The Weight in Enumeration

  • Johannes SchmidtEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


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-formulæ. 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 also complexity classifications for the optimization problems known as \(\textsc {Min}\hbox {-}\textsc {Ones}\) and \(\textsc {Max}\hbox {-}\textsc {Ones}\) which are in the B-formula setting two different tasks.


Computational complexity Enumeration Non-decreasing weight Polynomial delay Post’s lattice MaxOnes 



The author would like to thank Johan Thapper for combinatorial support.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Jönköping International Business SchoolJönköping UniversityJönköpingSweden

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