Optimal Long Code Test with One Free Bit

For arbitrarily small constants ε, δ > 0, we present a long code test with one free bit, completeness 1 − ε and soundness δ. Using the test, we prove the following two inapproximability results:
1. Assuming the Unique Games Conjecture of Khot [17], given an n-vertex graph that has two disjoint independent sets of size ( 1/2 − ε)n each, it is NP-hard to find an independent set of size δn.
2. Assuming a (new) stronger version of the Unique Games Conjecture, the scheduling problem of minimizing weighted completion time with precedence constraints is inapproximable within factor 2 − ε.

By: Nikhil Bansal; Subhash Khot

Published in: RC24848 in 2009


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