### Table 2. Bicriteria spanning tree results for treewidth-bounded graphs.

"... In PAGE 6: ... As before, the rows are indexed by the budgeted objective. All algorithmic results in Table2 also extend to Steiner trees in a straightforward way. Our results for treewidth-bounded graphs have an interesting application in the context of find- ing optimum broadcast schemes.... In PAGE 19: ...1 Exact Algorithms Theorem 8.1 Every problem in Table2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table 2 when restricted to the class of treewidth-bounded graphs.... In PAGE 19: ...1 Every problem in Table 2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table2 when restricted to the class of treewidth-bounded graphs. The basic idea is to employ a dynamic programming strategy.... In PAGE 23: ...7 For the class of treewidth-bounded graphs, there is an FPAS for the (Diame- ter, Total cost, Spanning tree)-bicriteria problem with performance guarantee (1; 1 + ). As mentioned before, similar theorems hold for the other problems in Table2 and all these results extend directly to Steiner trees. 8.... ..."

### Table 2. Bicriteria spanning tree results for treewidth-bounded graphs.

"... In PAGE 6: ... As before, the rows are indexed by the budgeted objective. All algorithmic results in Table2 also extend to Steiner trees in a straightforward way. Our results for treewidth-bounded graphs have an interesting application in the context of find- ing optimum broadcast schemes.... In PAGE 19: ...1 Exact Algorithms Theorem 8.1 Every problem in Table2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table 2 when restricted to the class of treewidth-bounded graphs.... In PAGE 19: ...1 Every problem in Table 2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table2 when restricted to the class of treewidth-bounded graphs. The basic idea is to employ a dynamic programming strategy.... In PAGE 23: ...7 For the class of treewidth-bounded graphs, there is an FPAS for the (Diame- ter, Total cost, Spanning tree)-bicriteria problem with performance guarantee (1; 1 + ). As mentioned before, similar theorems hold for the other problems in Table2 and all these results extend directly to Steiner trees. 8.... ..."

### Table 1: Preprocessing for weighted treewidth

"... In PAGE 18: ... We discuss the results by means of four tables. Table1 shows the re sults of preprocessing for (i) only the Simplicial rule, (ii) all described rules, and (iii) all described rules with initially low set to the MMNW+ lower bound. To illustrate the quality of the MMNW+ bound in comparison to the MMNW bound both values are presented in Table 2.... In PAGE 18: ... In the moralisation of a directed graph, for each pair of arcs with a common tail, an edge is added between the heads of the arc, and then all directions of arcs are dropped. Thus, in Table1 , the size of the graph after moralisation is shown (see [6] for the orig inal sizes). Note that vertices with many incoming edges in the probabilistic network create a large clique in the moralisation.... In PAGE 18: ... Note that vertices with many incoming edges in the probabilistic network create a large clique in the moralisation. In Table1 , we show the size of the networks after each of the preprocessing strategies. Moreover the ... In PAGE 20: ... To increase readability, the last col umn reports the computation times for the last strategy. Table1 shows that application of the Simplicial rule only already results in substantial graph size reductions in all cases. On average over 50% of the vertices are removed by preprocessing (with a minimum of 18% and a maximum of 87%).... In PAGE 20: ...ith the first rule in the order (i.e., Islet). In this way, it is avoided that, for example, a simplicial vertex is processed by the Almost Simplicial rule. As already observed in Table1 , the majority of the vertices is preprocessed by the Simplicial rule and its specialisations Islet and Twig. Notice that Islet is only applied if singletons are detected in the graph.... ..."

### Table 1: Optimal upper bounds for the clique number, inductiveness, and chromatic number of the square of a chordal / non-chordal outerplanar graph G.

2004

"... In PAGE 4: ... We examine in detail the low-degree cases, lt; 7, and derive best possible upper bounds on the maximum clique and chromatic numbers, as well as inductiveness of squares of outerplanar graphs. These bounds are illustrated in Table1 . We treat the special case of chordal outerplanar graphs separately, and further classify all chordal outerplanar graphs G for which the inductiveness of G2 exceeds or the clique or chromatic number of G2 exceed + 1.... In PAGE 17: ...orollary 4.10 together with Theorems 4.3 and 4.5 complete the proof of Theorem 4.1 as well as the entries in Table1 in the chordal case for 2 f2; 3; 4; 5; 6g. Observation 4.... ..."

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### Table 1: Optimal upper bounds for the clique number, inductiveness, and chromatic number of the square of a chordal / non-chordal outerplanar graph G.

