A Comparison of Lex Bounds for Multiset Variables in Constraint Programming

TL;DR

This paper compares eight different multiset variable representations in constraint programming, showing that certain orderings provide tighter bounds and improve pruning and runtime over traditional subset bounds.

Contribution

It introduces and empirically evaluates eight new multiset variable representations based on various orderings, demonstrating their effectiveness over subset bounds.

Findings

Length-variety-(co)lex representations give tighter bounds.

Certain orderings improve pruning efficiency.

Enhanced representations lead to better runtime performance.

Abstract

Set and multiset variables in constraint programming have typically been represented using subset bounds. However, this is a weak representation that neglects potentially useful information about a set such as its cardinality. For set variables, the length-lex (LL) representation successfully provides information about the length (cardinality) and position in the lexicographic ordering. For multiset variables, where elements can be repeated, we consider richer representations that take into account additional information. We study eight different representations in which we maintain bounds according to one of the eight different orderings: length-(co)lex (LL/LC), variety-(co)lex (VL/VC), length-variety-(co)lex (LVL/LVC), and variety-length-(co)lex (VLL/VLC) orderings. These representations integrate together information about the cardinality, variety (number of distinct elements in the multiset), and position in some total ordering. Theoretical and empirical comparisons of expressiveness and compactness of the eight representations suggest that length-variety-(co)lex (LVL/LVC) and variety-length-(co)lex (VLL/VLC) usually give tighter bounds after constraint propagation. We implement the eight representations and evaluate them against the subset bounds representation with cardinality and variety reasoning. Results demonstrate that they offer significantly better pruning and runtime.