  The principle characteristics of biased greedy random walks (BGRWs) on
two-dimensional lattices with real-valued quenched disorder on the lattice
edges are studied. Here, the disorder allows for negative edge-weights. In
previous studies, considering the negative-weight percolation (NWP) problem,
this was shown to change the universality class of the existing, static
percolation transition. In the presented study, four different types of BGRWs
and an algorithm based on the ant colony optimization (ACO) heuristic were
considered. Regarding the BGRWs, the precise configurations of the lattice
walks constructed during the numerical simulations were influenced by two
parameters: a disorder parameter rho that controls the amount of negative edge
weights on the lattice and a bias strength B that governs the drift of the
walkers along a certain lattice direction. Here, the pivotal observable is the
probability that, after termination, a lattice walk exhibits a total negative
weight, which is here considered as percolating. The behavior of this
observable as function of rho for different bias strengths B is put under
scrutiny. Upon tuning rho, the probability to find such a feasible lattice walk
increases from zero to one. This is the key feature of the percolation
transition in the NWP model. Here, we address the question how well the
transition point rho_c, resulting from numerically exact and "static"
simulations in terms of the NWP model can be resolved using simple dynamic
algorithms that have only local information available, one of the basic
questions in the physics of glassy systems.