### Table 2: Lower bounds for weighted treewidth

"... In PAGE 18: ... Table 1 shows the re sults of preprocessing for (i) only the Simplicial rule, (ii) all described rules, and (iii) all described rules with initially low set to the MMNW+ lower bound. To illustrate the quality of the MMNW+ bound in comparison to the MMNW bound both values are presented in Table2 . Table 3 records the number of vertices preprocessed by each of the rules.... In PAGE 20: ... We only performed computations with MMNW+ as initial lower bound. In Table2 the MMNW and MMNW+ lower bounds are compared. Although computation times are somewhat higher, the increase of the lower bound is substantially for many instances.... ..."

### Table 2: Upper bounds on the MCSLB for selected instances

2004

"... In PAGE 31: ...Table 2: Upper bounds on the MCSLB for selected instances In some cases the MCS-bound equals the best treewidth upper bound (bold values, cf. Table2 if MCS-UB is larger) and thus the reported value is the treewidth of those graphs. In total 30 instances could be solved to optimality by this lower bound, wheres with the degeneracy only 15 instances could be solved to optimality.... In PAGE 31: ... All three methods as well as the maximum degree (G), the actual best value achieved (cf. Table 1) and the best treewidth upper bound for selected instances are reported in Table2 . The maximum degree of each graph is reported since the algorithm to compute u(v) is initialised with the degree dG(v).... In PAGE 31: ... The maximum degree of each graph is reported since the algorithm to compute u(v) is initialised with the degree dG(v). Table2 shows that in several cases the final maximum of u(v) over all vertices is significantly smaller than the maximum degree. Only in cases where the maximum degree is close to the treewidth, only minor improvement could be achieved.... ..."

Cited by 10

### Table 1 shows the sizes of the graphs, and the results obtained for the treewidth lower bounds without contraction. These bounds are the exact parameters apart from the values for the three RD-heuristics. As the computation times for , 2 and R are negligible, we omit them in the table. Also the D can be computed within a fraction of a second. The computational complexity of 2D is O(n) larger than the one of D which is re ected in the CPU times for this parameter.

"... In PAGE 15: ... Table1... In PAGE 16: ...rie M. C. A. Koster, Thomas Wolle, and Hans L. Bodlaender Table 2 shows the results for the same graphs as in Table1 . Furthermore, in Table 2, we give the treewidth lower bounds according to the parameters that involve contraction.... In PAGE 16: ...4 Discussion. The results of algorithms and heuristics that do not involve edge-contractions ( Table1 ) show that the degeneracy lower bounds (i.e.... In PAGE 16: ...ethods involving computing lower bounds many times (e.g. branch amp; bound). Even though the 2D algorithm has much higher running times than the other algorithms in Table1 , it is still much faster than some heuristics with contraction. Furthermore, we expect that its running time could be improved by a more e cient implementation.... In PAGE 16: ... No further investigations about parameters without contraction have been carried out as the parameters with contraction are of considerably more interest. We can see that when using edge-contractions, the treewidth lower bounds can be signi cantly improved (compare Table 2 with Table1 ). The results show that values for 2C are typically equal or only marginal better than the value for C.... ..."

### Table 2. Results for the Chordal Ring. The (star) indicates an optimal bound.

2006

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### Table 1: The edges of the inequality graph for elimination of upper-bounds checks.

2000

"... In PAGE 5: ...length stBC := A01 while: limitBD := AUB4limitBC,limitBF) stBD := AUB4stBC,stBF) if (stBD lt; limitBD) CU stBE BMBP APB4stBDB5 limitBE BMBP APB4limitBDB5 stBF := stBE + 1 limitBF := limitBE A0 1 jBC := stBF for: jBD := AUB4jBC,jBG) if (jBD lt; limitBF) CU jBE BMBP APB4jBDB5 limitBG BMBP APB4limitBFB5 check A[jBE] jBF BMBP APB4jBEB5 tBC := jBF + 1 check A[tBC] tBD BMBP APB4tBCB5 jBG := jBF + 1 goto for CV goto while CV Figure 3: The running example before and after the conversion into e-SSA form. Recall that Table1 defines the edges of BZC1 for the elimination of upper-bound checks; the elimination of lower-bound checks is based on analogous, but separate, inequality graph. Example 2 (The Inequality Graph BZC1) Figure 4 shows the in- equality graph BZC1 for the running example.... In PAGE 10: ...e., those generated by individual program statements shown in Table1 , but also constraints deduced by a global program anal- ysis, for example, by global value numbering [AWZ88]. When two SSA variables DACX and DACY are found to be value congruent, their equivalence can be reflected in BZC1 by a constraint edge DACX AX DACY with weight BC.... ..."

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