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.

翻译：对称子模最大化是一类重要的组合优化问题，包括图与超图上的MAX-CUT问题。针对一般约束下该问题的当前最优算法具有0.432的近似比。该算法采用了涉及采样过程的典型连续贪心技术，因此存在查询复杂度高且本质随机化的缺陷。本文提出了多种在不同约束下最大化对称子模函数的高效确定性算法。具体而言，对于基数约束，我们设计了一种确定性算法，其达到0.432的近似比且仅需O(kn)次查询；此前最佳确定性算法仅能达到0.385-ε的近似比，且需要O(kn(10/(9ε))^(20/(9ε)-1))次查询。对于拟阵约束，我们设计的确定性算法达到1/3-ε的近似比，使用O(kn log ε^(-1))次查询；而此前最佳确定性算法虽能达到相同近似比，但需要多达O(ε^(-1)n^4)次查询。对于具有较大宽度的打包约束，我们设计的确定性算法达到0.432-ε的近似比，使用O(n^2)次查询；据我们所知，此前该约束下尚无确定性算法。最后，所提算法可适配单背包约束，以O(n^4)次查询实现0.432的近似比；此前最佳确定性算法仅能达到0.316-ε的近似比，且需要Õ(n^3)次查询。