  We consider fixpoint algorithms for two-player games on graphs with
$\omega$-regular winning conditions, where the environment is constrained by a
strong transition fairness assumption. Strong transition fairness is a widely
occurring special case of strong fairness, which requires that any execution is
strongly fair with respect to a specified set of live edges: whenever the
source vertex of a live edge is visited infinitely often along a play, the edge
itself is traversed infinitely often along the play as well. We show that,
surprisingly, strong transition fairness retains the algorithmic
characteristics of the fixpoint algorithms for $\omega$-regular games -- the
new algorithms have the same alternation depth as the classical algorithms but
invoke a new type of predecessor operator. For Rabin games with $k$ pairs, the
complexity of the new algorithm is $O(n^{k+2}k!)$ symbolic steps, which is
independent of the number of live edges in the strong transition fairness
assumption. Further, we show that GR(1) specifications with strong transition
fairness assumptions can be solved with a 3-nested fixpoint algorithm, same as
the usual algorithm. In contrast, strong fairness necessarily requires
increasing the alternation depth depending on the number of fairness
assumptions. We get symbolic algorithms for (generalized) Rabin, parity and
GR(1) objectives under strong transition fairness assumptions as well as a
direct symbolic algorithm for qualitative winning in stochastic
$\omega$-regular games that runs in $O(n^{k+2}k!)$ symbolic steps, improving
the state of the art. Finally, we have implemented a BDD-based synthesis engine
based on our algorithm. We show on a set of synthetic and real benchmarks that
our algorithm is scalable, parallelizable, and outperforms previous algorithms
by orders of magnitude.
