Min-Ones问题的预处理：二分法
Preprocessing of Min Ones Problems: A Dichotomy
摘要：
一个参数化问题由一个经典问题和一个额外的组成部分，即所谓的参数。这种观点允许预处理的正式定义：给定一个参数化实例（I，k），多项式核化计算大小和参数由k中的多项式限定的等价实例（I '，k'）。我们给出了最小一个约束满足问题的完整分类，即，Min Ones SAT（r），相对于允许或不允许多项式核化（除非NP \subseteq coNP/poly）。为此，我们引入了可合并性的概念。如果约束语言Γ的所有关系都是可合并的，那么基于非零闭核的向日葵核化的新变体适用。我们得到了一个具有O（k^{d+1}）个变量和多项式总大小的核，其中d是Γ中约束的最大元数，与一般性较低且可以说较简单的d-命中集问题的O（k^{d-1}）个顶点的界相比，这是很好的。否则，在Γ中任何不可合并的关系允许我们构造一个对数成本选择公式，即，一个n元选择公式，具有O（log n）真局部变量。由此我们可以使用Bodlaender等人以及Fortnow和Santhanam最近的结果来构造我们的下界，证明没有多项式核化，除非NP \subseteq coNP/poly和多项式层次崩溃到第三层。
英文摘要：
A parameterized problem consists of a classical problem and an additional component, the so-called parameter. This point of view allows a formal definition of preprocessing: Given a parameterized instance (I,k), a polynomial kernelization computes an equivalent instance (I',k') of size and parameter bounded by a polynomial in k. We give a complete classification of Min Ones Constraint Satisfaction problems, i.e., Min Ones SAT(Γ), with respect to admitting or not admitting a polynomial kernelization (unless NP \subseteq coNP/poly). For this we introduce the notion of mergeability. If all relations of the constraint language Γare mergeable, then a new variant of sunflower kernelization applies, based on non-zero-closed cores. We obtain a kernel with O(k^{d+1}) variables and polynomial total size, where d is the maximum arity of a constraint in Γ, comparing nicely with the bound of O(k^{d-1}) vertices for the less general and arguably simpler d-Hitting Set problem. Otherwise, any relation in Γthat is not mergeable permits us to construct a log-cost selection formula, i.e., an n-ary selection formula with O(log n) true local variables. From this we can construct our lower bound using recent results by Bodlaender et al. as well as Fortnow and Santhanam, proving that there is no polynomial kernelization, unless NP \subseteq coNP/poly and the polynomial hierarchy collapses to the third level.
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