Evolving Labelings of Graceful Graphs

A graceful labeling of a graph G = (V, E) is an assignment of labels to the vertices V of G subject to constraints arising from the structure of the graph. A graph is called graceful if it admits a graceful labeling. As a combinatorial problem, it has applications in coding theory, communications networks, and optimizing circuit layouts. Several different approaches, both heuristic and complete, for finding graceful labelings have been developed and analyzed empirically. Most such algorithms have been established in the context of verifying the conjecture that trees are graceful. In this paper, we present the first rigorous running time analysis of a simple evolutionary algorithm applied to finding labelings of graceful graphs. We prove that an evolutionary algorithm can find a graceful labeling in polynomial time for all paths, stars, and complete bipartite graphs with a constant-sized partition. We also empirically compare the running time of a simple evolutionary algorithm against a complete constraint solver.