A New QEA Computing Near-Optimal Low-Discrepancy Colorings in the Hypergraph of Arithmetic Progressions

We present a new quantum-inspired evolutionary algorithm, the attractor population QEA (apQEA). Our benchmark problem is a classical and difficult problem from Combinatorics, namely finding low-discrepancy colorings in the hypergraph of arithmetic progressions on the first n integers, which is a massive hypergraph (e.g., with approx. 3.88 ×1011 hyperedges for n = 250 000). Its optimal low-discrepancy coloring bound \(\Theta(\sqrt[4]{n})\) is known and it has been a long-standing open problem to give practically and/or theoretically efficient algorithms. We show that apQEA outperforms known QEA approaches and the classical combinatorial algorithm (Sarkozy 1974) by a large margin. Regarding practicability, it is also far superior to the SDP-based polynomial-time algorithm of Bansal (2010), the latter being a breakthrough work from a theoretical point of view. Thus we give the first practical algorithm to construct optimal colorings in this hypergraph, up to a constant factor. We hope that our work will spur further applications of Algorithm Engineering to Combinatorics.