A large number of NP-hard graph problems can be solved in $f(w)n^{O(1)}$ time and space when the input graph is provided together with a tree decomposition of width $w$, in many cases with a modest exponential dependence $f(w)$ on $w$. Moreover, assuming the Strong Exponential-Time Hypothesis (SETH) we have essentially matching lower bounds for many such problems. They main drawback of these results is that the corresponding dynamic programming algorithms use exponential space, which makes them infeasible for larger $w$, and there is some evidence that this cannot be avoided. This motivates using somewhat more restrictive structure/decompositions of the graph to also get good (exponential) dependence on the corresponding parameter but use only polynomial space. A number of papers have contributed to this quest by studying problems relative to treedepth, and have obtained fast polynomial space algorithms, often matching the dependence on treewidth in the time bound. E.g., a number of connectivity problems could be solved by adapting the cut-and-count technique of Cygan et al. (FOCS 2011, TALG 2022) to treedepth, but this excluded well-known path and cycle problems such as Hamiltonian Cycle (Hegerfeld and Kratsch, STACS 2020). Recently, Nederlof et al. (SIDMA 2023) showed how to solve Hamiltonian Cycle, and several related problems, in $5^τn^{O(1)}$ randomized time and polynomial space when provided with an elimination forest of depth $τ$. We present a faster (also randomized) algorithm, running in $4^τn^{O(1)}$ time and polynomial space, for the same set of problems. We use ordered pairs of what we call consistent matchings, rather than perfect matchings in an auxiliary graph, to get the improved time bound.

翻译：大量NP难图问题可在输入图连同宽度为$w$的树分解一起给出时，以$f(w)n^{O(1)}$时间和空间求解，且在许多情况下$f(w)$关于$w$具有适度的指数依赖关系。此外，在强指数时间假设（SETH）下，许多此类问题存在本质上匹配的下界。这些结果的主要缺陷在于相应的动态规划算法使用指数空间，这使得它们对于较大的$w$不可行，且有证据表明这一限制难以避免。这促使我们采用图中更具限制性的结构/分解，在相应参数上获得良好的（指数）依赖关系的同时仅使用多项式空间。多项研究通过分析树深相关问题对此目标做出贡献，并获得了快速的多项式空间算法，其时间边界中的参数依赖关系通常与树宽算法相匹配。例如，许多连通性问题可通过将Cygan等人（FOCS 2011, TALG 2022）的"剪裁与计数"技术适配到树深来求解，但这排除了诸如哈密顿环（Hegerfeld和Kratsch, STACS 2020）等著名路径与环问题。最近，Nederlof等人（SIDMA 2023）展示了当给出深度为$τ$的消除森林时，如何以$5^τn^{O(1)}$随机化时间和多项式空间求解哈密顿环问题及若干相关问题的算法。针对同一问题集，我们提出了一种更快速的（同样为随机化）算法，运行时间为$4^τn^{O(1)}$且使用多项式空间。我们采用有序对（称为相容匹配）而非辅助图中的完美匹配，从而获得改进的时间边界。