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Girard's Light linear logic (LLL) characterized polynomial time in theproof-as-program paradigm with a bound on cut elimination. This logic relied ona stratification principle and a "one-door" principle which were generalizedlater respectively in the systems L^4 and L^3a. Each system was brought withits own complex proof of Ptime soundness.  In this paper we propose a broad sufficient criterion for Ptime soundness forlinear logic subsystems, based on the study of paths inside the proof-nets,which factorizes proofs of soundness of existing systems and may be used forfuture systems. As an additional gain, our bound stands for any reductionstrategy whereas most bounds in the literature only stand for a particularstrategy.

On paths-based criteria for polynomial time complexity in proof-nets