Baby PIH: Parameterized Inapproximability of Min CSP
Venkatesan Guruswami, Xuandi Ren, Sai Sandeep
TL;DR
This work introduces Baby PIH, a parameterized inapproximability result for Min-CSPs in the form of $r$-list satisfiability. Under the hypothesis $\mathsf{W[1]} \neq \mathsf{FPT}$, it proves that distinguishing a satisfiable $2$CSP from one not even $r$-list satisfiable is hard, via a combinatorial bipartite direct-product reduction that runs in time polynomial in the number of variables and alphabet size and yields Baby PCP as a corollary. It further develops the Average Baby PIH framework, showing that the direct-product approach does not directly yield average-list hardness and presenting a counterexample; it then links average hypotheses to consequences for $k$-ExactCover under rectangular constraints, suggesting a path to stronger inapproximability results. The discussion outlines open problems, including whether average Baby PIH can be derived from clique hardness results or established through direct product testing theorems, highlighting key future directions toward a full PIH result. Overall, the paper provides a combinatorial, input-size-aware route to Baby PIH and clarifies the limitations and potential of averaging variants and product-testing methods in the parameterized hardness landscape.
Abstract
The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only $(1-\varepsilon)$-satisfiable (where the parameter is the number of variables) for some constant $0<\varepsilon<1$. We consider a minimization version of CSPs (Min-CSP), where one may assign $r$ values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the $r \times r$ pairs of values assigned to its variables (call such a CSP instance $r$-list-satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every $r \ge 1$, it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even $r$-list-satisfiable. We refer to this statement as "Baby PIH", following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size, and thus implies the Baby PCP theorem as well.