### Table 1: Median nodes searched by the Davis-Putnam procedure for random 3-sat problems at n = 50 and l=n = 4:3 using di erent heuristics.

1996

"... In PAGE 24: ... For the satis ability experiments, we implemented minimize- and maximize- branching heuristics for the Davis-Putnam procedure. In Table1 , we show how these heuristics compare to Mom apos;s heuristic on hard random 3-sat problems from the middle of the phase transition. The results support the thesis that, for soluble problems, it pays to minimize and for insoluble problems, it pays to maximize .... ..."

Cited by 99

### Table 2. Comparison of the runtimes of the basic Davis-Putnam QBF procedure, the same with the new propagation algorithm restricted to polynomial runtime, and the unrestricted new propagation algorithm, on a number of QBF. basic DP/QBF new algo. O(p(n)) new algo. O(2n) problem

2001

Cited by 26

### Table 1: Mean and percentiles in branches explored by the Davis-Putnam procedure on 10,000 satis able prob- lems from the constant probability model with 200 vari- ables. At the time of submission, Dfs had searched over 200,000,000 branches on 2 problems without success. We hope to report the nal result at the conference.

1997

"... In PAGE 5: ... To solve these problems, we use the Davis- Putnam procedure, branching on the rst literal in the shortest clause. Table1 demonstrates that Dds reduces the severity of ehps compared to Dfs. Other methods to tackle ehps include sophisticated backtracking pro- cedures like con ict-directed backjumping [Smith and Grant, 1995] and learning techniques like dependency- directed backtracking [Bayardo and Schrag, 1996] which identify quickly the insoluble subproblems after an early... ..."

Cited by 60

### Table 1: Mean and percentiles in branches explored by the Davis-Putnam procedure on 10,000 satisfiable prob lems from the constant probability model with 200 vari ables. At the time of submission, DFS had searched over 200,000,000 branches on 2 problems without success. We hope to report the final result at the conference.

1997

"... In PAGE 5: ... To solve these problems, we use the Davis- Putnam procedure, branching on the first literal in the shortest clause. Table1 demonstrates that DDS reduces the severity of ehps compared to DFS. Other methods to tackle ehps include sophisticated backtracking pro- cedures like conflict-directed backjumping [Smith and Grant, 1995] and learning techniques like dependency- directed backtracking [Bayardo and Schrag, 1996] which identify quickly the insoluble subproblems after an early ... ..."

Cited by 60

### Table 2 Locations of Tests in Gallo amp; Urbani (1989) parameter values used, the number of test formulas which were satis able (out of 10), and the time in seconds for Gallo and Urbani apos;s implementation of the Davis-Putnam Procedure. Also shown are our own experimental estimates (based on large samples and accurate to two decimal places) of the ratio at which 50% of the formulas will be satis able for the given values of n and m. Gallo and Urbani chose m based on estimates of where about one half of the formulas would be satis able, but comparison with the known 50% ratios in Table 2 shows they were somewhat o , probably due to estimating from small sample sizes. We do not know exactly where the peak is for the procedure used in this experiment, but the procedure is almost identical to our DP, and our experience suggests the peaks for the two procedures will be very close. If we 13

1996

"... In PAGE 13: ...ransition ratio for each k (eg., setting m = 4:25n when k = 3). In this sec- tion we consider whether this approach is adequate in the common situation in which the experimenter cannot precisely determine the location of peak hardness. To begin, we consider an experiment by Gallo and Urbani [11], in which per- formance of several SAT procedures were compared on 5 sets of random 3-SAT formulas, for which partial results are shown in Table2 . The table gives the... In PAGE 14: ...ig. 8. Median DP steps for random 3-SAT: selected values of n. assume this to be true, then the choices for c in Table2 shift from somewhat above the peak, to somewhat below the peak, crossing it near the 3rd data point. Thus, looking at the rst 3 points gives an over-estimate of the growth in time with n, and looking at the last 3 points we gives an under-estimate of growth.... ..."

Cited by 19

### Table 2: Branches explored by the Davis-Putnam pro cedure on 10,000 satish apos;able random 3-SAT problems at L/N = 3.5. Best results in each row are underlined.

1997

"... In PAGE 5: ... Each problem is solved using the Davis-Putnam procedure, branching as before on the first literal in the shortest clause. Results are given in Table2 . As expec ted, on such under-constrained and soluble problems, DDS and ILDS are superior to DFS, with DDS offering an advantage over ILDS especially on larger and harder problems.... ..."

Cited by 60

### Table 1: Three strategies used for solving 1,000 instances of satis able 3-SAT problems at L=N = 3:5. A 10 minutes time slot is given for each problem. \DFS quot; is the standard Davis-Putnam- Loveland (DPL) procedure. \Restart quot; refers to the restart of the DPL procedure 10 times or until a solution is found; each restart is allowed to run for at most one minute. \Jump quot; is the DPL procedure with the new random jump strategy. The columns under \failed quot; give the numbers of failed cases (out of 1,000). The random jump strategy succeed in every run and the \failed quot; column is omitted from the table. The average time used and the leaf nodes explored are given under \time quot; (in seconds) and \nodes quot;, respectively. The times were collected on a Linux PC (300MHz; 128M memory) running SATO. The column \R quot; provides the average number of restarts for each problem. The column \skipped quot; provides the average number of branches skipped by our strategy.

2002

Cited by 4

### Table 1: Three strategies used for solving 1,000 instances of satis able 3-SAT problems at L=N = 3:5. A 10 minutes time slot is given for each problem. \DFS quot; is the standard Davis-Putnam- Loveland (DPL) procedure. \Restart quot; refers to the restart of the DPL procedure 10 times or until a solution is found; each restart is allowed to run for at most one minute. \Jump quot; is the DPL procedure with the new random jump strategy. The columns under \failed quot; give the numbers of failed cases (out of 1,000). The random jump strategy succeed in every run and the \failed quot; column is omitted from the table. The average time used and the leaf nodes explored are given under \time quot; (in seconds) and \nodes quot;, respectively. The times were collected on a Linux PC (300MHz; 128M memory) running SATO. The column \R quot; provides the average number of restarts for each problem. The column \skipped quot; provides the average number of branches skipped by our strategy.

2002

Cited by 4

### Table 4.9: Davis-Putnam solution statistics: 32-input, 1-output variables

1992

Cited by 35

### Table 4.6: Davis-Putnam solution statistics: 16-input, 1-output variables

1992

Cited by 35