### Table 4: Simulations for the exact ratios and the upper bounds of RALG in Theorem 11.

"... In PAGE 12: ... The branch amp; bound algorithm is very time consuming so we were able to run it only for small values of (N, K). See the fourth, fifth columns in Table4 for the average and worst ALG(L) OPT(L) ratios, respectively. The sixth column is the percentage of instances whose ratio was one, meaning that ALG(L) found an optimal solution.... In PAGE 12: ... The number could go up to nearly 50% in the worst case. In the final three columns of Table4 , we try to demonstrate numerically the upper bounds in this paper. Since RALG describes the asymptotic behaviour of the algorithm, we tested ALG with input lists having as many as 1000 items.... In PAGE 13: ... each run of the algorithm, we collect the values of c (number of type-c bins) and d (number of type-d bins) and use d c to replace M in (17). As can be read from Table4 , the numerical upper bounds spread from approximately one (column 6) to several tens (the last column) and the average of them falls around 4. Let us take as an example the case of 5000 runs for N = 1000, K = 7.... ..."

### TABLE I TIME SEPARATION BOUNDS FOR SYSTEM OF FIG. 7. Mi DENOTES THE EXACT UPPER BOUND OF ai ? ai?1 FOR i 2.

1998

Cited by 8

### Table 1: Two proven upper bounds and the conjectured exact bound

"... In PAGE 9: ... So if this conjecture is correct, the family of complete graphs K4k+4 is an example of a family of bar k-visibility graphs with the maximum number of edges. Table1 shows the two proven upper bounds on the number of edges in a bar k-visibility graph, together with the conjectured exact bound. 4 Thickness of Bar k-Visibility Graphs By Theorem 4, K8 is a bar 1-visibility graph, and thus there are non-planar bar 1-visibility graphs.... ..."

### Table 1. Upper and lower estimates for exact upper bounds: short path lengths. We see that while we have an optimal upper bound for DM(1; 2), other metrics require longer paths. The next table gives the tighter upper and lower estimates that are within 2 % from the exact.

### Table 2. Upper and lower estimates for exact upper bounds: long path lengths Now, we turn to the lower bounds. To nd them we nd the upper bounds for DM(2; 1), DM(2; 3), DM(2; 4), DM(5; 2).

### Table 1 Lower and upper bounds for tr(A?1) Matrix (order) \Exact quot; MC estimation Lower bound Upper bound

"... In PAGE 7: ...y Wathen [17]. In our numerical experiment, we let nx = ny = 10. Then the matrix A is of order n = 341 and = 0:25 and = 4:5. The bounds for tr(D1 2 A?1D12 ) and tr(ln(D?12 AD?1 2 )) are tabulated in Table1 and 2. The Kantorovich apos;s upper bound for tr(D12 A?1D1 2 ) is 1:70974 103 and the Robinson and Wathen apos;s lower and upper bounds are 4:53280 102 and 9:19952 102, respectively.... ..."

### Table 2 Lower and upper bounds for tr(ln(A)) Matrix (order) \Exact quot; MC estimation Lower bound Upper bound

### Table 3: Upper bounds when each variable occurs exactly once.

1999

"... In PAGE 5: ... None of our lower bound proofs are valid under this restriction, and we have improved some of the upper bounds. See Table3 for a summary. The improved upper bounds all rely on the determinant polynomial being multi-a ne when no variable occurs twice.... ..."

Cited by 7