claimroutineinferred_dep
Rational approximation of local law
Local law rounds to a rational table within total variation
makes the output instance rational
Claim·§8.1 Constraint satisfaction and Vertex Cover·inside corGap hardness from a fixed SDP integrality gap
Statement (paper)
For the fixed invocation (fixed , , and alphabet ), the local probability law on the finite set depends only on , and there is a rational probability vector with and ; this rational table is fixed finite data of the reduction.
In plain English
For the fixed invocation, the local probability law on the finite set depends only on , and it can be replaced by a rational probability vector whose total variation distance from is less than .
Why it's here
The verifier from the previous claim may use irrational probabilities, which cannot be written down in a polynomial-time reduction. Since lives on a finite set determined only by the fixed data, one can round it to a rational table of constant size, which is hardcoded into the reduction. The bound is what the later claims use to show that rationalizing changes every assignment's value by less than , keeping the total loss within .
Classified routine:
Density of the rationals: a probability vector on a fixed finite set can be rounded to a rational probability vector within any positive total variation distance. Here this bookkeeping is applied to the local law from the preceding fixed-instance verifier claim.