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Sometimes Hoarding is Harder than Cleaning: NP-hardness of Maximum Blocked-Clause Addition

18 pagesPublished: May 26, 2024

Abstract

Adding blocked clauses to a CNF formula can substantially speed up SAT-solving, both in theory and practice. In theory, the addition of blocked clauses can exponentially reduce the length of the shortest refutation for a formula [17, 19]. In practice, it has been recently shown that the runtime of CDCL solvers decreases significantly for certain instance families when blocked clauses are added as a preprocessing step [10,22]. This fact is in contrast to, but not in contradiction with, prior results showing that Blocked- Clause Elimination (BCE) is sometimes an effective preprocessing step [14,15]. We suggest that the practical role of blocked clauses in SAT-solving might be richer than expected. Concretely, we propose a theoretical study of the complexity of Blocked-Clause Addition (BCA) as a preprocessing step for SAT-solving, and in particular, consider the problem of adding the maximum number of blocked clauses of a given arity k to an input formula F. While BCE is a confluent process, meaning that the order in which blocked clauses are eliminated is irrelevant, this is not the case for BCA: adding a blocked clause to a formula might unblock a different clause that was previously blocked. This order-sensitivity turns out to be a crucial obstacle for carrying out BCA efficiently as a preprocessing step. Our main result is that computing the maximum number of k-ary blocked clauses that can be added to an input formula F is NP-hard for every k ≥ 2.

Keyphrases: blocked clause addition, cnf formulas, np hardness, preprocessing, sat solving

In: Nikolaj Bjørner, Marijn Heule and Andrei Voronkov (editors). Proceedings of 25th Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 100, pages 408-425.

BibTeX entry
@inproceedings{LPAR2024:Sometimes_Hoarding_is_Harder,
  author    = {Bernardo Subercaseaux},
  title     = {Sometimes Hoarding is Harder than Cleaning: NP-hardness of Maximum Blocked-Clause Addition},
  booktitle = {Proceedings of 25th Conference on Logic for Programming, Artificial Intelligence and Reasoning},
  editor    = {Nikolaj Bjørner and Marijn Heule and Andrei Voronkov},
  series    = {EPiC Series in Computing},
  volume    = {100},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/ZtQzZ},
  doi       = {10.29007/p5mc},
  pages     = {408-425},
  year      = {2024}}
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