### Table 1. Exact and Approximate distance oracles for weighted undirected graphs.

2006

Cited by 6

### Table 1. Exact and Approximate distance oracles for weighted undirected graphs.

2006

Cited by 6

### Table 1: All available exact and approximate distance oracles for weighted undirected graphs

2001

Cited by 108

### Table 1: All available exact and approximate distance oracles for weighted undirected graphs

2001

Cited by 108

### TABLE I. ALL AVAILABLE EXACT AND APPROXIMATE DISTANCE ORACLES FOR WEIGHTED UNDIRECTED GRAPHS

### Table 4.2 I/O-e cient algorithms for problems on undirected graphs.

### Table 4.2 I/O-e cient algorithms for problems on undirected graphs.

### Table 4: Comparison over GraphBase undirected graphs for variable and value symmetries.

2004

"... In PAGE 39: ... For variable and value symmetries, a total of 233 undirected random in- stances were treated. We evaluated variable and values sym- metries separately and then together in Table4 . This table shows that, as expected, value symmetries and variable sym- metries each increase the number of solved instances.... In PAGE 48: ...5 The Impact of Restarts After seeing that symmetry breaking is bene cial even at the tremendous costs that SSB ltering incurs, we are curious to see how restarts affect the landscape. We show the perfor- mance of the restarted algorithms in Figure 5 on the bench- mark sets with 15 variables and in Table4 on the benchmark sets with 30 variables. The comparison of Figure 5 with Figure 4 shows that the algorithm that is unaware of symmetries can bene t greatl from restarts.... In PAGE 48: ... The fact that this was not visible in Figure 4 is due to the time limit (that we needed to impose to conduct our experiments within reason- able time) which arti cially decreases the variance of the slow algorithm NO. Table4 also shows very clearly that NR per- forms far better than NO. We get a similar picture when comparing the performance of SO and SR.... In PAGE 48: ... When we perform full SSB ltering, we see that restarts do not help on the benchmark with 15 variables and values. Only as we tackle large and very hard problems, FR starts to outperform FO as can be seen in Table4 when we consider instances with a global GCC. This leads to a suprising conclusion: Just breaking value symmetry in combination with restarts is in many cases com- petitive with full SSB! Compare FR and SR on global AllD- ifferent instances in Figure 5, or on global GCC instances in Table 4, for example.... In PAGE 48: ... Only as we tackle large and very hard problems, FR starts to outperform FO as can be seen in Table 4 when we consider instances with a global GCC. This leads to a suprising conclusion: Just breaking value symmetry in combination with restarts is in many cases com- petitive with full SSB! Compare FR and SR on global AllD- ifferent instances in Figure 5, or on global GCC instances in Table4 , for example. Of course, SSB is still the clear winner in the critical region on our large benchmark set with 30 vari- ables, but the good performance of restarted sibling- ltering is still astonishing.... In PAGE 50: ...5K - 42 112 - 90 90 - 349K 12 X - 0 X - 0 X - 0 X - 0 X - 0 X - 0 Table 3: Times per choice point in micro seconds and number of search nodes (averages over 50 instances per data point). AllDifferent GCC One-Shot Restarted One-Shot Restarted FO SO NO FR SR NR FO SO NO FR SR NR 2 100 100 98 100 100 100 46 62 86 78 100 100 3 100 100 90 100 100 100 36 36 78 46 100 100 4 86 98 76 76 92 100 41 40 55 41 88 47 5 60 72 40 56 56 88 30 30 24 28 54 24 6 52 23 4 54 18 51 14 10 2 12 8 10 7 61 41 2 65 20 9 14 4 2 10 12 6 8 80 84 0 82 52 0 4 0 0 14 0 0 9 96 100 0 98 76 0 24 2 0 34 0 0 10 100 100 36 100 100 0 26 10 0 76 2 82 11 100 100 96 100 100 100 52 50 48 70 100 100 12 100 100 100 100 100 100 100 100 100 100 100 100 Table4 : Histogramm for the benchmark sets with 30 variables and values, 12 variables per constraint, and biased variable selection. The rst column gives the number of values per constraint, the numbers in the table the percentage of instances... ..."

