### Table 1: Dimacs graph coloring benchmarks

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 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 39: ... Regarding the mean time, the full variable sym- metry breaking constraint has a clear advantage. This mean Table1 : Comparison over GraphBase undirected graphs. All solutions 5 min.... 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 1. Some special families of graphs

"... In PAGE 4: ... If M is a square matrix we will only write ME for the minor of M given by the rows and columns indexed by R. In Table1 of Chapter 2, Section 1 some more notational conventions are listed.... In PAGE 9: ...8 Bullet u Bullet v Bullet r Bullet w gt; gt; gt; gt; gt; Figure 1 An example digraph ~ G Bullet u Bullet v Bullet r Bullet w Figure 2 The underlying graph G of the digraph in Figure 2 Notation Meaning V(G), p set and number of vertices of G E(G), q set and (weigthed) number of edges of G NG(v), No G(v), Ni G(v) neighbourhood, successors and predecessors of vertex v in G dG(v), do G(v), di G(v) (weighted) degree, out- and indegree of vertex v in G T(G), t(G) set and number of spanning trees of G FR(G), fR(G) set and number of spanning forests of G with roots in R H G H is a subgraph of G C(G), c(G) set and number of components of G Table1 . Some graph theoretic notations In a digraph the set of successors (predecessors) No G(v) (Ni G(v) is the set of vertices succeeding (preceding) a given vertex.... ..."

### 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 1. Performances on the graph coloring problem

"... In PAGE 10: ... In particular, we select the family DSJC of randomly-generated graphs proposed in [14]. Table1... ..."

### Table 3: Graph Coloring: E ciency of Localizer

in Localizer

"... In PAGE 31: ... The results are given both for random and cooked graphs and the frequencies are similar for both Localizer and the C implementation. E ciency Table3 compares the e ciency of Localizer with the C implementation on the same problems. Each row reports the average time of the two implementations for the 100 graphs in each class and computes the slowdown of Localizer.... ..."

### Table 2 - Experimental Results for Graph Coloring.

"... In PAGE 13: ... The selection of the design of the IIR filter is used for the evaluation of the new procedure as a design space exploration tool. Table2 provides the experimental results for the application of the probabilistic iterative improvement procedure to the Graph Coloring problem. Testing was performed on instances from [Con93, Dim93].... ..."

### Table 1: Dimacs graph coloring benchmarks

2006

"... In PAGE 6: ... 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.... ..."

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### Table 1. Graph-Coloring Model Parameter

"... In PAGE 2: ... In each cycle, independent of whether the color decision process is activated or not, the node may randomly select any color with a probability specified in the Reset Probability (RP) parameter. Table1 lists all available model parameters that may be var- ied in the exploration of the emergent system dynamics. In the following section three we present a software infrastructure that supports a systematic exploration of this parameter space.... In PAGE 3: ... 4.1 Parameter Space Table1 lists the parameters of our agent model of distributed graph coloring. These parameters may be grouped into problem parameters, solution parameters, and environmental parameters.... ..."

### Table 1: DIMACS graph coloring bench-

2004

"... In PAGE 21: ... We use two families in this work, named mulsol, zeroin Mycielski graphs. Instances of triangle-free graphs based on the Mycielski (My- cielski, 1955) transformation, called myciel Table1 gives the name, size (number of vertices and edges) and the chromatic number for each benchmark. We use a maximum value of K = 20 for K coloring.... In PAGE 23: ...0e+164 941 167 NU+SC 437K 777925 3193 5.0e+148 597 47 Table 2: CNF formula sizes, symmetry detection results and runtimes, totaled for 20 benchmarks from Table1 , with K = 20. NU = null-color elimina- tion; CA = cardinality-based; LI = lowest-index; SC = selective coloring.... ..."

Cited by 5

### Table 1: DIMACS graph coloring bench-

2004

"... In PAGE 21: ... We use two families in this work, named mulsol, zeroin Mycielski graphs. Instances of triangle-free graphs based on the Mycielski (My- cielski, 1955) transformation, called myciel Table1 gives the name, size (number of vertices and edges) and the chromatic number for each benchmark. We use a maximum value of K = 20 for K coloring.... In PAGE 23: ...0e+164 941 167 NU+SC 437K 777925 3193 5.0e+148 597 47 Table 2: CNF formula sizes, symmetry detection results and runtimes, totaled for 20 benchmarks from Table1 , with K = 20. NU = null-color elimina- tion; CA = cardinality-based; LI = lowest-index; SC = selective coloring.... ..."

Cited by 5