  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.
