### 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: Linear growth of d-DNNF compilations for plan- ning problems with bounded treewidth.

"... In PAGE 6: ... Having variable orders of bounded width implies that the size of the d-DNNF compilation will grow at most linearly with the horizon, which will allow us to ultimately scale the proposed approach to large horizons. To give a concrete idea of this linear growth and the scalability of this part of the computation, we report in Table1 the results of compiling the SLIPPERY-GRIPPER domain (an extra action clean has been added as in (Hyafil amp; Bacchus 2003) ) using C2D for horizons 1 to 20 and some selected large horizons. The state variables have been existentially quantified from all results using the -exist option for C2D.... ..."

### 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. Complexity of instance checking and conjunctive query entailment

"... In PAGE 13: ... A matching lower bound can be taken from [8] (which relies on the presence of general TBoxes and already applies to the instance problem), and thus we obtain P- completeness. 5 Summary and Outlook The results of our investigation are summarized in Table2 . In all cases the lower bounds apply to instance checking and the upper bounds to conjunctive query entailment.... ..."

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### 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 4.2 shows the sizes of the graphs, treewidth upper bounds and the results obtained from the treewidth lower bound algorithms without edge-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 a factor of O(n) larger than the one of D which is re ected in the CPU times for this parameter.

### Table II. Summary of conjunctive answer algorithms. Running times in milliseconds. Dataset Query size Synthetic ChefMoz GovTrack

2006

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### Table 1. The first column is from [11]. Partial k-trees are the same as graphs of bounded treewidth; the polynomial-time result for this row is shown in Section 4. Empty entries mean that this has not been studied yet.

2005

"... In PAGE 5: ... The proof for the claim that G has a vertex cover of size k iff Gprime has a power dominating set of size k uses a similar argument as the proof of Theorem 1. intersectionsqunionsq Altogether, Table1... In PAGE 12: ... 5 Conclusion There are several avenues for future work. Table1 contains several empty en- tries concerning the complexity of Power Dominating Set in particular graph classes we did not address but which have been addressed for Dominating Set [11]. In particular, is there a significant difference between the complex- ities of DS and PDS? How do fixed-parameter tractability results for DS in planar graphs transfer to PDS? Are there nontrivial data reduction rules for PDS? So far, we are not aware of a graph class where DS is polynomial-time solvable and PDS is NP-complete or vice versa.... ..."

Cited by 7

### Table 1. The first column is from [11]. Partial k-trees are the same as graphs of bounded treewidth; the polynomial-time result for this row is shown in Section 4. Empty entries mean that this has not been studied yet.

2005

"... In PAGE 5: ... The proof for the claim that G has a vertex cover of size k iff Gprime has a power dominating set of size k uses a similar argument as the proof of Theorem 1. intersectionsqunionsq Altogether, Table1... In PAGE 12: ... 5 Conclusion There are several avenues for future work. Table1 contains several empty en- tries concerning the complexity of Power Dominating Set in particular graph classes we did not address but which have been addressed for Dominating Set [11]. In particular, is there a significant difference between the complex- ities of DS and PDS? How do fixed-parameter tractability results for DS in planar graphs transfer to PDS? Are there nontrivial data reduction rules for PDS? So far, we are not aware of a graph class where DS is polynomial-time solvable and PDS is NP-complete or vice versa.... ..."

Cited by 7