### Table 1: Comparative performance results for different rule combinations. The abbreviations are: MP for max-product, SP for sum-product and C1, C2 are the first and second constraint-sets described in Section 5.

"... In PAGE 11: ...MP, the additional 54 constraints, defined on the previous section, by C1 and the additional set of nine permutation constraints by C2. Table1 shows the percentage of Sudoku puzzles that can be complectly solved using various combinations of the constraints. Table 1 also shows for the failure cases the average number of cells that were revealed until we arrived to a stopping-set.... In PAGE 11: ... Table 1 shows the percentage of Sudoku puzzles that can be complectly solved using various combinations of the constraints. Table1 also shows for the failure cases the average number of cells that were revealed until we arrived to a stopping-set. The last column presents the average processing time of solving a single Sudoku puzzle, measured in mili-seconds.... In PAGE 11: ... It can be seen that using the decision rules derived from the factor graph representation and the first rule-set described in the this section, the contribution of second rule-set is neglected. The second part of Table1 shows the performance results of Table 1: Comparative performance results for different rule combinations. The abbreviations are: MP for max-product, SP for sum-product and C1, C2 are the first and second constraint-sets described in Section 5.... In PAGE 11: ...the sum-product algorithm and the results of applying the sum-product algorithm on stopping-sets obtained from several combinations of decision rules. Table1 shows that the best strategy in term of performance (and also in term of computational complexity comparing with the sum-product) is applying the sum-product algorithm on the stopping-set obtained by the (extended version of) max-product. The short cycles that appear in the Sudoku factor-graph cause that the sum-product algorithm can quickly amplify non-correct assignments.... ..."

### Table 1: C4BD approximation error of single node marginals for the fully connected graph C3BL and the 4 nearest neighbour grid with 9 nodes, with varying potential and coupling strengths B4CSpotBN CScoupB5. Three different variational methods are compared: MF/Tree derives a lower bound with mean field approximation for A8BV and tree-reweighted belief propagation for A8; MF/SDP derives a lower bound with the SDP relaxation used for A8; Tree/MF derives an upper bound using tree- reweighted belief propagation for A8BV and mean field for A8. SDP denotes the heuristic use of the dual parameters in the SDP relaxation, with no provable upper or lower bounds.

2004

"... In PAGE 7: ... To assess the accuracy of each approximation, we use the C4BD error, defined as BD D2 D2 CG D7BPBD CYD4AIB4CG BE BVB5 A0 CQ D4AIB4CG BE BVB5CY (36) where CQ D4AI denotes the estimated marginal. The results are shown in Table1 for the single node case, and in Table 2... ..."

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### Table 2: PM Message Passing

2000

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### Table 1: Message Passing Summary

1999

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### Table 1: Message Passing Summary

1999

Cited by 8

### Table 1: Message Passing Summary

1999

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### Table 1: Message Passing Summary

1999

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### Table 1. Sequence of message passing

2005

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