Coloring Queens with Thousands of Encodings

2026-09-23

AI Summary

本文通过比较数千种编码方法，研究了n×n棋盘上皇后图的着色问题，确定了影响求解器性能的关键因素。

Abstract

In The Art of Computer Programming, Knuth benchmarked 10 encoding techniques for computing the chromatic number of the queen's graph: the minimum number of colors needed to color the squares of an $n \times n$ chessboard so that no two squares sharing a row, column, or diagonal receive the same color. In this paper, we extend his analysis much further by comparing thousands of encodings for the same problem, which allows us to identify additional factors that are important for solver performance. We obtain 1584 encodings for this problem by varying (a) the constraints that encode which color is assigned to each cell, (b) the constraints that forbid the same color appearing in a row, column, or diagonal line, and (c) the symmetry-breaking constraints. We find that the three most impactful encoding factors are (i) the choice of symmetry-breaking constraints, (ii) enabling so-called clique hints, and (iii) enforcing that each cell is assigned exactly one color through blocked clauses. Furthermore, while Knuth proposed clique hints as an advantage of the order encoding, we show in fact that they can be effectively employed for the one-hot encoding as well.

Problem

• coloring
• queens graph
• chromatic number
• encoding techniques
• solver performance

Innovation

• symmetry-breaking constraints
• clique hints
• blocked clauses