Stochastic Lookahead with Bipartite Polarity for Graceful Labeling of Dense Graphs

This paper presents a multi-threaded randomized solver for the graceful labeling problem on dense graphs, specifically targeting grid and lattice structures. While the problem is NP-complete and standard backtracking typically fails beyond N ≈ 35 nodes, this approach successfully scales to N=1000. Key Contributions: Algorithm: Combines bipartite polarity constraints (soft α-labeling), stochastic lookahead pruning (R4), and a shared-memory "Totem" for progress inheritance across rapid restarts. Performance: Achieves sub-second solutions for 6 × 10 grids (N=60) on a mobile processor (Snapdragon 8 Gen 2). Breakthrough Result: Reports the first verified graceful labeling of a 10 × 10 × 10 cubic lattice (N=1000, M=2700), verified with all edge differences distinct. Constraint Density Paradox: Identifies a phenomenon where denser graphs solve faster due to more effective lookahead pruning. This work demonstrates that carefully engineered randomized heuristics can push the computational frontier of graph labeling by orders of magnitude using consumer hardware.