  We formulate an irreversible Markov chain Monte Carlo algorithm for the
self-avoiding walk (SAW), which violates the detailed balance condition and
satisfies the balance condition. Its performance improves significantly
compared to that of the Berretti-Sokal algorithm, which is a variant of the
Metropolis-Hastings method. The gained efficiency increases with the spatial
dimension (D), from approximately $10$ times in 2D to approximately $40$ times
in 5D. We simulate the SAW on a 5D hypercubic lattice with periodic boundary
conditions, for a system with a linear size up to $L=128$, and confirm that as
for the 5D Ising model, the finite-size scaling of the SAW is governed by
renormalized exponents $\nu^*=2/d$ and $\gamma/\nu^*=d/2$. The critical point
is determined, which is approximately $8$ times more precise than the best
available estimate.
