Dinur-style proof of the Reconfiguration Inapproximability Hypothesis (RIH)

Establish a Dinur-style proof of the Reconfiguration Inapproximability Hypothesis (RIH)—the assertion that Gap_{1,1-ε} q-CSP_W Reconfiguration is NP-hard for some ε in (0,1) and integers q, W—by developing approximation-preserving versions of degree reduction and gap amplification for Maxmin Binary CSP Reconfiguration that work from subconstant gaps.

Background

The paper develops an alphabet reduction for Maxmin Binary CSP Reconfiguration, complementing prior degree-reduction and gap-amplification tools. However, unlike Dinur’s combinatorial PCP proof, some of the available reconfiguration reductions are only gap-preserving and require an already constant gap, which prevents a direct Dinur-style iterative amplification starting from subconstant gaps.

The authors point out that an approximation-preserving degree reduction (that reduces degree without shrinking the gap when the starting gap is subconstant) and an approximation-preserving gap amplification step for Maxmin Binary CSP Reconfiguration appear necessary to carry out a Dinur-like proof of RIH in the reconfiguration setting.