Symmetric Operations on Domains of Size at Most 4

TL;DR

This paper characterizes all idempotent clones with symmetric operations up to arity four, confirming a conjecture that such clones contain symmetric operations of all arities for small sets, aiding in solving constraint satisfaction problems.

Contribution

It provides a complete classification of these clones for sets of size at most four, proving the conjecture for small domains.

Findings

All such clones contain symmetric operations of every arity for sets of size at most four.

The conjecture holds true for domains with size up to four.

This characterization aids in understanding when linear programming relaxations solve CSPs.

Abstract

To convert a fractional solution to an instance of a constraint satisfaction problem into a solution, a rounding scheme is needed, which can be described by a collection of symmetric operations with one of each arity. An intriguing possibility, raised in a recent paper by Carvalho and Krokhin, would imply that any clone of operations on a set $D$ which contains symmetric operations of arities $1, 2, \ldots, \lvert D \rvert$ contains symmetric operations of all arities in the clone. If true, then it is possible to check whether any given family of constraint satisfaction problems is solved by its linear programming relaxation. We characterize all idempotent clones containing symmetric operations of arities $1, 2, \ldots, \lvert D \rvert$ for all sets $D$ with size at most four and prove that each one contains symmetric operations of every arity, proving the conjecture above for $\lvert D \rvert \leq 4$.