The Opacity of 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]. This paper studies the nontransparency of backbones. 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. Further, 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.