Breaking Value Symmetry

TL;DR

This paper investigates the computational limitations of static and dynamic methods for handling value symmetries in constraint problems, showing that both approaches face significant complexity challenges.

Contribution

It demonstrates that static symmetry-breaking methods are NP-hard and dynamic methods can be exponential, highlighting fundamental computational limits.

Findings

Static symmetry-breaking is NP-hard.

Dynamic methods can take exponential time.

Both approaches have inherent computational limitations.

Abstract

One common type of symmetry is when values are symmetric. For example, if we are assigning colours (values) to nodes (variables) in a graph colouring problem then we can uniformly interchange the colours throughout a colouring. For a problem with value symmetries, all symmetric solutions can be eliminated in polynomial time. However, as we show here, both static and dynamic methods to deal with symmetry have computational limitations. With static methods, pruning all symmetric values is NP-hard in general. With dynamic methods, we can take exponential time on problems which static methods solve without search.