Yes, afaik there is meaningful relevance between logic programming and type systems - Curry-Howard; Dependent types; SMT solvers (related to SAT solvers mentioned in this discussion). However, constraint problems discussed (scheduling, travel planning, puzzles) would require quite verbose and complex encoding to solve with a type system and significant sophistication - Idris or Coq aren't for the faint-hearted.

Type systems would absolutely choke on trying to solve things like: Knowlege representation and querying - imagine medical diagnosis systems with thousands symptoms and conditions; Dynamic constraint solving - where constraints emerge at runtime; Exploratory search problems requiring bfs/dfs of large solution spaces - e.g., route finding with changing conditions; NLP tasks - grammar parsing and understanding; etc.