### 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.

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 21: Non-approximability of various levels of the Function Bounded NP Query Hierarchy. y Applies only to complete problems.

"... In PAGE 89: ...P query hierarchy. Though the correspondence is not exact ([Kre88, p. 492]; [CP91, p. 243]), there is a pattern of approximability and non-approximability (see Table21 ). This pattern may assume greater signi cance in the light of future discoveries of lower limits on approximability.... ..."

### Table 21: Non-approximability of various levels of the Function Bounded NP Query Hierarchy. y Applies only to complete problems.

"... In PAGE 82: ...P query hierarchy. Though the correspondence is not exact ([Kre88, p. 492]; [CP91, p. 243]), there is a pattern of approximability and non-approximability (see Table21 ). This pattern may assume greater signi cance in the light of future discoveries of lower limits on approximability.... ..."

### TABLE 1.- SUMMARY OF DFP/T PERFORMANCE ON SIX TEST PROBLEMS WITH UPPER AND LOWER BOUND CONSTRAINTS

1975

"... In PAGE 37: ...TABLE1 V.- THE X-SPACE AND Z-SPACE STARTING POINTS FOR SEVEN TEST PROBLEMS WITH LINEAR CONSTRAINTS Problem Starting points (z-space) Rosenbrock (III) (0.... ..."

### Table 1. Quantitative Comparison of the Approximation Algorithms. average minimum maximum

1997

"... In PAGE 13: ... In addition, the size of their output has to be taken into account to esti- mate the gain in runtime. Table1 summarizes the average, minimum, and maximum numbers of focal elements both in the original data and their approximations for each candidate algorithm after the fth combination. Remarks: 1.... In PAGE 13: ... For technical reasons the algorithms were not run with identical test data. However, the statistical data in Table1 indicate that the aver- age problem size was approximately equal for all candidates.... In PAGE 14: ... This is due to the fact that only relatively few focal elements with extremely low values are removed from the input data. The average size of a mass function approx- imated with this method is 440 focal elements, the maximum is even more than 1800 (see Table1 ). As a consequence the gain in runtime is the least among all candidates|taking into account both the time to compute the combinations and the approximation itself.... ..."

Cited by 14

### Table 3 Improving on CONC2 Problem

"... In PAGE 16: ... Note that CONCORDE was not able to find any violated constraints for this x using its default separation routines, which in particular included its separation heuristics for comb constraints. The results are shown in Table3 . Column 2 lists the optimal tour value, column 3 lists the lower bound CONC2 value and column 4 lists the new lower bound ob- tained using the new DP-constraints found by DP-CUTFINDER (note that OTL denotes that the optimal tour value was reached).... In PAGE 16: ... For pcb442 and rat783, adding the violated DP-constraints found by DP-CUTFINDER had no effect on the lower bound. Another interesting observation from Table3 deals specifically with the prob- lems kroD100, kroB200 and pr439. For all three of these points there was an increase in the lower bound, yet none of the violated DP-constraints we found for these problems were tight for the final solution.... ..."

### Table 4. Experimental results on optimal test access architecture design under power constraints: (a) S1 (b) S2.

2000

"... In PAGE 5: ... Gi can be obtained from power models for core i. Experimental results for power-constrained test access archi- tecture design for S1 and S2 are shown in Table4 . For our ex- periments, we approximated Gi by the number of gates in core i.... In PAGE 5: ... On the other hand, for higher values of W, the testing time is affected substantially. For example, in Table4 (a), for W 24 and power budget of 300 units, the testing time does not decrease with an increase in W due to power constraints. In some cases, the ILP problem may even be infeasible for higher test widths, e.... In PAGE 5: ...g. in Table4 (b) with W = 48 and power budget of 300 units for S2. Comparing with Table 2, we note that the width distribution is also significantly different due to power constraints.... In PAGE 5: ... This is achieved using a width distribution of (10,10) and test bus assignment (2,2,2,2,2,2,2,2,2,1). However, as seen from Table4 . for test width W = 24, the test bus assignment has to be changed to meet power constraints, and the minimum testing time increases to 471900 cycles.... ..."

Cited by 44

### Table 4. Experimental results on optimal test access architecture design under power constraints: (a) S 1 (b) S 2 .

"... In PAGE 5: ... G i can be obtained from power models for core i. Experimental results for power-constrained test access archi- tecture design for S 1 and S 2 are shown in Table4 . For our ex- periments, we approximated G i by the number of gates in core i.... In PAGE 5: ... On the other hand, for higher values of W, the testing time is affected substantially. For example, in Table4 (a), for W 24 and power budget of 300 units, the testing time does not decrease with an increase in W due to power constraints. In some cases, the ILP problem may even be infeasible for higher test widths, e.... In PAGE 5: ...g. in Table4 (b) with W =48 and power budget of 300 units for S 2 . Comparing with Table 2, we note that the width distribution is also significantly different due to power constraints.... In PAGE 5: ... This is achieved using a width distribution of (10,10) and test bus assignment (2,2,2,2,2,2,2,2,2,1). However, as seen from Table4 . for test width W =24, the test bus assignment has to be changed to meet power constraints, and the minimum testing time increases to 471900 cycles.... ..."

### 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.

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 2: Improved Upper Bounds

"... 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.... ..."