.... ..."

Cited by 5

### Table 3 Primal Dual

"... In PAGE 10: ... If the ith primal variable yi is urs, the ith dual constraint is an equality constraint. Table3 gives a more complete relationship between nonnormal primal and dual problems. Table 3 Primal Dual... ..."

### Table 4: Results for solving the dual GP problems.

1996

"... In PAGE 28: ... However, they only present results for solving small-sized GP problems and the results presented in [36] indicate that these methods may not be as e cient as \direct quot; primal-dual methods. In Table4 results for the GP dual (49) of the test problems are presented. First, the table shows the number of constraints and variables before and after presolve.... ..."

### Table 4: Results for solving the dual GP problems.

"... In PAGE 28: ... However, they only present results for solving small-sized GP problems and the results presented in [36] indicate that these methods may not be as e cient as \direct quot; primal-dual methods. In Table4 results for the GP dual (49) of the test problems are presented. First, the table shows the number of constraints and variables before and after presolve.... ..."

### Table 5: Accuracy results for the dual GP problems.

1996

### Table 5: Accuracy results for the dual GP problems.

### Table 5 Actual and projected times neded to compare large sequences on comodity hardware. Single procesor times are on actual values, cluster times are based on an ideal linear spedup for distributing the sequence acros the cluster.

"... In PAGE 7: ... 7.2 Problem Size Table5 lists the times to compare sequences of diferent lengths at the dual procesor rate using comod- ity hardware. Asuming a linear spedup is achievable on a cluster of such machines, we also show extrapolated re- sults for large sequences.... ..."

Cited by 1

### Table 3: Primal Dual algorithm

"... In PAGE 9: ... The running time increases with the accuracy needed. The next Theorem states the running time and the correctness of the algorithm shown in Table3 . The proof is omitted here due to lack of space, but is similar to the one in [31].... In PAGE 9: ... Theorem 4. The algorithm in Table3 computes a (1 ) 3 optimal solution to the ow scaling problem in time polynomial in Q; L; n and 1 , where Q is the number of com- modities, L is the number of constraining sets, and n is the number of nodes. 6.... ..."

### Table 1: Relationships Between Primal and Dual Problems (page 241 in [3])

"... In PAGE 3: ...2 Rules for Formulating the Dual In the dual of a linear program, there is one dual variable corresponding to one primal constraint, and one dual constraint for each primal variable. Table1 gives the rules in formulating the dual of a primal LP. Connection with Lagrange dual function (in x5.... ..."

### Table 4.1: Dual algebraic approaches for primal and dual multidimensional transporta- tion problems of several types.

2002