P–NP Threshold for Synchronizing Road Coloring

Abstract

The parameterized Synchronizing-Road-Coloring Problem (in short: SRCP$_C^\ell$) in its decision version can be formulated as follows: given a digraph G with constant out-degree ℓ, check if G can be synchronized by some word of length C for some synchronizing labeling. We consider the family $\{SRCP_C^\ell\}_{C,\ell}$ of problems parameterized with constants C and ℓ and try to find for which C and ℓ SRCP$_C^\ell$ is NP-complete. It is known that SRCP$_C^3$ is NP-complete for C ≥ 8. We improve this result by showing that it is so for C ≥ 4 and for ℓ ≥ 3. We also show that SRCP$_C^\ell$ is in P for C ≤ 2 and ℓ ≥ 1. Hence, we solve SRCP almost completely for alphabet with 3 or more letters. The case C = 3 is still an open problem.