### Table 1: Comparison over GraphBase undirected graphs. All solutions 5 min.

2004

"... In PAGE 9: ...6 Edges Nodes Assign golf222 464/2810 206/1020 150/946 golf322 1290/7629 548/2681 423/2559 steiner5 333/5898 157/2080 102/1977 steiner6 6215/65830 2226/22321 2035/21980 Table1 : Size of the graphs obtained for social golfers and Steiner triplets with/without pruning. Of course, care needs to be taken when inconsistent assignments have been used to build the graph.... In PAGE 17: ... 5.2 Dimacs graph coloring benchmarks Table1 shows the results of the methods on some graph col- oring benchmarks of Dimacs. It gives the number of nodes of the search tree and the CPU time for each method.... In PAGE 17: ...1c 84 - - - - 28044984 3096.0 (500-121275) Table1 : Dimacs graph coloring benchmarks succeed to solve 9 benchmarks among the 22 proposed. For space reason, we report here the results on the most rele- vant Dimacs problems to compare DSATUR and our method, but it is important to inform the reader that all the others DIMACS problems which are solved by DSATUR are also solved by SFC-weak-dom with a comparable performance.... In PAGE 23: ... We would therefore expect the search for lex-inspired early backtracks to be expensive, with not many useful domain deletions being returned. This ex- pectation is realised in our results given in Table1 . The better results for GAP-SBDD are in part because GAP- SBDD has special support for problems with Boolean variables.... In PAGE 24: ...21 Table1 : GAP-SBDD vs GAPLex. Problem class: BIBDs modelled as binary matrices.... In PAGE 31: ... Solv- ing Psym can still be longer than solving P because ltering thanks to the constraints of Crest also prunes the search tree. d dn T(n, m = d) 2 2n O( 2n pn) 3 3n O(3n n ) n nn O( pn (2 )n2 nn) Table1 : Comparison between the size of the search space of a CSP and the orbit size of a canonical solution in the case of variable symmetries. For instance, the line d = 3 shows that if the complexity of computing the canonical solutions of Psym is lower than O(3n n ) and the number of these canon- ical solutions is lower than n, then the overall complexity of solving P is reduced.... In PAGE 39: ...) are posted, and nally the full variable symmetry (FVS) that breaks all variable symmetries. Results are shown in Table1 and 2. In those runs, the prepro- cessing time has not been considered.... In PAGE 46: ... In our example, those are the val- ues 1; 2, and 4. Figure 2 (B) shows the data structure after another variable has been instantiated by adding (X4; 2) to UNBIASED BIASED 15 15 30 VPC AllDiff GCC AllDiff GCC AllDiff GCC 2 100 100 100 100 100 98-100 3 100 100 100 100 100 52-100 4 100 98-100 100 92 84-96 14-80 5 100 100 88 66 52-82 0-54 6 100 98-100 68 18-32 26-76 0-50 7 94 96-100 26 4-18 6-40 0-50 8 90 88-94 18 0-6 0-16 0-34 9 84 92 0 0-2 0 0-32 10 48 58 0 0 0 0-22 11 16 14 0 0 0 0-22 12 4 0 0 0 0 0 Table1 : Percentages of feasible solutions in the different benchmark sets for different numbers of values per constraint (VPC). We give ranges where even the best algorithm hit the time limit of 600 seconds.... In PAGE 47: ... The number of variables per constraint is xed at 12 while the number of values per constraint runs from 2 to 12, thus giving us a range of differently constrained instances. Table1 shows the percentage of feasible instances out of 50 randomly generated ones. In addition, we vary the constraint over all variables and values (GCC or AllDiffer- ent), and we select variables either uniformly or in a biased fashion, while values are always selected uniformly.... In PAGE 62: ... For this reason, symmetries are broken using SBDS in [15]. We present in Table1 . and Table 2.... In PAGE 63: ...60 Table1 : Results for computing all solutions for graceful graphs Graph SBDS DLC SOL BT sec. SOL BT sec.... ..."

### Table 4: Results for the vertex packing problem.

1996

"... In PAGE 29: ...he graph between 0.1 and 0.9. The code used by the authors was limited in the sense that the cutting plane algorithm was run only in the root node, and that only primitive branching rules were available. In Table4 we report the results for the 0.2 density instances.... ..."

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