HOLZEM – Formal Definition, Algorithm and Correctness Proof for Pathfinding in Finite Graphs

HOLZEM – A Greedy Pathfinding Algorithm for Finite Weighted Graphs This paper introduces HOLZEM, a greedy algorithm designed to find paths in finite weighted directed graphs. The algorithm selects the locally minimum available outgoing edge at each vertex. If the selected edge leads to an already visited vertex or a dead end, the corresponding edge is permanently blocked, and the search restarts from the initial vertex. The paper provides a formal mathematical definition of HOLZEM, describes its execution rules and state transitions, and establishes two supporting lemmas. The main theorem proves the conditional completeness and finite termination of the algorithm: if a path from the starting vertex A to the target vertex Z exists in the original finite graph, HOLZEM will reach Z after finitely many steps. Furthermore, the paper presents an upper bound on the number of edge traversals and demonstrates the algorithm through an example. The work focuses on mathematical correctness, reachability preservation and termination. It does not claim that HOLZEM necessarily finds the shortest or minimum-weight path