### 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. 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: Tree-width obtained after triangulation and triangulation time (in s) for graphs from CALMA archive.

2005

"... In PAGE 4: ... For instance, for n = 50 and e = 300, the tree-width observed in [14] is about 27. Table1 presents empirical results1 for some graphs of the CALMA archive2 (real-world frequency assignment 1All the experimentations are performed on a Linux-based PC with a Pentium IV 2.4 GHz and 512 MB of memory.... In PAGE 10: ...95 0.66 Table1 0: [CSP] Runtime for solving partial random structured CSP with (a) MRC and SC heuristics, (b) MRC and MRS heuristics. 6 Discussion and Conclusion In this article, we have studied several heuristics with a view to improve the efficiency of CSP solving methods based on a tree-decomposition of the constraint network.... ..."

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### Table 1: Results for small Sym-G instances: BPC algorithm and the compact linear MIP formulation solved with Cplex.

2004

"... In PAGE 15: ... In the BPC algorithm, if the time limit is reached while solving the linear relaxation, the gap is given in brackets in the troot column. According to Table1 , the small-size Sym-G instances (with up to 40 nodes) are solved very rapidly and the performance of Cplex 8.1 on the compact linear MIP formulation and of the BPC algorithm are comparable.... ..."

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### Table 3: Results for Asym-G instances: compact linear MIP formulation solved with Cplex.

2004

"... In PAGE 16: ... Note that the situation for Asym-G differs slightly: most instances have an integer optimum with the same cost as the linear relaxation optimal solution, but we have found some instances with an integer optimum with larger cost. Computational results for Asym-G are reported in Table3 . The compact flow-based formu- lation is remarkably tight and effective also in this case.... ..."

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### Table 1.6. Solving (a) a 16-team round-robin scheduling problem (CSP formulation) and (b) the logistics.d instance (SAT formulation) for a range of cutoff values.

### 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 2: Results for medium-to-large size Sym-G instances: BPC algorithm and the compact linear MIP formulation solved with Cplex.

2004

"... In PAGE 15: ...1 on the compact linear MIP formulation and of the BPC algorithm are comparable. The computational results obtained for larger Sym-G instances are reported in Table2 . Our compact flow-based formulation turns out to be very tight so as to yield the optimal solution within less than a minute even for the large problem n147 .... In PAGE 16: ... Let us take the three examples res1, stein1 and n147. For the Sym-G case from Table2 , we see that res1 was solved to optimality by generating 2843 paths and 3433 cuts. The same statistics when we pass to Asym-G uncertainty problem are 4972 and 5952, respectively.... ..."

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### 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.... ..."

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