### Table 8: Approximate Linear Programming Algorithm for the Decentralized Problem.

2005

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### Table 2. Rotation functions for the MIN 2-SAT approximation algorithms. (a) The piecewise linear rotation function CUBECA used for the 1.115682-approximation algorithm. (b) The piecewise linear rotation function CUBECC used for the 1.103681-approximation algorithm.

"... In PAGE 9: ...1 An Approximation Algorithm for MIN 2-SAT using CAC7CCCU The algorithm first solves the semidefinite programming of Figure 1. It then rounds the solution using the rounding procedure CAC7CCCUBECA, which rotates the vectors using the piecewise linear rotation function CUBECA in Table2 (a). The algo- rithm achieves a performance ratio of at most BDBMBDBDBHBIBKBE, and the rotation func- tion CUBECA is nearly optimal in the CAC7CC family as BDBMBDBDBHBIBJBG BO CXD2CUCU ACBEB4CAC7CCCUB5 BO ACBEB4CAC7CCCUBECAB5 BO BDBMBDBDBHBIBKBE The lower bound may be obtained in the following way.... In PAGE 9: ...2 An Approximation Algorithm for MIN 2-SAT using CC C0CABXCBC0CV The algorithm first solves the semidefinite programming of Figure 1. It then rounds the solution using the rounding procedure CC C0CABXCBC0CVBECC , with CVBECC B4DCB5 BP A0CP CRD3D8 CUBECC B4CPD6CRCRD3D7 DCB5D4BD A0 DCBE, where CP BP BDBMBH and CUBECC is the piece- wise linear function given in Table2... ..."

### Table 2a. Computing approximate linear programming solutions (n = 50): CPU time

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"... In PAGE 26: ...Table2 b. Computing approximate linear programming solutions (n = 50): number of generated columns name 0.... ..."

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### Table 3a. Computing approximate linear programming solutions (n = 100): CPU time

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"... In PAGE 28: ...Table3 b. Computing approximate linear programming solutions (n = 100): number of generated columns name 1 % 0.... ..."

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### Table 1: Execution and checkpoint parameters of the parallel programs.

1993

"... In PAGE 6: ... The four programs include two computer-aided circuit design applications, Test Generation and Logic Synthesis, and two search applications, Knight Tour and N-Queen. The execution times are between 25 and 45 minutes (see Table1 ). The predetermined minimum basic checkpoint in- terval is chosen to be 2 minutes.... In PAGE 6: ... In particular, we choose Q = 2 to estimate the induction ra- tio. The exact value of Q for each program is listed in Table1 . Although Q is slightly greater than 2 for the first two programs, the numbers listed in the row of Under-2 percentage show that a very high percentage of the ba- sic checkpoint intervals are covered by Q = 2 which thus serves as a good approximation.... ..."

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### Table 6: Approximate computation times.

"... In PAGE 10: ... This suggests that there exists more optimal (linear and non-linear) combinations of these functions. Computational issues Table6 lists the times taken for the computational tasks outlined in this paper. Times are given per 1000 MHz Pen- tium III processor and for a cluster of 64 such processors when the algorithm can run in parallel.... ..."

### Table 6: Approximate computation times.

2002

"... In PAGE 10: ... This suggests that there exists more optimal (linear and non-linear) combinations of these functions. Computational issues Table6 lists the times taken for the computational tasks outlined in this paper. Times are given per 1000 MHz Pen- tium III processor and for a cluster of 64 such processors when the algorithm can run in parallel.... ..."

### Table 2b. Computing approximate linear programming solutions (n = 50): number of generated columns

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"... In PAGE 24: ...Table2 a. Computing approximate linear programming solutions (n = 50): CPU time name 0.... ..."

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### Table 2b. Computing approximate linear programming solutions (n = 50): number of generated columns

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"... In PAGE 25: ...Table2 a. Computing approximate linear programming solutions (n = 50): CPU time name 0.... ..."

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### Table 3b. Computing approximate linear programming solutions (n = 100): number of generated columns

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"... In PAGE 26: ...Table3 a. Computing approximate linear programming solutions (n = 100): CPU time name 1 % 0.... ..."

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