5. Parallel Algorithm:

The parallel algorithm could be useful if we wish to reduce the number of exceptions. In fact, we have implemented in $C^3$ some variations of the parallel algorithm which uses the three algorithms JANUS, Simplex and Fourier-Motzkin. For example, we try the Fourier-Motzkin method when JANUS has an exception.

The implementation of POLYBENCH permits us to verify that in 6_tab:PerfectClub_satisfiability_exception_100, the five overflow exceptions from JANUS and the exceptions from other three algorithms come from different constraint systems. It means if we use the parallel algorithm using JANUS and another algorithm, we can resolve all the tests. For instance, for this database, the parallel algorithm of JANUS and Simplex has zero exception. For the SPEC95 sampled database, JANUS and Simplex share the same $39$ overflow exceptions, which represents the number of exceptions that their parallel algorithm has. If we use the parallel algorithm of the four algorithms, we have only one exception, instead of $52$ exceptions with JANUS.

In the case of filtered databases, we can for example use the Fourier-Motzkin in 6_tab:PerfectClub_satisfiability_exception_filter and the Double Description method 6_tab:SPEC95_satisfiability_exception_filter for the constraint systems that JANUS cannot deal with.