A polynomial Turing compression (PTC) for a parameterized problem $L$ is a polynomial time Turing machine that has access to an oracle for a problem $L'$ such that a polynomial in the input parameter bounds each query. Meanwhile, a polynomial (many-one) compression (PC) can be regarded as a restricted variant of PTC where the machine can query the oracle exactly once and must output the same answer as the oracle. Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam (STOC 2008) initiated an impressive hardness theory for PC under the assumption coNP $\not\subseteq$ NP/poly. Since PTC is a generalization of PC, we define $\mathcal{C}$ as the set of all problems that have PTCs but have no PCs under the assumption coNP $\not\subseteq$ NP/poly. Based on the hardness theory for PC, Fernau et al. (STACS 2009) found the first problem Leaf Out-tree($k$) in $\mathcal{C}$. However, very little is known about $\mathcal{C}$, as only a dozen problems were shown to belong to the complexity class in the last ten years. Several problems are open, for example, whether CNF-SAT($n$) and $k$-path are in $\mathcal{C}$, and novel ideas are required to better understand the fundamental differences between PTCs and PCs. In this paper, we enrich our knowledge about $\mathcal{C}$ by showing that several problems parameterized by modular-width ($mw$) belong to $\mathcal{C}$. More specifically, exploiting the properties of the well-studied structural graph parameter $mw$, we demonstrate 17 problems parameterized by $mw$ are in $\mathcal{C}$, such as Chromatic Number($mw$) and Hamiltonian Cycle($mw$). In addition, we develop a general recipe to prove the existence of PTCs for a large class of problems, including our 17 problems.

翻译：多项式图灵压缩（PTC）是针对参数化问题$L$的一种多项式时间图灵机，该机器可访问问题$L'$的预言机，且每次查询以输入参数的多项式为界。而多项式（多对一）压缩（PC）可视为PTC的受限变体，其中机器只能对预言机进行一次查询，且必须输出与预言机相同的答案。Bodlaender等人（ICALP 2008）以及Fortnow和Santhanam（STOC 2008）在假设coNP $\not\subseteq$ NP/poly的条件下，开创了PC的硬性理论。由于PTC是PC的推广，我们定义$\mathcal{C}$为所有在假设coNP $\not\subseteq$ NP/poly下具有PTC但无PC的问题集合。基于PC的硬性理论，Fernau等人（STACS 2009）发现了$\mathcal{C}$中的首个问题——Leaf Out-tree($k$)。然而，由于过去十年间仅有十余个问题被证明属于该复杂类，人们对$\mathcal{C}$的了解仍非常有限。若干问题尚未解决，例如CNF-SAT($n$)和$k$-path是否属于$\mathcal{C}$，而理解PTC与PC的根本差异需要新颖的见解。本文通过证明若干以模宽（$mw$）参数化的问题属于$\mathcal{C}$，丰富了我们对这一复杂类的认知。具体而言，我们利用已被充分研究的图结构参数$mw$的性质，证明了17个以$mw$参数化的问题（如Chromatic Number($mw$)和Hamiltonian Cycle($mw$)）属于$\mathcal{C}$。此外，我们还发展了一种通用方法，用于证明包括这17个问题在内的广泛问题类存在PTC。