  We consider general discrete Markov Random Fields(MRFs) with additional
bottleneck potentials which penalize the maximum (instead of the sum) over
local potential value taken by the MRF-assignment. Bottleneck potentials or
analogous constructions have been considered in (i) combinatorial optimization
(e.g. bottleneck shortest path problem, the minimum bottleneck spanning tree
problem, bottleneck function minimization in greedoids), (ii) inverse problems
with $L_{\infty}$-norm regularization, and (iii) valued constraint satisfaction
on the $(\min,\max)$-pre-semirings. Bottleneck potentials for general discrete
MRFs are a natural generalization of the above direction of modeling work to
Maximum-A-Posteriori (MAP) inference in MRFs. To this end, we propose MRFs
whose objective consists of two parts: terms that factorize according to (i)
$(\min,+)$, i.e. potentials as in plain MRFs, and (ii) $(\min,\max)$, i.e.
bottleneck potentials. To solve the ensuing inference problem, we propose
high-quality relaxations and efficient algorithms for solving them. We
empirically show efficacy of our approach on large scale seismic horizon
tracking problems.
