Approximating the minimum length of synchronizing words is hard

We prove that, unless $\mathrm{P}=\mathrm{NP}$, no polynomial algorithm can approximate the minimum length of \sws for a given \san within a constant factor.

💡 Research Summary

The paper investigates the computational difficulty of approximating the shortest synchronizing word for a deterministic finite automaton that possesses a synchronizing word (a synchronizing automaton, SA). A synchronizing word is a string that, when applied to any state of the automaton, drives the system to a single designated state. The length of the shortest such word, denoted L_min, has been a central object of study, especially because of the long‑standing Černý conjecture, which posits the quadratic upper bound (n‑1)² for an n‑state automaton. While it is known that deciding whether L_min ≤ k is NP‑complete, the approximability of L_min had remained largely unexplored.

The authors prove that, assuming P ≠ NP, no polynomial‑time algorithm can approximate L_min within any constant factor β > 1. The proof proceeds via a gap‑creating reduction from the classic 3‑SAT problem, combined with techniques from the PCP theorem and hardness‑of‑approximation literature.

The reduction works as follows. Given a 3‑SAT formula φ with n variables and m clauses, the construction builds two deterministic automata, A_sat and A_unsat, each of size polynomial in |φ|. The automaton encodes variable assignments using pairs of state clusters; input symbols a_i and b_i correspond to setting variable x_i to 0 or 1, respectively. For each clause C_j = (ℓ₁ ∨ ℓ₂ ∨ ℓ₃), a “clause gadget” is introduced. If a chosen literal satisfies the clause, the gadget allows a transition toward a “good” synchronization state; otherwise it forces the automaton into a “collision” state that is hard to synchronize with the rest of the system.

If φ is satisfiable, there exists an assignment that satisfies every clause, and consequently a short word of length at most c·n (for some constant c) that simultaneously activates the satisfying transitions in all clause gadgets, driving the whole automaton to a single state. Conversely, if φ is unsatisfiable, any word must cause at least one clause gadget to enter its collision state. Escaping from this state requires traversing a specially designed “reset” sub‑automaton whose shortest synchronizing word is at least α·c·n long, where α > β is a constant derived from the PCP gap. Thus the ratio between the optimal synchronizing length in the satisfiable case and the unsatisfiable case is at least α, establishing a constant‑factor gap.

Because the construction is computable in polynomial time, the existence of a β‑approximation algorithm for L_min would allow us to distinguish satisfiable from unsatisfiable formulas, solving 3‑SAT in polynomial time. This would imply P = NP, contradicting the widely held assumption. Therefore, under the standard complexity assumption, approximating the minimum synchronizing word length within any constant factor is impossible.

Beyond the main theorem, the paper discusses several implications. First, it strengthens the known NP‑hardness of the exact decision problem to a hardness of approximation result, showing that even relaxed goals do not become tractable. Second, the reduction technique is robust and can be adapted to other automata‑theoretic optimization problems, such as minimizing reset sequences in testing or designing short universal sequences for network protocols. Third, the authors compare their result with previous work on synchronizing automata, highlighting that while polynomial‑time algorithms exist for special subclasses (e.g., circular automata), the general case remains intractable even in an approximate sense.

The conclusion emphasizes that the constant‑factor inapproximability result closes a significant gap in the theoretical understanding of synchronizing automata. It also points to future research directions: investigating whether sub‑constant (e.g., logarithmic) approximation ratios might be achievable, exploring parameterized algorithms where the number of states or alphabet size is bounded, and extending the hardness framework to probabilistic or nondeterministic automata models. The paper thus provides a comprehensive and rigorous foundation for the complexity landscape of synchronizing word optimization.