If you wish to avoid any kind of herbicide, the only option is to manually pull/dig the plant up (perhaps hiring help in that case) or to exhaust it by either continually cutting foliage off or covering it to block its light. For perennial weeds that are well-established, this starving of the plant by cutting or covering might take a long time (maybe weeks or months).

Every time a plant is forced to replace all of its foliage, it uses-up some of the energy stored in its roots. By making a weed resprout multiple times, you'll eventually exhaust those energy reserves and it will starve to death. Remove any new foliage that appears after each cut-back promptly in order to avoid prolonging the process, since the longer leaves remain on the plant, the more the plant can replenish some of those root reserves to keep regrowing.

The alternative method that achieves the same goal -- denying the plant the opportunity to photosynthesize -- is to smother it with a light-blocking cover. This is less useful of a tactic when a weed is growing very close to desirable plants, or when it could creep out from under the cover to get leaves back into sunlight. Curly Dock is at least a clumping plant, but close to a rose shrub may be understandably hard to cover-up effectively enough to deny the dock light while not interfering with rose growth or good air circulation in the rose foliage to keep them as disease-free as possible. Cutting all dock foliage off first and then covering the clump can help minimize the amount of area you need to cover, but depending on how abundant the weeds are in that area, this still might be a harder approach to use successfully compared to the mowing-down method above.

Going forward, if the dock is growing in soil with no mulch cover, using 2-3 inches of a mulch layer atop bare soil will discourage most weed seeds from germinating and establishing. (Make sure mulch doesn't cover the rose stems.)

Miri

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