Is there a signature scheme which doesn't rely on the difficulty of factoring/dlp which generates short signatures? The Lamport signature scheme, for example, doesn't rely on the difficulty of any problem and it only depends on the existence of one-way functions.
Is there an alternative scheme which also doesn't rely on the difficulty of factoring/discrete logarithm, but provides shorter signatures?
There are actually quite a few of these. Interest has been raised on this topic mainly due to the "post-quantum" security of such schemes. Also, Lamport is only a one-time signature, and we want a full-blown signature schemes.

For just one example see, SPHINCS: https://sphincs.cr.yp.to/sphincs-20150202.pdf.