  Symmetric submodular maximization is an important class of combinatorial
optimization problems, including MAX-CUT on graphs and hyper-graphs. The
state-of-the-art algorithm for the problem over general constraints has an
approximation ratio of $0.432$. The algorithm applies the canonical continuous
greedy technique that involves a sampling process. It, therefore, suffers from
high query complexity and is inherently randomized. In this paper, we present
several efficient deterministic algorithms for maximizing a symmetric
submodular function under various constraints. Specifically, for the
cardinality constraint, we design a deterministic algorithm that attains a
$0.432$ ratio and uses $O(kn)$ queries. Previously, the best deterministic
algorithm attains a $0.385-\epsilon$ ratio and uses $O\left(kn
(\frac{10}{9\epsilon})^{\frac{20}{9\epsilon}-1}\right)$ queries. For the
matroid constraint, we design a deterministic algorithm that attains a
$1/3-\epsilon$ ratio and uses $O(kn\log \epsilon^{-1})$ queries. Previously,
the best deterministic algorithm can also attain $1/3-\epsilon$ ratio but it
uses much larger $O(\epsilon^{-1}n^4)$ queries. For the packing constraints
with a large width, we design a deterministic algorithm that attains a
$0.432-\epsilon$ ratio and uses $O(n^2)$ queries. To the best of our knowledge,
there is no deterministic algorithm for the constraint previously. The last
algorithm can be adapted to attain a $0.432$ ratio for single knapsack
constraint using $O(n^4)$ queries. Previously, the best deterministic algorithm
attains a $0.316-\epsilon$ ratio and uses $\widetilde{O}(n^3)$ queries.
