Hiding solutions in model RB: Forced instances are almost as hard as unforced ones

TL;DR

This paper demonstrates that forcing solutions in model RB does not significantly reduce problem difficulty, as forced instances are asymptotically similar to unforced ones in solution count and distribution.

Contribution

The paper provides rigorous proof that forced RB instances have similar solution counts and distributions as unforced instances, indicating hidden solutions do not simplify problem solving.

Findings

Expected number of solutions in forced RB matches unforced RB asymptotically

Distribution of forced RB instances is similar to unforced ones asymptotically

Hardness of solving forced RB instances is comparable to unforced instances

Abstract

In this paper we study the forced instance spaces of model RB, where one or two arbitrary satisfying assignments have been imposed. We prove rigorously that the expected number of solutions of forced RB instances is asymptotically the same with those of unforced ones. Moreover, the distribution of forced RB instances in the corresponding forced instance space is asymptotically the same with that of unforced RB instances in the unforced instance space. These results imply that the hidden assignments will not lead to easily solvable formulas, and the hardness of solving forced RB instances will be the same with unforced RB instances.