The first problem proved to be NP-complete was

Stephen Cook (1971) and independently Leonid Levin proved that SAT — the question of whether a Boolean formula in CNF has a satisfying assignment — is NP-complete. This was the foundational result every other NP-completeness proof reduces from SAT (or from a problem reduced from SAT). The Halting Problem is undecidable, not NP-complete Hamilton predates the theory Dijkstra worked on shortest paths.