The Weight in Enumeration

TL;DR

This paper classifies the complexity of enumerating solutions of B-formulae based on their weight, considering various orderings and weighted variants, revealing new computational insights and complexity results.

Contribution

It provides a complete complexity classification for enumerating B-formula models by weight, including weighted variants and their impact on tractability.

Findings

Enumeration by weight varies in complexity depending on B and ordering.

Weighted variants can make previously tractable problems intractable.

Classifications include Min-Ones and Max-Ones optimization problems.

Abstract

In our setting enumeration amounts to generate all solutions of a problem instance without duplicates. We address the problem of enumerating the models of B-formulae. A B-formula is a propositional formula whose connectives are taken from a fixed set B of Boolean connectives. Without imposing any specific order to output the solutions, this task is solved. We completely classify the complexity of this enumeration task for all possible sets of connectives B imposing the orders of (1) non-decreasing weight, (2) non-increasing weight; the weight of a model being the number of variables assigned to 1. We consider also the weighted variants where a non-negative integer weight is assigned to each variable and show that this add-on leads to more sophisticated enumeration algorithms and even renders previously tractable cases intractable, contrarily to the constraint setting. As a by-product we obtain complete complexity classifications for the optimization problems known as Min-Ones and Max-Ones which are in the B-formula setting two different tasks.