We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: Binary CSP parameterized by the vertex cover number is $\mathrm{W}[3]$-complete. More generally, for every positive integer $d$, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is $\mathrm{W}[2d+1]$-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd $t$, we introduce a parameterized complexity class $\mathrm{S}[t]$ with $\mathrm{W}[t]\subseteq \mathrm{S}[t]\subseteq \mathrm{A}[t]$, defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems.

翻译：我们研究了以约束图的顶点覆盖数、树深度以及若干基于调制器的相关参数为参数的二元CSP问题的参数化复杂度。主要发现如下：以顶点覆盖数为参数的二元CSP问题是$\mathrm{W}[3]$-完备的。更一般地，对于每个正整数$d$，以到树深度为$d$的图的调制器大小为参数的二元CSP问题是$\mathrm{W}[2d+1]$-完备的。这为W-层次的奇数层次提供了新的自然问题完备族。我们引入了一个新的复杂度类XSLP，并定义二元CSP问题以树深度为参数时对该类完备。我们给出了XSLP的两种等价刻画：其一将XSLP与一种在共非确定性和空间复杂度上具有特定限制的交替图灵机模型相关联；其二将XSLP与具有适当受限全称量词的一阶逻辑模型检验问题相联系。有趣的是，XSLP机器刻画证明中使用了通用树的概念，该概念在近期关于奇偶博弈的研究中占据重要地位。我们描述了一个介于W-层次与A-层次之间的新的复杂度层次：对于每个奇数$t$，我们引入参数化复杂度类$\mathrm{S}[t]$，满足$\mathrm{W}[t]\subseteq \mathrm{S}[t]\subseteq \mathrm{A}[t]$，其定义使用了插值于顶点覆盖数与树深度之间的参数。我们预期，所研究的诸多类在未来将有助于精确刻画图问题各种结构化参数化的复杂度。