  In this paper, we introduce the problem of finding an orientation of a given
undirected graph that maximizes the burning number of the resulting directed
graph. We show that the problem is polynomial-time solvable on
K\H{o}nig-Egerv\'{a}ry graphs (and thus on bipartite graphs) and that an almost
optimal solution can be computed in polynomial time for perfect graphs. On the
other hand, we show that the problem is NP-hard in general and W[1]-hard
parameterized by the target burning number. The hardness results are
complemented by several fixed-parameter tractable results parameterized by
structural parameters. Our main result in this direction shows that the problem
is fixed-parameter tractable parameterized by cluster vertex deletion number
plus clique number (and thus also by vertex cover number).

Data Structures and Algorithms