  Tightness is a generalisation of the notion of convexity: a space is tight if
and only if it is "as convex as possible", given its topological constraints.
For a simplicial complex, deciding tightness has a straightforward exponential
time algorithm, but efficient methods to decide tightness are only known in the
trivial setting of triangulated surfaces.
  In this article, we present a new polynomial time procedure to decide
tightness for triangulations of $3$-manifolds -- a problem which previously was
thought to be hard. Furthermore, we describe an algorithm to decide general
tightness in the case of $4$-dimensional combinatorial manifolds which is fixed
parameter tractable in the treewidth of the $1$-skeletons of their vertex
links, and we present an algorithm to decide $\mathbb{F}_2$-tightness for weak
pseudomanifolds $M$ of arbitrary but fixed dimension which is fixed parameter
tractable in the treewidth of the dual graph of $M$.
