# Conjunctions of Among Constraints

## TL;DR

This paper investigates conditions for efficient, complete domain filtering algorithms for conjunctions of among constraints, unifying and generalizing previous approaches through a flow-based algorithm.

## Contribution

It introduces a systematic analysis of when complete filtering is feasible for conjunctions of among constraints and presents a unifying flow-based filtering algorithm.

## Key findings

- Derived a domain flow-based filtering algorithm
- Identified necessary restrictions on scope and range for meaningful filtering
- Unified and generalized previous filtering results

## Abstract

Many existing global constraints can be encoded as a conjunction of among constraints. An among constraint holds if the number of the variables in its scope whose value belongs to a prespecified set, which we call its range, is within some given bounds. It is known that domain filtering algorithms can benefit from reasoning about the interaction of among constraints so that values can be filtered out taking into consideration several among constraints simultaneously. The present pa- per embarks into a systematic investigation on the circumstances under which it is possible to obtain efficient and complete domain filtering algorithms for conjunctions of among constraints. We start by observing that restrictions on both the scope and the range of the among constraints are necessary to obtain meaningful results. Then, we derive a domain flow-based filtering algorithm and present several applications. In particular, it is shown that the algorithm unifies and generalizes several previous existing results.