Solving Linear Constraints in Elementary Abelian p-Groups of Symmetries

TL;DR

This paper explores a polynomial-time method for solving certain symmetry constraints in CSPs using linear algebra, focusing on elementary Abelian p-groups, and demonstrates its limitations and potential for practical use.

Contribution

It introduces a polynomial algorithm for symmetry constraints in elementary Abelian p-groups when k=p=2, and analyzes its limitations for other parameters.

Findings

Algorithm applies when k=p=2

NP-hardness for other values of k and p

Potential for efficient symmetry exploitation in CSPs

Abstract

Symmetries occur naturally in CSP or SAT problems and are not very difficult to discover, but using them to prune the search space tends to be very challenging. Indeed, this usually requires finding specific elements in a group of symmetries that can be huge, and the problem of their very existence is NP-hard. We formulate such an existence problem as a constraint problem on one variable (the symmetry to be used) ranging over a group, and try to find restrictions that may be solved in polynomial time. By considering a simple form of constraints (restricted by a cardinality k) and the class of groups that have the structure of Fp-vector spaces, we propose a partial algorithm based on linear algebra. This polynomial algorithm always applies when k=p=2, but may fail otherwise as we prove the problem to be NP-hard for all other values of k and p. Experiments show that this approach though restricted should allow for an efficient use of at least some groups of symmetries. We conclude with a few directions to be explored to efficiently solve this problem on the general case.