### Table 6: Computational statistics of the MILP model of Maravelias and Grossmann for Pr3, for variable processing times.

2004

"... In PAGE 28: ... Feasible solutions are found for certain time grids, but when finer time grids are used the model becomes intractable. To illustrate, we report in Table6 the computational statistics of the MILP model of Maravelias and Grossmann, for various time grids, for instance Pr3. As shown, models with up to 11 time points are solved with reasonable computational effort, but when more than 12 time points are used, the model becomes intractable.... In PAGE 29: ... For the maximization of profit we obtained better solutions than the ones obtained by the MILP model, and with significantly less computational effort for instances Pr1 and Pr2. As explained above, the MILP for instance Pr3 yields a suboptimal solution (profit = 16) in 356 CPU seconds but it becomes intractable for finer time partitions ( Table6 ) and no better solution can be found. The proposed scheme, in contrast, yields a much better solution (profit = 20.... ..."

Cited by 7

### Table 1: Computational complexities for algorithms to compute the graph network measures.

"... In PAGE 4: ...Table 1: Computational complexities for algorithms to compute the graph network measures. Table1 summarises the computational complexity of the algorithms we need consider here in terms of the number of vertices N. In general we are severely limited in use of the algorithm for counting or enumerating the elementary circuits due to its intractability for anything but small networks of around 60 vertices and mean degree of around 6.... ..."

### Table 1. Values of tm i when: (a) zi = 10, Zk = 50, 1 g = 10, and PB = 0:01; and (b) zi = 5, Zk = 50, 1 g = 10, and PB = 0:16.

"... In PAGE 18: ... The expected delay of item di is evaluated by computing t5 i , that is truncating at the fth term the series giving ti. Indeed, as shown in Table1 for zi = 10, Zk = 50, 1 g = 10, and PB = 0:01 and for zi = 5, Zk = 50, 1 g = 10, and PB = 0:1, at the fth term the series giving ti is already stabilized up to the fourth decimal digit. Since the data allocation problem is computationally intractable when items have non-unit lengths, lower bounds for non-unit length instances are derived by transforming them into unit length instances, as explained in Subsection 2.... ..."

### Table 3. The complexities of the problems of computing large models (large-bound problems, the cases of = \= quot; and \ quot;). The problems speci ed by = \ quot; and concerning the existence of models are in P. Similarly, the problems speci ed by = \ quot; and involving Horn programs are solvable in polynomial time. Lastly, the problem ST 0 =(H) is in P, as well. These problems are in P even without xing k and eliminating it from input. All other problems in this group have higher complexity and, in all likelihood, are xed-parameter intractable. One of the problems, M0 =(N), is W[1]-complete. Most of the remaining problems are W[2]-complete. Surprisingly, some problems are even harder. Three problems concerning supported models are W[3]-complete. For two problems involving stable models, ST 0 =(A) and ST 0 (A),

### Table 4. Mean FDC over all instances in each testset, compared with the mean D0D7CR.

in Search Space Features Underlying the Performance of Stochastic Local Search Algorithms for MAX-SAT

"... In PAGE 7: ... Influence of variability and granularity of clause weights on FDC. The results re- ported in Table4 indicate that, with the exception of our set of unweighted instances, there are small, but systematic differences in FDC depending on the variablity and gran- ularity of clause weights: Mean FDC values increase monotonically with granularity and variability of clause weights. Compared to the respective ACL results, the variation of FDC values within each test-set is considerably higher (as reflected in the higher 5 Since measuring CB exactly was computationally intractable for the instance types and sizes used here, we determined putatively optimal solution qualities as described in [17] and ap- proximated CB for each instance using the set of unique solutions of that quality obtained from... ..."

### Table 2: aBDD Sampling for multi-output circuits. Note, multiplier c6288 is ommitted since it is provably intractable for OBDDs.

"... In PAGE 5: ... Compared with cube based sampling approaches, our method is also superior (Figure 3) since our method does not have the large deviation problem. Experiment 3 ( Table2 ): We performed another set of experiments to show the efficacy of window based methods on multiple output functions. It is known that sifting works very well for ISCAS85 cir- cuits [11] and for many circuits, there may not be scope for significant improvement.... ..."

### Table 1 gives an estimate of the number of transistors required to implement delayed issue with 5 queues of depth 8, 64 registers, with all queues moving in lockstep (second column), and with two sets of queues that can move independently (third column). Shown are transistor counts for the buffers themselves and for the expensive computations. The remaining components (sense amps, row decoders, shifters, random logic, etc.) consist of a few hundred transistors each and are lumped together in overhead . We conservatively estimate the overhead at 10,000 transistors for a single set of queues and 15,000 transistors for two sets of queues. This gives totals of 67,016 transistors and 104,784 transistors respectively. By comparison, transistor counts for dynamic out-of-order issue logic range from 141,000 for a small window size of 20 instructions [Farrell98] to as many as 850,000 for two 28 entry reorder buffers [Gaddis96].

2000

"... In PAGE 10: ...overhead 10,000 15,000 Total 67,016 104,784 Table1 : Estimated transistor counts for 5 queues of depth 8, 64 registers With all queues moving in lockstep, the instructions, addresses, and bit vectors rd, wd can all be stored in register SRAM with an index pointing to the head of the queues. However, with multiple sets of queues that can move independently, the bit vectors rds, wds must be physically arranged by depth or the computation of the hst becomes intractable.... ..."

### Table 2 Comparison of the proposed method versus four other heuristics developed for SLPa;b

"... In PAGE 13: ... However, the proposed algorithm is computationally e2cient than [8] as the second is intractable for problems with twelve machines and above. Table2 shows the results obtained through the proposed algorithm and three other heuristics for problems in the second set. As seen in this table, the proposed ant algorithm has obtained the best solution for all the eight problems outperforming all the heuristics in terms of the OFV.... ..."

### Table 1: Some vertex subset properties expressed as ( ; )-sets, with tractabil- ity/intractability cuto value for the partitioning problem (1 means tractable for all k values). Reference * if cuto value proved here.

1998

"... In PAGE 1: ... Gen- eralized dominating sets, introduced by Telle in [8] and de ned formally in the next section, are parameterized by two sets and of nonnegative integers. Many well-studied vertex subset properties with applications in facility location and network communication can be expressed as ( ; )-sets [8, 1, 2, 3], see Table1 . In this paper we present a systematic study of the complexity of partitioning vertices of a given graph G into k ( ; )-sets, for varying values of the parameters k; ; .... In PAGE 4: ... This is not surprising, as the number of partitions of n vertices into k classes is not polynomial in n for any k 2. 3 NP-completeness results The values of and used to describe the vertex subset properties listed in Table1 are con- ned to f0g; f0; 1g; f1g; N and P. For = f0g we know from Fact 5 that the corresponding partition problems are easy.... ..."

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