Complexity Assessments for Decidable Fragments of Set Theory. I: A Taxonomy for the Boolean Case*

2021 ◽  
Vol 181 (1) ◽  
pp. 37-69
Author(s):  
Domenico Cantone ◽  
Andrea De Domenico ◽  
Pietro Maugeri ◽  
Eugenio G. Omodeo
Keyword(s):  
Set Theory ◽  
Polynomial Time ◽  
Linear Time ◽  
Von Neumann ◽  
Disjoint Sets ◽  
Np Complete ◽  
Multi Level ◽  
Proof Verification ◽  
Automated Proof ◽  
Decision Algorithms

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪ ∩ \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates ‘• = Ø’ and ‘Disj(•, •)’, meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘• ≠ Ø and ‘¬Disj(•, •)’. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

James Cummings and Ernest Schimmerling, editors. Lecture Note Series of the London Mathematical Society, vol. 406. Cambridge University Press, New York, xi + 419 pp. - Paul B. Larson, Peter Lumsdaine, and Yimu Yin. An introduction to Pmax forcing. pp. 5–23. - Simon Thomas and Scott Schneider. Countable Borel equivalence relations. pp. 25–62. - Ilijas Farah and Eric Wofsey. Set theory and operator algebras. pp. 63–119. - Justin Moore and David Milovich. A tutorial on set mapping reflection. pp. 121–144. - Vladimir G. Pestov and Aleksandra Kwiatkowska. An introduction to hyperlinear and sofic groups. pp. 145–185. - Itay Neeman and Spencer Unger. Aronszajn trees and the SCH. pp. 187–206. - Todd Eisworth, Justin Tatch Moore, and David Milovich. Iterated forcing and the Continuum Hypothesis. pp. 207–244. - Moti Gitik and Spencer Unger. Short extender forcing. pp. 245–263. - Alexander S. Kechris and Robin D. Tucker-Drob. The complexity of classification problems in ergodic theory. pp. 265–299. - Menachem Magidor and Chris Lambie-Hanson. On the strengths and weaknesses of weak squares. pp. 301–330. - Boban Veličković and Giorgio Venturi. Proper forcing remastered. pp. 331–362. - Asger ToÖrnquist and Martino Lupini. Set theory and von Neumann algebras. pp. 363–396. - W. Hugh Woodin, Jacob Davis, and Daniel RodrÍguez. The HOD dichotomy. pp. 397–419.

2014 ◽  
Vol 20 (1) ◽  
pp. 94-97
Author(s):  
Natasha Dobrinen
Keyword(s):  
New York ◽  
Set Theory ◽  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
London Mathematical Society ◽  
Equivalence Relations ◽  
Classification Problems ◽  
Von Neumann ◽  
Borel Equivalence Relations

Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdul Majeed ◽  
Mohsin Kamran ◽  
Noreen Asghar
Keyword(s):  
Caputo Derivative ◽  
Numerical Solutions ◽  
Linear Time ◽  
Initial Boundary ◽  
Telegraph Equation ◽  
Test Problems ◽  
Von Neumann ◽  
B Spline ◽  
Nonlinear Part ◽  
Spline Basis

Abstract This article focusses on the implementation of cubic B-spline approach to investigate numerical solutions of inhomogeneous time fractional nonlinear telegraph equation using Caputo derivative. L1 formula is used to discretize the Caputo derivative, while B-spline basis functions are used to interpolate the spatial derivative. For nonlinear part, the existing linearization formula is applied after generalizing it for all positive integers. The algorithm for the simulation is also presented. The efficiency of the proposed scheme is examined on three test problems with different initial boundary conditions. The influence of parameter α on the solution profile for different values is demonstrated graphically and numerically. Moreover, the convergence of the proposed scheme is analyzed and the scheme is proved to be unconditionally stable by von Neumann Fourier formula. To quantify the accuracy of the proposed scheme, error norms are computed and was found to be effective and efficient for nonlinear fractional partial differential equations (FPDEs).

