### Table 2: Possible Multiobjective NP-Complete Functions NP-Complete Problem Examples

"... In PAGE 4: ... This may prevent general comparison between various MOEAs, but the problems apos; inherent di culty should present the desired algorithmic challenges and complement numeric test suite MOPs. Table2 suggests possible NP- Complete MOPs for inclusion. To date, only two non-nu- merical MOP examples are found in the MOEA literature: one is a multiobjective NP-Complete example (a multiob- 0 0.... ..."

### Table 1. Summary of results. NPCmeans that the problem is NP-complete.

1997

"... In PAGE 3: ... Since we can prove that it is NP-complete to decide whether a 2-connected planar graph of maximum degree 2 T ? 1 has a T -spanning tree, this result establishes a complete characterization of the T -spanning tree problem for k-connected planar graphs of maximum degree G. Table1 summarizes the results (it assumes that G gt; T 2). Organization of the paper Section 2 provides basic terminology.... ..."

Cited by 5

### Table 1. Summary of results. NPCmeans that the problem is NP-complete.

1997

"... In PAGE 3: ... Since we can prove that it is NP-complete to decide whether a 2-connected planar graph of maximum degree 2 T ? 1 has a T -spanning tree, this result establishes a complete characterization of the T -spanning tree problem for k-connected planar graphs of maximum degree G. Table1 summarizes the results (it assumes that G gt; T 2). Organization of the paper Section 2 provides basic terminology.... ..."

Cited by 5

### Table 1. Summary of results. NPCmeans that the problem is NP-complete.

1997

"... In PAGE 3: ... Since we can prove that it is NP-complete to decide whether a 2-connected planar graph of maximum degree 2 T ? 1 has a T -spanning tree, this result establishes a complete characterization of the T -spanning tree problem for k-connected planar graphs of maximum degree G. Table1 summarizes the results (it assumes that G gt; T 2). Organization of the paper Section 2 provides basic terminology.... ..."

Cited by 5

### Table 1: New parameterized complexity results for several NP-complete variants of Vertex Cover treated in this work. The parameter k is the size of the desired vertex cover, m denotes the number of edges, and n denotes the number of vertices.

2006

"... In PAGE 3: ... The parameter k is the size of the desired vertex cover, m denotes the number of edges, and n denotes the number of vertices. surveyed in Table1 . In our presentation, n denotes the number of vertices and m denotes the number of edges of the input graph.... In PAGE 20: ...Conclusion We extended the parameterized complexity picture for natural variants and gen- eralizations of Vertex Cover. Table1 in Section 2 summarizes our new results. Notably, whereas the fixed-parameter tractability of Vertex Cover immedi- ately follows from a simple search tree strategy, this does not appear to be the case for all of the problems studied here.... ..."

### Table 1: The complexity of FCSPs Problem Restriction Sequential Complexity Parallel Complexity FCSP decision binary relations NP-complete

"... In PAGE 5: ... In addition to the theoretical results, we have simulated the ne-grain distributed algorithm based on logical time assumptions and experimented with the coarse-grain distributed algo- rithm on a network of transputers. Table1 summarizes current knowledge on the complexity of FCSPs; our results are marked with (y). The rest of the paper is organized as follows.... ..."

### Table 6.2: Input-output complexity of computing interconnection queries. D is always assumed to conform to S. If S is a tree, the complexity is always polynomial in D and the output.

2004

Cited by 2

### Table 1: Summary of our approximation results. The entry marked with * had earlier been proven by Koutsoupias et al. [21]. They had also shown that computing the optimal search path is NP-complete for (planar) graphs. It is also NP-complete for polygons with holes, whereas it is not known to be NP-complete for trees and polygons without holes.

2004

Cited by 5

### Table 1 Summary of complexity results for some edge modification problems. Boldfaced results are obtained in this work, NPC indicates an NP-complete problem, P a polynomial problem, and ? an open problem.

"... In PAGE 2: ... In this work we prove new NP-completeness results for these problems in some classes of graphs, such as interval, circular-arc, permutation, circle, bridge, weakly chordal and clique-Helly graphs. Table1 summarizes the known complexities of edge mod- ification problems in different graph classes, including those obtained in this work (which are boldfaced). Some preliminary results of this work appear in [5].... In PAGE 27: ... Proof: Circular-arc, interval, chordal, perfect, comparability and permutation graphs verify the hypotheses of Theorem 24. The results of Table1 and The- orem 24 imply this corollary. a50 Bipartite edge modification problems can be defined in analogous way to edge modification problems.... In PAGE 29: ...chain deletion are NP-complete. Proof: Chain graphs verify the hypotheses of Theorem 28, hence the results of Table1 and Theorem 28 imply this corollary. a50 Note that the complexity of chain editing is still unknown (see Table 1).... In PAGE 29: ... Proof: Chain graphs verify the hypotheses of Theorem 28, hence the results of Table 1 and Theorem 28 imply this corollary. a50 Note that the complexity of chain editing is still unknown (see Table1 ). We only know that this problem is reducible in polynomial time to biclique-Helly chain bipartite editing.... ..."

### Table 1 below summarizes the main complexity results regarding the initial secrecy problem and the secrecy problem for bounded protocols obtained in pre- vious sections of this chapter (X-c means that the problem is complete for the complexity class X, and ? indicates an open problem). The NP-completeness of the initial secrecy problem for 1-session bounded protocols with freshness check follows from [57], and the NP-completeness of the initial secrecy problem for 1- session bounded protocols without freshness check was established in [25]. These results also follow from Corollary 4.1.4. Moreover, Corollary 4.1.4 proves the NP-completeness of the secrecy problem for 1-session bounded protocols with or without freshness check. The DEXP-complete result was proposed in [24] under the MSR formalism (Corollary 4.1.3 provides a simple proof under the formalism used in thesis) and the NEXP-complete results follow from Theorem 4.1.3.

2006

"... In PAGE 73: ... Table1 : Complexity results for the initial secrecy problem and the secrecy problem for bounded protocols 4.1.... ..."