Symmetry Breaking with Polynomial Delay

TL;DR

This paper demonstrates that adding polynomially many lexleader constraints to conservative CSPs allows for efficient solution enumeration with polynomial delay, effectively breaking symmetries without exponential overhead.

Contribution

It proves that polynomially many lexleader constraints enable polynomial delay solution enumeration in conservative CSPs, balancing symmetry breaking and computational efficiency.

Findings

Solution enumeration with polynomial delay is possible with added lexleader constraints.

Adding polynomially many lexleader constraints often excludes exponentially many solutions.

The approach maintains efficiency even without complete symmetry breaking.

Abstract

A conservative class of constraint satisfaction problems CSPs is a class for which membership is preserved under arbitrary domain reductions. Many well-known tractable classes of CSPs are conservative. It is well known that lexleader constraints may significantly reduce the number of solutions by excluding symmetric solutions of CSPs. We show that adding certain lexleader constraints to any instance of any conservative class of CSPs still allows us to find all solutions with a time which is polynomial between successive solutions. The time is polynomial in the total size of the instance and the additional lexleader constraints. It is well known that for complete symmetry breaking one may need an exponential number of lexleader constraints. However, in practice, the number of additional lexleader constraints is typically polynomial number in the size of the instance. For polynomially many lexleader constraints, we may in general not have complete symmetry breaking but polynomially many lexleader constraints may provide practically useful symmetry breaking -- and they sometimes exclude super-exponentially many solutions. We prove that for any instance from a conservative class, the time between finding successive solutions of the instance with polynomially many additional lexleader constraints is polynomial even in the size of the instance without lexleaderconstraints.