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Symmetry
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Symmetry in Solving NP-Hard Problems
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Symmetry in Solving NP-Hard Problems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Computer".

Deadline for manuscript submissions: 31 October 2025 | Viewed by 296

Special Issue Editors

Dr. Luka Fürst

E-Mail Website
Guest Editor
Faculty of Computer and Information Science, University of Ljubljana, Večna pot 113, 1000 Ljubljana, Slovenia
Interests: graph grammars; graphs; algorithms in general; machine learning
Dr. Uroš Čibej

E-Mail Website
Guest Editor
Faculty of Computer and Information Science, University of Ljubljana, Večna pot 113, 1000 Ljubljana, Slovenia
Interests: graph theory; isomorphisms; combinatorial problems

Special Issue Information

Dear Colleagues,

The P=NP question is one of the most intriguing open problems in computer science. If the answer to the question is negative, as most researchers believe, then it makes sense to talk about NP-complete problems (the hardest problems in NP) and NP-hard problems (problems that are at least as hard as the hardest problems in NP). These complexity classes include a wide range of problems from diverse areas, such as combinatorial optimization (e.g., integer programming), graph theory, automata and language theory, logic, and games and puzzles.

Unless P=NP, NP-hard problems cannot be solved by polynomial-time algorithms. However, for some problems, a clever use of symmetry may give rise to an algorithm that is, in general, still exponential, but may be feasible for many real-world inputs. In this Special Issue of Symmetry, we solicit papers that make use of symmetry (or asymmetry) to tackle NP-hard problems. Problems may be solved exactly (using strategies such as complete search, branch-and-bound, and dynamic programming) or approximately (using a variety of approaches, including, but not limited to, simulated annealing, evolutionary algorithms, and particle swarm optimization) and may come from a broad spectrum of domains.

Dr. Luka Fürst
Dr. Uroš Čibej
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • symmetry
  • NP-hard problems
  • NP-complete problems
  • combinatorial optimization
  • graph theory
  • complete search
  • dynamic programming
  • metaheuristics
  • evolutionary algorithms

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Published Papers (1 paper)

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Research

40 pages, 1068 KiB  
Article
Tableau with Holes: Clarifying NP-Completeness
by Edgar Graham Daylight
Symmetry 2025, 17(5), 677; https://doi.org/10.3390/sym17050677 (registering DOI) - 29 Apr 2025
Abstract
In the context of defining NP-completeness, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula ψ, defined as [...] Read more.
In the context of defining NP-completeness, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula ψ, defined as ψ=ψcell∧ψrest. The component ψcell enforces the constraint that each cell in the tableau contains exactly one symbol, while ψrest incorporates constraints governing the step-by-step behavior of N on w. Intuitively, ψrest appears to pose a much greater challenge for satisfiability. This raises the question of whether the distinction between ψcell being a 3cnf formula, rather than a cheap 2cnf formula, actually matters. We show that if, hypothetically, ψrest can be succinctly represented as a Horn formula, then satisfying ψ can be achieved efficiently in Kf(n,k) steps, where N operates within O(nk) steps and both k and K are constants. Asymptotically, f(n,k)≈n23k. Our method has the potential for iterative application. Technically, we trim ψcell down to a 2cnf–Horn formula, whose satisfiability allows for empty cells, or “holes,” in the tableau. This modified tableau represents exponentially many paths of N on w, rather than a single accepting path p. While a tableau with holes conceptualizes the satisfiability of ψtrim—a trimmed-down version of ψ—it does not directly address the satisfiability of ψ. Therefore, we introduce an external user who efficiently employs backtracking to fill in specific holes, ultimately verifying the satisfiability of the original ψ. Full article
(This article belongs to the Special Issue Symmetry in Solving NP-Hard Problems)
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