### Table 1 Combinatorial optimization problems and their geometric equivalents

"... In PAGE 10: ... Similarly, if we color the graph with a minimum number of colors, then this is equivalent with dividing the collection of lines into a minimum number of subcollections such that each subcollection contains no parallel pairs of lines. Table1 gives an overview of problems in graph theory and their geometric equivalent. Since in this paper we focus on parallel line grouping, we propose two combi- natorial algorithms that can be used to partition a graph of parallel pairs into subgraphs which are or which resemble cliques.... ..."

### Table 8 Protocol theory complexity overview Bounded # Roles Unbounded # Roles

1982

"... In PAGE 35: ...3. Protocol complexity matrix Table8 shows a summary of the complexity results for the main theorems pre- sented in this paper. The two main columns consider the case of whether the number of roles is bounded or unbounded.... In PAGE 36: ... Because the number is fixed, the nonces can be assumed to have been produced during initialization, and not within the roles themselves. The two rows of Table8 consider whether the term size k is fixed in all instances of the problem, or whether the term size is allowed to vary as a parameter of the problem. For each entry of the matrix in Table 8, we show the complexity result for that case, using P to indicate the problem is in polynomial time, NPC forNP-complete, DEXPC forDEXP-complete, and Undec .... In PAGE 36: ... The two rows of Table 8 consider whether the term size k is fixed in all instances of the problem, or whether the term size is allowed to vary as a parameter of the problem. For each entry of the matrix in Table8 , we show the complexity result for that case, using P to indicate the problem is in polynomial time, NPC forNP-complete, DEXPC forDEXP-complete, and Undec . for Undecidable.... In PAGE 36: ... Table 9 is a more detailed summary of the complexity results, where we show more detail about the results for the upper and lower bounds. The columns are the same as in Table8 , but now the two main rows consider whether the intruder is allowed to generate fresh values or not. These rows are further subdivided into the cases where the roles can perform disequality tests which would allow them to determine whether two fresh values are different from each other.... In PAGE 36: ... If disequality is not allowed, then this test is not performed. Table 9 shows the complexity results for these cases, using the same notation as for Table8 . The numeric references indicate the propositions about specific lower or upper bounds which we discuss in the following sections.... ..."

Cited by 2

### Table 9 Protocol theory complexity matrix Bounded # Roles Unbounded # Roles

1982

"... In PAGE 36: ...DEXPC forDEXP-complete, and Undec . for Undecidable. The entries also indi- cate which theorem is applicable in each case. Table9 is a more detailed summary of the complexity results, where we show more detail about the results for the upper and lower bounds. The columns are the same as in Table 8, but now the two main rows consider whether the intruder is allowed to generate fresh values or not.... In PAGE 36: ... If disequality is not allowed, then this test is not performed. Table9 shows the complexity results for these cases, using the same notation as for Table 8. The numeric references indicate the propositions about specific lower or upper bounds which we discuss in the following sections.... In PAGE 46: ....5.2. Ssize=k is Undecidable (Theorem 1) For the case of unbounded roles and unbounded existentials (the rightmost col- umn in Table9 ), we turn to results from Database theory, where the complexity results for Embedded Implicational Dependencies (EIDs) [17] and Datalog [22] can be applied. Embedded Implication Dependencies are exactly Horn clauses with ex- istentials and equality, as defined in Section 5.... In PAGE 55: ...301 We have identified an open problem for the complexity of the secrecy case with disequality, unbounded roles, and bounded nonces (the ??? box in Table9 ). We con- jecture that the additional power of the disequality test makes this case undecidable.... ..."

Cited by 2

### Table 3: Combinatorial complexity of trees First plan

2000

"... In PAGE 8: ... Let us take the small naive example of dimen- sioned sketch on Fig. 8 and consider then two construction plans solving this dimen- sioned sketch given on Table3 . The maximum number of nodes per level is indicated, and D1CZCRCXD2D7 refers to the function building the four circles (in general) tangent to three lines.... In PAGE 10: ... AF Procedure CRD3D1D4D9D8CTCND7D3D0B4BWB5 computes values of definition BW, using the value of its definition arguments and its multifunction. With the worst construction plan given on Table3 there is no failure, only 44 calculi are computed, instead of 90 foreseen: values for D0BG, D0BH and D4BG are computed one time (3 calculi) despite of 16 branches in the associated search tree to this construction plan (48 calculi). Even with a naive backtracking, this method allows to compute only necessary calculi.... ..."

Cited by 1

### Table 3: Combinatorial complexity of trees First plan

"... In PAGE 8: ... Let us take the small naive example of dimen- sioned sketch on Fig. 8 and consider then two construction plans solving this dimen- sioned sketch given on Table3 . The maximum number of nodes per level is indicated, and D1CZCRCXD2D7 refers to the function building the four circles (in general) tangent to three lines.... In PAGE 10: ... AF Procedure CRD3D1D4D9D8CTCND7D3D0B4BWB5 computes values of definition BW, using the value of its definition arguments and its multifunction. With the worst construction plan given on Table3 there is no failure, only 44 calculi are computed, instead of 90 foreseen: values for D0BG, D0BH and D4BG are computed one time (3 calculi) despite of 16 branches in the associated search tree to this construction plan (48 calculi). Even with a naive backtracking, this method allows to compute only necessary calculi.... ..."

### Table 1. Worst-case combinatorial complexities of V (njf) F

"... In PAGE 2: ... (A region may consist of multiple cells.) We summarize in Table1 the major results of this paper: the bounds on the largest diagram complexities for these distance functions. (For some distance... ..."

### Table 3. Mixed integer program for obtaining a combinatorial lower bound on the optimal number of moves for the stacking problem

### Table 1: Combinatorial optimization problems and their autocorrelation functions.

### Table 19. General Problems Encountered in Negotiating Formal Agreements by Organisational Role

2007

"... In PAGE 45: ... Figure 10. General Problems in Negotiating Formal Agreements 24% 40% 36% 1% 24% 40% 30% 6% 23% 34% 36% 7% 19% 31% 41% 9% 14% 38% 37% 10% 9% 23% 55% 12% 14% 20% 49% 17% 18% 28% 35% 19% 11% 28% 40% 21% 8% 11% 55% 27% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Prevented project proceeding Other party had leverage Eroded trust Too costly Difficulties with industry Differing expectations Became too complex Difficulties with university Difficulties with government Unreasonable delays in project commencement Never Rarely Sometimes Often One-way ANOVAs were conducted to examine differences in frequencies of negotiation issues by organisational role (see Table19 ). Respondents who have legal and contract roles (compared to those in research roles) more often encountered the problem of the other party having all the leverage (mean=2.... ..."

### Table 1: Combinatorial complexities of the visibility map in a 3D scene for various types of view elements under different models of visibility.

2005

"... In PAGE 3: ... The main contribution of our paper is that we show exactly how hard computing shadows in general 3D scenes can be. Table1 gives a summary of our results. All displayed results are worst-case combinatorial complexities.... In PAGE 15: ...ated, i.e., what is the fraction of the points on e that illuminate them. Also, as can be seen in Table1 , there are a few gaps left between lower and upper bounds. Finally, it would be interesting to look into output-sensitive algorithms to compute the different types of visibility maps.... ..."