Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on $n$-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth $d$ and using $k$ labels, we can solve - Independent Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $O(dk^2\log n)$ space; - Max Cut in time $n^{O(dk)}$ using $O(dk\log n)$ space; and - Dominating Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $n^{O(1)}$ space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with $d$ if one wishes to keep the space complexity polynomial.

翻译：动态规划在各种图分解上是参数化复杂度中最基础的技术之一。然而，即使我们考虑像路径分解或树分解这样简单的概念，此类动态规划使用的空间在分解宽度上是指数级的，并且有充分理由相信这是必要的。但研究表明，在低树深度图中，可以在不牺牲比有界宽度树分解算法更优的时间复杂度的前提下，设计出达到多项式空间复杂度的算法。这里，树深度是一个图参数，直观而言，它同时考虑了树分解的深度和宽度，而非仅宽度。受此启发，我们考虑允许有界深度和标签数量的团表达式图，即低灌木深度（sd）图。在此，sd是团宽度的有界深度类比，正如td是树宽度的有界深度类比。我们证明，在此设定下，限制分解深度是改善空间复杂度的决定因素。具体而言，我们证明：在配备深度为d、使用k个标签的树模型（sd的底层分解概念）的n顶点图上，我们可以：- 独立集问题在时间2^{O(dk)}·n^{O(1)}内使用O(dk^2 log n)空间求解；- 最大割问题在时间n^{O(dk)}内使用O(dk log n)空间求解；- 支配集问题通过随机算法在时间2^{O(dk)}·n^{O(1)}内使用n^{O(1)}空间求解。我们还基于最长公共子序列复杂性的某种假设，建立了一个下界，表明至少在IS情形下，若要保持空间复杂度为多项式，时间复杂度参数因子的指数必须随d增长。