The Opacity of Backbones

TL;DR

This paper investigates the nontransparency of backbones in boolean formulas, showing that even obvious backbones can be computationally intractable to identify under common cryptographic assumptions.

Contribution

It demonstrates that the intractability of finding backbones persists despite their apparent obviousness, under assumptions like the hardness of integer factoring.

Findings

Obvious backbones can be computationally hard to find.

Intractability persists under assumptions like integer factoring hardness.

Results extend to settings where P ≠ NP ∩ coNP or NP ∩ coNP problems are frequently hard.

Abstract

This paper approaches, using structural complexity theory, the question of whether there is a chasm between knowing an object exists and getting one's hands on the object or its properties. In particular, we study the nontransparency of so-called backbones. A backbone of a boolean formula $F$ is a collection $S$ of its variables for which there is a unique partial assignment $a_S$ such that $F[a_S]$ is satisfiable [MZK+99,WGS03]. We show that, under the widely believed assumption that integer factoring is hard, there exist sets of boolean formulas that have obvious, nontrivial backbones yet finding the values, $a_S$, of those backbones is intractable. We also show that, under the same assumption, there exist sets of boolean formulas that obviously have large backbones yet producing such a backbone $S$ is intractable. Furthermore, we show that if integer factoring is not merely worst-case hard but is frequently hard, as is widely believed, then the frequency of hardness in our two results is not too much less than that frequency. These results hold more generally, namely, in the settings where, respectively, one's assumption is that P $\neq$ NP $\cap$ coNP or that some problem in NP $\cap$ coNP is frequently hard.