### Table 1. Typical computational time required to solve some hard random 3-SAT instances for di erent ratios and sizes N. Slightly below threshold, some rare instances may require much longer computational time (Sect. 3.4.1).

2001

"... In PAGE 6: ... The number of nodes added per minute ranges from 300,000 (typically obtained for =3:5) to 100,000 ( = 10) since unit propagation is more and more frequent as increases. The order of mag- nitude of the computational time needed to solve an in- stance are listed in Table1 for ratios corresponding to hard instances. These times limit the maximal size N of the instances we have experimentally studied and the num- ber of samples over which we have averaged.... In PAGE 6: ... These times limit the maximal size N of the instances we have experimentally studied and the num- ber of samples over which we have averaged. Some rare instances may be much harder than the typical times in- dicated in Table1 . For instance, for =3:1andN = 500, instances are usually solved in about 4 minutes but some samples required more than 2 hours of computation.... ..."

### Table 1.a present the number of symmetric benchmarks (the column #P gives the number of instances, #S the number of symmetric instances and % the percentage of symmetrical instances) with respect to the complete benchmarks collection. A large number of them contains symmetries (i.e. 38.5% of instances exhibit symmetries). As we can see, the easy problems are highly symmetric (80% of instances), whereas the medium category contains less symmetries (29.9%). Let us note that medium category is composed of a large fraction of random generated instances that rarely contains symmetries (see table 1.b). More interestingly, about half of the hard benchmarks repre- senting the most difficult QBFs instances (not solved in the last competition) contains symmetries (47.9%). Breaking such symmetries might help QBF solvers to handle such hard instances. Table

2004

"... In PAGE 5: ... Table1 . Results on QBF03 Evaluation problems 2 summarise the first experimental comparison between our hybrid solver Figure2 that detect and exploit symmetries and the OpenQBF solver i.... ..."

Cited by 1

### Table 1 F1 comparison for some rare categories

2004

"... In PAGE 6: ... The improvement should be more dramatic for categories with only few training examples to learn the corresponding classifiers. Table1 lists a performance comparison between binary MFoM (Gao, et al., 2003) and MC MFoM learning on 5 categories in which only less than 10 training instances are available.... ..."

Cited by 10

### Table 14 - RARE Implementation Options

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"... In PAGE 6: ...able 13 - RARE Example Heuristics with Strategies and Metrics............................................................................77 Table14 - RARE Implementation Options.... In PAGE 90: ... Table14 lists the functional areas associated with implementation options and their tradeoffs. Implementation options are ordered beginning with the currently preferred option for each functional area based on the benefits/drawbacks identified.... ..."

### Table 4.2 For the two examples quot;dense switchbox quot; and quot;augmented dense switchbox quot; marked with an asterisk, the execution of the branch and cut algorithm was stopped after the time given in the last column, because no further progress could be achieved. We believe that the values given in column 2 are optimal, but we are not yet able to prove this with the cutting plane algorithm. All other problem instances are solved to optimality. The running times in the last column are surely quite high. This is due to the fact that we were interested in finding an optimal solution. On the other hand, a provable quality guarantee of 5% can be given after at most 5 minutes for all these problem instances, which shows that our methodology is approaching practical usability. In fact, standard routing algorithms are rarely able to provide any quality guarantee at all.

### Table 4.4: Computational results using model (RQ). Given are times and numbers of minor iterations required to nd the best (optimal) feasible as- signment resp. a global minimum (if any). The times are with resp. without time for rounding the fractional solutions incorporated. Also, the number of feasible assignments (if any) that can be constructed from the nal (fractional) minimizer and the number of frequencies they use are given. more rare that a global minimum was found. This only happened for a few (relatively small) instances. In some cases a local minimum was found in which only the constraint on the number of frequencies to be used was violated, i.e. for the nal solution y = (x; z)T we found that xTQx = 0 while zTRx gt; 0. Still, the 25

### Table ii: List of typed non Latin variables 2In a few cases we will use these symbols denoting variables as constant symbols for auxiliary constants within the same domain as the corresponding variables, too. But these constants will only be used within a limited scope, for instance in proofs when we specialise universal quanti ed expressions or in the statement of theorems in order to denote the values of auxiliary terms. Therefore no misunderstandings should occur because of this. We also believe that the introduction of new symbols for these rare cases would turn the reading of the text more di cult.

### Table 9 Summary of characteristics of journals

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"... In PAGE 10: ... With respect to reference disciplines, our data shows that CS research seldom relies on research in other disciplines and in the rare instances that it does, it relies primarily on mathematics. Table9 presents a summary of the most important research characteristics in each of the 13 journals. The data indicate that while CS research addresses a diverse range of topics, there is a high degree of consistency in terms of the research approaches, research methods, and levels of analysis used to study these topics.... ..."