Polynomial Turing compressions for some graph problems parameterized by modular-width

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 $L^{\prime }$ such that a polynomial in the input parameter bounds each query. Meanwhile, a polynomial 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 ⊈ NP/poly. Let $C$ be the set of all problems with PTCs but without PCs assuming coNP ⊈ NP/poly. Fernau et al. (STACS 2009) identified Leaf Out-tree(k) as the first problem in $C$. However, little is known about $C$, with only a dozen problems confirmed in it over the last fifteen years. Open questions remain, such as whether CNF-SAT(n) and k-path are in $C$, requiring novel ideas to clarify the differences between PTCs and PCs.

In this paper, we enrich our knowledge about $C$ by demonstrating that 17 problems parameterized by modular-width (mw), such as Chromatic Number(mw) and Hamiltonian Cycle(mw), belong to $C$. Additionally, we develop a general recipe to prove the existence of PTCs for a class of problems, including these 17.