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Table 1. Development of upper bound for maximum clique.

in Combining Semidefinite and Polyhedral Relaxations for Integer Programs
by C. Helmberg, S. Poljak, F. Rendl, H. Wolkowicz
"... In PAGE 8: ... The most expensive operation within each iteration is the factorization of a positive de nite matrix of size jV j + #non-edges + #ineq. The computational results are displayed in Table1 . The largest clique found so far is of size 23.... ..."

Table 4: Branch and cut results: feasible instances. Infeasible instances The instances not having a feasible solution are much harder to tackle by a branch and cut framework, since it is more di cult to nd linear constraints that improve the lower bound. The lack of combinatorial bounds also translates into very weak linear programming relaxations. Some constraints that have been incorporated in the framework are based on cliques in the interference graph in which the minimal distance between each of the links in the clique is (much) larger than one and the penalties of the constraints are all large. Also, we have been able to add constraints that, given we assign to a link from a certain set of frequencies, force either another link to be assigned or at least one constraint to be violated. Up to now, experiments have been performed for two test problems, which we proved to be infeasible (Table 5). The lower bounds, however, are very poor. For CELAR07 we obtained only poor solutions, since our strategies so far are designed to nd lower bounds, instead of good solutions. The time required to generate the lower bounds and best value is approximately 30 minutes.

in EUCLID CALMA Radio Link Frequency Assignment Project Report 2.2.1: Implementation and Testing of Polyhedral Techniques and Interior Point Methods
by K. I. Aardal, A. Hipolito, C. P. M. Van Hoesel, B. Jansen, C. Roos, T. Terlaky 1995
"... In PAGE 7: ... The generation of such solutions enormously reduces the amount of work to be performed. Table4 below lists some computational results using branch and cut on the feasible RLFAP instances using the pre-processed formulations in 3.... ..."
Cited by 2

Table 1: SDP approximation for EQP by iteratively adding sign constraints. The number and largest violation of sign constraints are displayed together with computation time and the lower bound.

in Semidefinite Programming in Combinatorial Optimization
by Michel X. Goemans, Franz Rendl 1997
"... In PAGE 11: ... Solve the SDP min 1 2L X such that diag(X) = e; Xe = me; X 0; and iteratively add only a few of the most violated sign constraints xij 0. Table1 contains some results using this approach. We note for instance that in the case of partitioning into k = 10 subsets, the initial lower bound of 275.... In PAGE 11: ... The value of the best feasible solution found is given in the column labeled `cut apos;. The computational behavior shown in Table1 is quite typical for SDP relaxations of combinatorial optimization problems. Most of the improvement is obtained in the rst few iterations.... ..."
Cited by 76

Table 1. Cliques found by construct and local

in On Maximum Clique Problems In Very Large Graphs
by J. Abello, P. M. Pardalos, M. G. C. Resende 1999
"... In PAGE 8: ... Because of the independent nature of the GRASP iterations and since our computer is configured with 20 processors, we created 10 threads, each independently running GRASP starting from a different random number generator seed. Table1 summarizes the first part of the experimental results. It shows, for each clique size found, the number of GRASP iterations that constructed or improved such solution, and from sizes 5 to 15, the number of distinct cliques that were found by the GRASP iterations.... ..."
Cited by 36

Table 5. Computational results Data Comp. Speedup # of Generated # of Evaluated # of Improved Name Time Estimation Subproblems Subproblems Solutions

in Solving the Maximum Clique Problem Using PUBB
by Yuji Shinano , Tetsuya Fujie, Yoshiko Ikebe, Ryuichi Hirabayashi
"... In PAGE 7: ... Ta- ble 4 displays data and the exact solution (maximum clique) of the problem instances. Table5 shows the result of the parallel branch-and- bound algorithm with 50 solvers. By the result of C250.... ..."

Table 5. Computational results Data Comp. Speedup # of Generated # of Evaluated # of Improved Name Time Estimation Subproblems Subproblems Solutions

in Solving the Maximum Clique Problem Using PUBB
by Yuji Shinano, Tetsuya Fujie, Yoshiko Ikebe, Ryuichi Hirabayashi
"... In PAGE 7: ... Ta- ble 4 displays data and the exact solution (maximum clique) of the problem instances. Table5 shows the result of the parallel branch-and- bound algorithm with 50 solvers. By the result of C250.... ..."

Table 17: Numerical results on maximum clique problems.

in Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm).
by Katsuki Fujisawa, Mituhiro Fukuda, Masakazu Kojima, Kazuhide Nakata 1997
"... In PAGE 24: ... SDPA started from the initial point (X0; y0; Z0) = (100I; 0; 100I) in all maximum clique problems. Table17 and 18 show numerical results. Recall that the number of constraints m is corresponding to \the number n(n ? 1)=2 ? jEj of node pairs having no edge quot; + 1: This implies that m can be of O(n2).... In PAGE 24: ...roblems. Table 17 and 18 show numerical results. Recall that the number of constraints m is corresponding to \the number n(n ? 1)=2 ? jEj of node pairs having no edge quot; + 1: This implies that m can be of O(n2). We see from Table17 that the computation time strongly depends on m. In Table 18, we see that part (II) depends more heavily on m than part(I).... ..."
Cited by 25

Table 1. Cliques found by construct and local

in AN OPTIMIZER IN THE TELECOMMUNICATIONS INDUSTRY
by Mauricio G. C. Resende
"... In PAGE 13: ... Because of the independent nature of the GRASP iterations and since our computer was configured with 20 processors, we created 10 threads, each independently running GRASP starting from a different random number generator seed. Table1 summarizes the first part of the experimental results. It shows, for each clique size found, the number of GRASP iterations that constructed or improved... ..."

Table 2: Improved Upper Bounds

in Iterative Relaxations for Iterative Flattening in Cumulative Scheduling
by Laurent Michel , Pascal Van Hentenryck
"... In PAGE 6: ... Algorithm IFlatIRelax was implemented in COMET, which is constraint-based language for local search (Michel amp; Van Hentenryck 2002; 2003). New Upper Bounds Table2 shows the lower and upper bound reported in (Nuijten amp; Aarts 1996) and used in the evaluation in (Cesta, Oddi, amp; Smith 2000). It also reports the new best upper bounds found by IFlatIRelax during the course of this research.... ..."

Table 2: Using the unfolded workflow in presence of constraints: cost/benefits

in A lightweight approach for QoS-aware service composition
by Gerardo Canfora, Massimiliano Di Penta 2004
"... In PAGE 8: ... In particular, a workflow containing a Loop over 3 services was considered, with the number of iterations varying from 3 to 20. As shown in Table2 , the unfolding approach can lead to an increase of the fitness function value, although this is almost always limited by the maximum time to be taken. In our experiments, we found improvements of the fitness function values varying from 7 to 11%, while the additional time needed for convergence was up to 2000%.... ..."
Cited by 4
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