A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the Boolean satisfiability problem (SAT), and the constraint satisfaction problem (CSP), there exist many results concerning upper and lower bounds for kernelizability of specific problems, but it is safe to say that we lack general methods to determine whether a given SAT problem admits a kernel of a particular size. This could be contrasted to the currently flourishing research program of determining the classical complexity of finite-domain CSP problems, where almost all non-trivial tractable classes have been identified with the help of algebraic properties. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard SAT and CSP problems, parameterized by the number of variables. Our main focus is on problems admitting linear kernels, as has, somewhat surprisingly, previously been shown to exist. We show that a CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP problem which is preserved by a Maltsev operation. We also study extensions of this towards SAT and CSP problems with kernels with O(n^c) constraints, c>1, based on embeddings into CSP problems preserved by a k-edge operation, k > c. These results follow via a variant of the celebrated few subpowers algorithm. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization of SAT problems with linear kernels, by showing that an algebraic condition that is shared by all problems with a Maltsev embedding is also necessary for the existence of a linear kernel unless NP is included in co-NP/poly.
The Complexity of Network Satisfaction Problems for Symmetric Relation Algebras with a Flexible Atom
Robin Hirsch在1996年提出了“真正的大复杂性问题”：对所有有限关系代数A的网络满意度问题的计算复杂性进行分类。我们提供了一个完整的分类的情况下，A是对称的，并有一个灵活的原子;在这种情况下，问题是NP完全或在P的分类任务可以减少到的情况下，A是整数。如果一个有限积分关系代数有一个柔性原子，则它有一个正规表示B。然后，我们可以研究的计算复杂性的网络满意度问题的A使用泛代数方法，通过分析的多态性B。我们还使用了一个拉姆齐型的结果Nešetziil和Rödl和复杂性二分的结果Bulatov保守的有限域约束满足问题。
Complexity Classification in Infinite-Domain Constraint Satisfaction
Université Paris Diderot – Paris 7（巴黎第七大学（巴黎狄德罗大学））；Université Paris 7（巴黎第七大学）；CNRS（法国国家科学研究中心）；Université Paris 6（巴黎第六大学）；Albert-Ludwigs-Universität Freiburg（阿尔伯特-路德维希-弗莱堡大学）；ENS Lyon（里昂高等师范学院）；INRIA（法国国家信息与自动化研究所）；ENS Cachan（卡尚高等师范学院）；Université Pompeu Fabra（庞培法布拉大学）；Université de Linköping（林雪平大学）
Max-3-Lin over Non-Abelian Groups with Universal Factor Graphs
具有n个变量和m个约束的约束满足问题的实例的因子图是[m]和[n]之间的二分图，描述哪个变量出现在哪个约束中。因此，CSP的实例完全由其因子图和谓词列表定义。我们展示了非阿贝尔群上的Max-3-LIN的不可逼近性（无论是在完美完备性的情况下还是在不完美完备性的情况下），具有与一般情况下相同的不可逼近因子，即使因子图是固定的。在此过程中，我们还表明，即使我们将Max-3-LIN实例中的线性方程限制为x * y * z = g的形式，这些最佳硬度结果也保持不变，其中x，y，z是变量，g是群元素。我们使用表示理论和傅立叶分析非阿贝尔群分析的减少。
University of California, Berkeley（加州大学伯克利分校）；Simons Institute for the Theory of Computing and the University of California, Berkeley（计算理论西蒙斯研究所和加州大学伯克利分校）；Eindhoven University of Technology, The Netherlands（埃因霍温理工大学）；Royal Holloway, University of London（伦敦大学皇家霍洛威学院）
