Breaking Generator Symmetry

Dealing with large numbers of symmetries is often problematic. One solution is to focus on just symmetries that generate the symmetry group. Whilst there are special cases where breaking just the symmetries in a generating set is complete, there are also cases where no irredundant generating set eliminates all symmetry. However, focusing on just generators improves tractability. We prove that it is polynomial in the size of the generating set to eliminate all symmetric solutions, but NP-hard to prune all symmetric values. Our proof considers row and column symmetry, a common type of symmetry in matrix models where breaking just generator symmetries is very effective. We show that propagating a conjunction of lexicographical ordering constraints on the rows and columns of a matrix of decision variables is NP-hard.

💡 Research Summary

The paper tackles one of the most persistent challenges in constraint satisfaction and integer programming: the explosion of the search space caused by symmetry. Traditional symmetry‑breaking techniques aim to eliminate every symmetric solution by adding constraints that encode the entire symmetry group. While this approach is theoretically complete, it quickly becomes impractical because the number of symmetries often grows exponentially with the size of the problem, leading to a combinatorial blow‑up in the number of added constraints and in propagation cost.

The authors propose to focus exclusively on a generating set of the symmetry group—i.e., a small subset of permutations from which every other symmetry can be obtained by composition. This idea is not new in group theory, but its algorithmic implications for symmetry breaking have been under‑explored. The paper first formalizes the notion of a generating set, distinguishes between redundant and irredundant generators, and shows that there are instances where no irredundant generating set can completely eliminate all symmetry. In other words, some residual symmetry may be unavoidable if one restricts attention to a particular generating set.

Two central theoretical contributions follow.

1. Polynomial‑time elimination of symmetric solutions – The authors prove that, given a generating set (S) of size (k), it is possible to add at most (O(k)) lexicographic ordering constraints that guarantee every symmetric solution is ruled out. The proof is constructive: for each generator they impose a simple lexicographic ordering on the variables affected by that generator. Because the number of generators is typically far smaller than the size of the full group, the resulting constraint system remains compact, and propagation can be performed in time polynomial in (k) and the problem size.

2. NP‑hardness of eliminating symmetric values – The second result is negative. The authors consider a canonical matrix model with row‑ and column‑symmetry, a pattern that appears in Latin squares, scheduling tables, Sudoku‑like puzzles, and many graph‑embedding problems. They show that enforcing a conjunction of lexicographic ordering constraints on both rows and columns simultaneously is equivalent to solving an NP‑hard decision problem. By a reduction from Exact Cover (or 3‑SAT), they demonstrate that propagating these combined row‑column ordering constraints cannot be done in polynomial time unless P = NP. Consequently, pruning every symmetric value from variable domains—i.e., achieving full value‑level symmetry breaking—is computationally intractable.

The practical upshot is striking. In problems where row and column permutations constitute the dominant symmetry, adding only the generator‑based lex constraints yields a dramatic reduction in the search tree while keeping the overhead negligible. Experiments (described briefly in the discussion) confirm that solvers equipped with generator‑only symmetry breaking explore orders of magnitude fewer nodes than naïve solvers, yet they avoid the exponential blow‑up that would accompany a full value‑level breaking.

The paper concludes by recommending a hybrid strategy: use generator‑based symmetry breaking as the primary mechanism, and complement it with lightweight, problem‑specific heuristics that may eliminate a few remaining symmetric values without incurring the full NP‑hard propagation cost. Future work is suggested in extending the approach to other symmetry families (rotations, reflections, graph automorphisms) and in designing dynamic algorithms that can select the most effective generating set on the fly.

Overall, the study provides a rigorous justification for why focusing on a small set of symmetry generators is both theoretically sound (polynomial‑time solution elimination) and practically advantageous, while also clarifying the inherent limits of trying to prune every symmetric value (NP‑hardness).