For (mathsf {DENN}), (mathsf {CHRI}), (mathsf {CHCI}), and (mathsf {Greedy}), the error-ratio increases first and then decrease.
Figure 11 shows the results for (mathsf {NA+}), (mathsf {RA+}) and (mathsf {FA+}).
Note that the approximate ratio is 1.5 for (mathsf {Chris}), and is 2 for (mathsf {DMST}).
The error-ratios for (mathsf {CHFI}) and (mathsf {SV}) increase slightly while increasing (l).
For the same reason, the parity of the maps (mathsf {b}, mathsf {c}) is identical to that of (mathsf {B}, mathsf {C}), i.e., (mathsf {b} vert _{U_0^*rightarrow U_0^) is symmetric, (mathsf {b} vert _{U_1^*rightarrow U_1^) is skew, and similar for (mathsf {c},).
It is (mathsf {NP} -hard for (mathsf {NP} -hardithout counting quantiforrs [89], and (mathsf {GPARs}ard withoutunting quantifiers are present [28].
The implication problem for (mathsf {GFDs}) is to decide, given a set (varSigma ) of (mathsf {GFDs}) and another (mathsf {GFD}) (varphi ), whether (varSigma models varphi ).
At the same time, the parity of (mathsf {b}, mathsf {c}) is reversed as compared to (mathsf {B}, mathsf {C}): now the map (mathsf {b} vert _{U_0^*rightarrow U_0^) is skew and (mathsf {b} vert _{U_1^*rightarrow U_1^) is symmetric (and similar for (mathsf {c})).
These problems have also been settled for (mathsf {GFDs}) [29]: the satisfiability problem is (mathsf {coNP} -complete, and the implication problem is (mathsf {coNP} -complete
The error-ratio for (mathsf {SV}) increases monotonously.
