Bounded-Depth Succinct Encodings and the Structure they Imply on Graphs

We study the complexity of graph problems succinctly encoded by bounded depth circuits and the existence of upward translation theorems for these models. While almost all succinct encodings have an upward translation theorem for some type of reduction, we prove that such theorems for CNF- and DNF-encoded graphs and for the most studied reductions do not exist. In contrast, we show that there are upward translation theorems for AC⁰ circuits of depth at least 3. This implies that the complexity of the succinct versions of problems complete for NP (under quantifier-free reductions), encoded by such circuits, have an exponential blow-up. We adapt these results to problems on explicitly given graphs with the same structural properties as graphs encoded by bounded depth circuits. We define a graph class hierarchy \(\mathcal {I}^{k}\) which consists of at most k alternating unions and intersections of edge-sets in \(\mathcal {I}^{0}\), a class which only consists of single bicliques. We show that the complexity of every NP-complete problem (under quantifier-free reductions) collapses to the second level: on graphs in \(\mathcal {I}^{2}\) the problem is already NP-complete. Finally, we show that by restricting \(\mathcal {I}^{2}\) to use only sub-logarithmic many intersections we get graphs for which Dominating Set is not NP-complete unless the Exponential Time Hypothesis is false. In contrast, a degree of O(log n) is enough for NP-completeness. Therefore, Dominating Set on \(\mathcal {I}^{2}\) graphs with intersection-degree O(log ^δ (n)) has either a spontaneous transition from P (for all δ < 1) to NP-complete (for δ = 1), or is NP-intermediate on the restricted graph class (for some δ < 1).