### Table 3: Avg and Max Data Graph Size (edge set ED)

2006

"... In PAGE 7: ... BFS is slightly better since all paths are explored in unison. In Table3 , we show the average (first number) and maximum (second number) size of the data graph for the same experiment. We notice that for k=1 the data graph size is on the average 207 with a maximum instance of 1189 (edges).... ..."

Cited by 3

### Table 3: Avg and Max Data Graph Size (edge set ED)

"... In PAGE 35: ... BFS is slightly better since all paths are explored in unison. In Table3 , we show the average (first number) and maximum (second number) size of the data graph for the same experiment. We notice that for k=1 the data graph size is on the average 207 with a maximum instance of 1189 (edges).... ..."

### Table 2. Initial edge-set for 100,000-city TSP Edge-set |E|-initial |E|-final CPU Time (seconds)

2002

"... In PAGE 40: ... Although we do not use repeated runs of Lin-Kernighan to obtain an initial tour, the Padberg-Rinaldi tour-union method remains an attractive idea since it provides an excellent sample of the edges of the complete graph (with respect to the TSP). In Table2 , we compare the Padberg-Rinaldi idea with the k-nearest set on our 100,000-city instance. The tours were obtained with short runs of the Chained Lin- Kernighan heuristic of Martin et al.... In PAGE 40: ...2. Adding and deleting edges The results in Table2 indicate the growth of the cardinality of the core edge-set as the combined cutting-plane and column generation algorithm progresses. To help limit this growth, we are selective about the edges that are permitted to enter the core and we also take steps to remove edges from the core if they do not appear to be contributing to the LP solution.... ..."

### Table 7. 1,000,000-city subtour bound: choice of initial edge-set Edge-set CPU Hours

2002

"... In PAGE 55: ... The tests reported in Table 6 were run with the initial edge-set consisting of 4 least-reduced-cost edges meeting each city, together with a tour generated by a greedy algorithm. In Table7 , we compare the total CPU time needed to obtain the 1,000,000-city sub- tour bound starting with three different initial edge-sets. The 10 Tours set is obtained using short runs of Chained Lin-Kernighan to generate the tours; the 4-Nearest set consists of the 4 least-cost edges meeting each city, together with a tour generated by a greedy algorithm; the Fractional 4-nearest is the set used in Table 6.... In PAGE 55: ... The 10 Tours set is obtained using short runs of Chained Lin-Kernighan to generate the tours; the 4-Nearest set consists of the 4 least-cost edges meeting each city, together with a tour generated by a greedy algorithm; the Fractional 4-nearest is the set used in Table 6. The best of the results reported in Table7 was obtained with the fractional 4-nearest edge-set. The running time can be improved further, however, by increasing the density of the set, as we report in Table 8.... ..."

### Table 1.1: Edge set of Example 1

"... In PAGE 20: ...425 G2 20 51 [6], Table 3 p.425 G3 20 46 [15], Table1 p.52 G4 21 48 [2], Figure 2 p.... ..."

### Table 2: MAX-DICUT: Mean number of iterations ( ips) with standard deviation. OB is the oblivious search, NOB the non{oblivious one, see text for details. The mean number of iterations of the di erent LS-based components is illustrated in Table 2. Let us note that LS-NOB requires between 22 % (density 0.1) and 38 % (density 0.9) more iterations than LS-OB. If OB is started from a NOB local optimum, the local optimum of the OB function is found in a very small number of additional iterations (line NOB amp;OB in Table 2). 3.2 MAX IND SET in cubic graphs De nition 6 Given a graph G = (V; E), V being the vertex set and E the edge set, the Maximum Independent Set problem consists of nding a subset I V such that no two vertices in I are joined by an edge in E ( i 2 I and j 2 I =) (i; j) 62 E) and whose cardinality is maximized.

### Table 3 The number of k{Feedback Edge Set obstructions for k 6. k-FES # connected with and also

1996

"... In PAGE 37: ... This obstruction is easily constructed by 4 vertex triples, as promised by our direct enumeration algorithm. Table3 shows a summary of how many k{Feedback Edge Set obstructions there are for k 6. The third column of the table gives the counts for the number... ..."

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### Table 3: The number of kn7bFeedback Edge Set obstructions for k n14 6.

1996

"... In PAGE 29: ... This obstruction is easily constructed by4vertex triples, as promised by our direct enumeration algorithm. Table3 shows a summary of how many kn7bFeedback Edge Set obstructions there are for k n14 6. The third column of the table gives the counts for the number of connected obstructions without vertices of degree 2.... ..."

Cited by 3

### Table 2: Example tabular chromosome used to encode for a degree-3 spanning tree in a 9-node graph via the RPM. At each step in generating the tree the chromosome is consulted: For each vertex i in the tree having degree di lt; d the allele a(i; di) is looked up in the chromosome in position (i; di). The allele represents the choice in the edge list li in table T which will be selected to go forward into the low-weight edge set L. The highlighted values in the table represent the values that will be looked up in the iteration where the next edge is being selected in the partially constructed tree given in Figure 1.

2000

"... In PAGE 15: ... That is, for vertex i having current degree di, we choose the a(i; di)th lowest cost edge from list li. The allele values a, 1 a n, for each vertex i at each degree level di lt; d are given as input to RPM in the form of a tabular chromosome or solution vector (see Table2 ). So, RPM is a means of decoding a chromosome into a valid low-weight d-ST.... In PAGE 17: ...Table 3: Updated edge choice lists for the partially constructed tree in Figure 1. The highlighted choices are those that are encoded for by the chromosome in Table2 . These edges are placed in the low-weight edge set L and the lightest is chosen.... In PAGE 17: ... i, the latter having a current degree, di. In Table2 , the alleles that will be used to help in choosing the next edge in the partially constructed tree in Figure 1 are shown in bold face. These allele values are then used to select the edges that will go into the low-cost edge set L.... ..."

Cited by 6

### Table I summarizes the performance of the BDD solver with four representative orderings. Column (a) corresponds to the default ordering used by BuDDy; this ordering cannot solve real benchmarks in a reasonable time. Column (b) is another example of a bad ordering, with h1 before v1. This ordering already allows the solver to finish on small inputs. The last two columns show the performance when using a good domain arrangement, without interleaving v1 and h1. The performance improvement is dramatic. The difference between last two columns shows the effect of interleaving v1 and v2. This effect is much less significant. The BDD for the edgeSet relation is smaller when v1 and v2 are interleaved, and we observed fewer garbage collections. The pointsTo relation has the same size with either ordering. On small inputs (s/t), the two orderings yield comparable performance. On large problem sets (ns/nt), interleaving v1 and v2 gives much better performance.

2003

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