  Radio relics in galaxy clusters are thought to be associated with powerful
shock waves that accelerate particles via diffusive shock acceleration (DSA).
Among the particles accelerated by DSA, relativistic protons should outnumber
electrons by a large factor. While the relativistic electrons emit synchrotron
emission detectable in the radio band, the protons interact with the thermal
gas to produce gamma-rays in hadronic interactions. Using simple models for the
propagation of shock waves through clusters, the distribution of thermal gas
and the efficiency of DSA, we find that the resulting hadronic $\gamma$-ray
emission lies very close or above the upper limits from the FERMI data on
nearby clusters. This suggests that the relative acceleration efficiency of
electrons and protons is at odds with predictions from DSA. The inclusion of
re-accelerated "fossil" particles does not seem to solve the problem. Our study
highlights a possible tension of the commonly assumed scenario for the
formation of radio relics and we discuss possible solutions to the problem.
