### Table 3. Statistics on maximal cliques and chromatic numbers for the initial interference graphs constructed by AS and OS.

"... In PAGE 8: ... 6.2 Interference Graphs Table3 compares AS and OS in terms of the initial interference graphs, i.... In PAGE 8: ... Statistics on maximal cliques and chromatic numbers for the initial interference graphs constructed by AS and OS. By comparing Column 4 of Table 1 to Columns 3 and 5 in Table3 , we find that the chromatic number, i.e.... In PAGE 8: ... This indicates clearly that the data set of a program can be successfully placed in an SPM with a size that is much smaller than that of the data set. From Columns 3 and 5 in Table3 , we can also compare AS and OS in terms of the chromatic numbers of their initial interfer- ence graphs. For each benchmark, the chromatic number for AS is significantly smaller than that for OS.... ..."

### TABLE 1. Complexity results for ten famous graph problems when restricted to the classes of graphs discussed in the text. The ten problems are INDEPENDENT SET, CLIQUE, PARTITION INTO CLIQUES, CHROMATIC NUMBER, CHROMATIC INDEX, HAMILTONIAN CIRCUIT, DOMINATING SET, SIMPLE (unweighted) MAX CUT, (unweighted) STEINER TREE IN GRAPHS, and GRAPH ISOMORPHISM. The first nine are known to be NP-complete for general graphs; the complexity of GRAPH ISOMORPHISM for general graphs is a long-standing open problem. The well-known VERTEX COVER problem is not included as its complexity will always be the same as that of INDEPENDENT SET; see [G amp;J, p.54]. The second column gives the com- plexity of determining membership in the given class (and constructing the associated model where appropriate, as in the case of intersection graphs). For a key to the abbreviations used as entries in the table, see the continuation of the table on the next page.

1985

Cited by 159

### Table 3: Comparison of OPAS1 Heuristics for Bipartite Join Graphs

1995

"... In PAGE 14: ... given join graph size (i.e. jV j), the performance gain of the COH heuristic over the FPH heuristic decreases as the edge ratio increases. Table 2: Comparison of OPAS1 Heuristics for General Join Graphs jV j Edge Ratio B(FPH) B(COH) B(FPH) - B(COH) (pages) (%) (pages) (pages) (pages) 5 396:7 341:6 55:1 10 443:8 403:5 40:3 15 461:1 430:2 30:9 500 20 471:5 446:2 25:2 25 477:6 455:9 21:6 30 481:9 463:5 18:4 5 883:0 801:2 81:8 10 938:9 883:8 55:1 15 958:4 918:8 39:6 1000 20 968:4 937:0 31:4 25 975:4 948:8 26:6 30 980:2 958:2 22:0 5 1872:8 1760:7 112:2 10 1933:2 1866:6 66:7 15 1955:7 1906:9 48:8 2000 20 1966:4 1929:2 37:2 25 1974:0 1943:2 30:8 30 1978:9 1953:1 25:8 The results for bipartite join graphs are shown in Table3 for cases when (a) jV1j = jV2j and (b) jV1j lt; jV2j. For both cases, the COH heuristic outperforms the FPH heuristic with the performance margin increasing with the edge ratio.... ..."

Cited by 1

### Table 1: Results for bipartite graphs with |Vi| vertices per bipartition and |E| edges.

2003

"... In PAGE 10: ...158 Graphbase [11] that were used in the experiments of Mutzel [13, 14]2. The results of our experiments are shown alongside the results of Mutzel [14] in Table1 . Each row in the table corresponds to the average values from applying the algorithm to 100 different graphs3.... In PAGE 10: ... It is, however, meaningful to compare the shapes of the |E| versus running time graphs. In the first 17 rows of Table1 , we see that the FPT implementation is quite efficient up to |E| = 55, finding exact solutions to all input graphs. After |E| = 55, the FPT implementation is able to obtain exact solutions to only a few input graphs for the maximum time of 600 seconds (10 minutes) per graph.... In PAGE 10: ... 2We note that Theorem1 does not require the input graph G to be bipartite; consequently, our implementation is not limited to bipartite graphs. 3The graphs for the experiments corresponding to the first 17 rows of Table1 can be repro- duced using the Stanford Graphbase [11]. We first generate 1700 random integers beginning with seed 5841.... ..."

Cited by 4

### Table 8: Lower and upper bounds for instances of equal-split assignment on bipartite graphs (time limit: 300s)

2007

"... In PAGE 116: ...Computational Experiments Table8 shows the comparison of lower and upper bounds returned by default CPLEX and when we add our cuts and heuristic. Results are presented in such a way that it is easy to see the improvement in both lower and upper bounds when cutting planes are added.... In PAGE 116: ...Entries in bold denote convergence to an optimal solution The results from Table8 show that the lower bounds returned by CPLEX at the end of the run are quite bad in general, the main reason being that the LP relaxation is very weak. By adding our cuts we were able to signiflcantly increase the lower bounds for almost... ..."

### Table A.12: Results on the PUC-instances. Type: Constructed difficult instances: hypercubes, from code covering, and bipartite graphs [RdAR+01].

### Table 1 instead of the bipartite graphs to be readable.

"... In PAGE 6: ... Eight and sixteen I/O transfers are applied for the first and second cases respectively. In each case, there are three patterns of I/O transfers as shown in Table1 . The edge tk=ei,j in Table 1 means a data (tk) is going to be transferred between the processor number i and disk number j.... In PAGE 6: ... t10 = e 23, t11= e 34, t12=e 54, t13=e 65, t14=e 67, t15=e 76, t16=e 87 16 8 6 t1=e 11, t2=e 22, t3=e 33, t4=e 44, t5=e 55, t6=e 66, t7=e 12, t8=e 21, t9=e 23, t10= e 34,t11= e 43, t12= e 54, t13= e 65, t14 = e75, t15= e73, t16= e 84 16 8 4 t1= e11, t2= e22, t3= e33, t4= e44, t5= e21, t6= e12, t7= e53, t8= e84, t9= e31, t10= e52, t11= e63, t12= e74, t13= e41, t14= e62, t15= e73, t16= e84 16 8 2 t1= e11, t2= e12, t3= e21, t4= e42, t5= e31, t6= e33, t7= e41, t8= e42, t9= e51, t10= e52, t11= e61, t12= e62, t13= e71, t14= e72, t15= e81, t16= e82 Table1 : No.... ..."

### Table 1: The construction of node groups as a rst step in the construction of the reduced graph.

"... In PAGE 7: ... So, the rst step in the construction of Gr is to iterate over all edges in Gm and construct at most three node groups for both nodes connected to the edge. This step is illustrated in Table1 for the example multirate graph taken from [1] as shown in Figure 4(a). In the gure, the edges ek have been labeled by their index k enclosed in a rectangle, following the style of [12].... ..."

### Table 2 Clustering coeSOcients of the market graph

2004