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　[16].　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-e　ratio　and　uses　O　kn(10　9e　)　20　9e-1)　queries　[12].　For　the　matroid　constraint,　we　design　a　deterministic　algorithm　that　attains　a　1/3-e　ratio　and　uses　O(kn　log　e-1)　queries.　Previously,　the　best　deterministic　algorithm　can　also　attain　1/3-e　ratio　but　it　uses　much　larger　O(e-1n4)　queries　[24].　For　the　packing　constraints　with　a　large　width,　we　design　a　deterministic　algorithm　that　attains　a　0.432-e　ratio　and　uses　O(n2)　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(n4)　queries.　Previously,　the　best　deterministic　algorithm　attains　a　0.316-e　ratio　and　uses　O(n3)　queries　[2].