A natural strengthening of an algorithm for the (promise) constraint satisfaction problem is its singleton version: we first fix a variable to an element from its domain, then run the algorithm, and remove the element from the domain if the answer is negative. Using the Hales-Jewett theorem, we characterize the power of the singleton versions of standard universal algorithms for the (promise) CSP over a fixed template in terms of the existence of polymorphisms with certain symmetries, which we call palette symmetric polymorphisms. By proving the existence of such polymorphisms we establish that the singleton version of the BLP+AIP algorithm solves all (multi-sorted) tractable CSPs over domains of size at most 7. We further show that already for domain size 8 there exists a relational structure arising from the dihedral group $\mathbf D_4$ that does not admit palette symmetric polymorphisms and cannot be solved by singleton BLP+AIP. By providing concrete CSP templates, we illustrate the limitations of linear programming, the power of the singleton versions, and the elegance of palette symmetric polymorphisms. Among tractable temporal templates, we exhibit a structure demonstrating that finiteness is crucial for the Hales-Jewett argument; nevertheless, by introducing generalized palette polymorphisms we establish tractability for each such template.

翻译：为了强化（承诺）约束满足问题的算法，其自然改进版本为单一算法：我们首先将某个变量固定为其域中的一个元素，然后运行该算法，若返回否定答案则从域中移除该元素。借助Hales-Jewett定理，我们以具有特定对称性的多态性（称为调色板对称多态性）的存在性为表征，刻画了在固定模板上标准通用算法的单一版本对于（承诺）CSP的能力。通过证明此类多态性的存在性，我们确立了BLP+AIP算法的单一版本能求解所有域大小至多为7的（多类）可解CSP。我们进一步证明，对于域大小为8的情况，存在一个由二面体群$\mathbf D_4$衍生出的关系结构，该结构不承认调色板对称多态性，且不能被单一版BLP+AIP求解。通过提供具体的CSP模板，我们阐明了线性规划的局限性、单一版本的强大能力以及调色板对称多态性的优雅性。在可解时间模板中，我们展示了一个结构，证明有限性对于Hales-Jewett论证至关重要；尽管如此，通过引入广义调色板多态性，我们确立了每个此类模板的可解性。