当前位置: X-MOL 学术SIAM J. Comput. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Topology Is Irrelevant (In a Dichotomy Conjecture for Infinite Domain Constraint Satisfaction Problems)
SIAM Journal on Computing ( IF 1.6 ) Pub Date : 2020-03-30 , DOI: 10.1137/18m1216213
Libor Barto , Michael Pinsker

SIAM Journal on Computing, Volume 49, Issue 2, Page 365-393, January 2020.
The tractability conjecture for finite domain constraint satisfaction problems (CSPs) stated that such CSPs are solvable in polynomial time whenever there is no natural reduction, in some precise technical sense, from the 3-SAT problem; otherwise, they are NP-complete. Its recent resolution draws on an algebraic characterization of the conjectured borderline: the CSP of a finite structure permits no natural reduction from 3-SAT if and only if the stabilizer of the polymorphism clone of the core of the structure satisfies some nontrivial system of identities, and such satisfaction is always witnessed by several specific nontrivial systems of identities which do not depend on the structure. The tractability conjecture has been generalized in the above formulation to a certain class of infinite domain CSPs, namely, CSPs of reducts of finitely bounded homogeneous structures. It was subsequently shown that the conjectured borderline between hardness and tractability, i.e., a natural reduction from 3-SAT, can be characterized for this class by a combination of algebraic and topological properties. However, it was not known whether the topological component is essential in this characterization. We provide a negative answer to this question by proving that the borderline is characterized by one specific algebraic identity, namely, the pseudo-Siggers identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our main theorem is also of independent mathematical interest, characterizing a topological property of any $\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).


中文翻译:

拓扑无关紧要(在无限域约束满足问题的二分法猜想中)

SIAM计算杂志,第49卷,第2期,第365-393页,2020年1月。
有限域约束满足问题(CSP)的可预测性推测说,只要在某些精确的技术意义上,自然不存在3-SAT问题的减少,那么这种CSP就可以在多项式时间内求解。否则,它们是NP完全的。它的最新解决方案基于推测的边界线的代数表征:如果且仅当结构核心的多态性克隆的稳定剂满足某些非平凡的身份系统时,有限结构的CSP不允许从3-SAT自然还原,这种满足总是通过不依赖于结构的几个特定的​​非平凡的身份系统来证明的。在上面的公式中,可延性猜想被概括为一类无限域CSP,即 有限界均匀结构还原的CSP。随后表明,对于此类,可以通过代数和拓扑性质的组合来表征硬度和可延展性之间的推测边界,即从3-SAT自然还原。但是,尚不清楚在此表征中拓扑部分是否必不可少。我们通过证明边界线具有一个特定的代数身份,即伪Siggers身份$ \ alpha s(x,y,x,z,y,z)\ approx \ beta s,来提供对该问题的否定答案。 (y,x,z,x,z,y)$。这完成了将无限域CSP二分法猜想简化为有限情况的拟议策略的步骤之一。我们的主要定理也具有独立的数学意义,
更新日期:2020-03-30
down
wechat
bug