Αλγόριθμοι σχεδίασης δικτύων τοπολογίας δέντρου με ποικίλους περιορισμούς χωρητικότητας

2013 ◽  
Author(s):  
Χρίστος Παππάς
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Branch And Cut ◽  
Capacitated Minimum Spanning Tree ◽  
Np Complete ◽  
Multi Level

Κατά τη σχεδίαση κεντρικοποιημένων δικτύων συχνά προκύπτει η ανάγκη για τη εύρεση δέντρων ελάχιστου κόστους. Ένα πρόβλημα που έχει μελετηθεί εκτενώς στη βιβλιογραφία είναι το πρόβλημα εύρεσης Ελάχιστου Δέντρου Επικάλυψης με Περιορισμό Χωρητικότητας (Capacitated Minimum Spanning Tree ή CMST). Στο πρόβλημα CMST στόχος είναι να σχεδιαστεί δίκτυο τοπολογίας δέντρου ελάχιστου κόστους, το οποίο να εξυπηρετεί την προώθηση της κίνησης που παράγει ένα σύνολο τερματικών κόμβων προς ένα κεντρικό κόμβο, με τον περιορισμό η συνολική κίνηση σε οποιαδήποτε ζεύξη να μην υπερβαίνει μία ενιαία προκαθορισμένη τιμή-χωρητικότητα. Ωστόσο, κατά τη σχεδίαση πραγματικών δικτύων συχνά επιλέγεται η εγκατάσταση ζεύξεων διαφορετικών χωρητικοτήτων. Γενικεύοντας το πρόβλημα CMST, έτσι ώστε να υπάρχει η δυνατότητα επιλογής από μία γκάμα διαφορετικών τύπων ζεύξεων, οι οποίοι διαφοροποιούνται μεταξύ τους ως προς τη χωρητικότητα αλλά και το κόστος, οδηγούμαστε στο πρόβλημα εύρεσης Ελάχιστου Δέντρου Επικάλυψης με Περιορισμούς Χωρητικότητας Πολλαπλών Επιπέδων (Multi-Level Capacitated Minimum Spanning Tree ή MLCMST). Η έρευνα γύρω από το πρόβλημα MLCMST είχε μέχρι σήμερα περιοριστεί σε μία συγκεκριμένη κλάση στιγμιότυπων όπου η παραγόμενη από κάθε κόμβο κίνηση είναι μοναδιαία αλλά και η μέγιστη επιτρεπτή χωρητικότητα λαμβάνει χαμηλές τιμές.Η παρούσα διδακτορική διατριβή έχει ως αντικείμενο τη μελέτη του προβλήματος MLCMST και την ανάδειξη αλγορίθμων που να αντιμετωπίζουν ένα ευρύ φάσμα στιγμιότυπων του. Αρχικά εξετάζεται η δυνατότητα επίλυσης προβλημάτων με τεχνικές μεικτού ακέραιου γραμμικού προγραμματισμού. Η πλήρης επίλυση των γραμμικών μοντέλων μέσα σε λογικά χρονικά πλαίσια αποδεικνύεται δυνατή μόνο για στιγμιότυπα περιορισμένου μεγέθους. Σε μεγαλύτερα προβλήματα, και δεδομένου ότι το πρόβλημα MLCMST ανήκει στη κλάση των NP-complete προβλημάτων, η προσπάθεια αναπόφευκτα μετατοπίζεται στην εύρεση ποιοτικών, αλλά όχι απαραίτητα βέλτιστων λύσεων. Βασιζόμενοι σε προηγούμενες εργασίες στον τομέα παρουσιάζουμε ευρετικούς αλγορίθμους αναβαθμίσεων, με στόχο την αντιμετώπιση στιγμιότυπων ποικίλων χαρακτηριστικών και μεγεθών. Εν συνεχεία, οι προτεινόμενοι αλγόριθμοι αναβαθμίσεων ενσωματώνονται σε αλγόριθμο Διακλάδωσης και Αποκοπής (Branch and Cut) δημοφιλούς πακέτου βελτιστοποίησης. Τέλος, εξετάζεται η εφαρμογή εξελικτικών αλγορίθμων στο πρόβλημα. Σε αυτή την προσέγγιση οι προτεινόμενοι αλγόριθμοι αναβαθμίσεων αξιοποιούνται ως τροφοδότες λύσεων καλής ποιότητας κατά την αρχικοποίηση των πληθυσμών.

scholarly journals MPF Problem over Modified Medial Semigroup Is NP-Complete

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 571 ◽  
Author(s):  
Eligijus Sakalauskas ◽  
Aleksejus Mihalkovich
Keyword(s):  
Power Function ◽  
Polynomial Time ◽  
Cryptographic Protocols ◽  
Binary Field ◽  
Dual Interpretation ◽  
Np Complete ◽  
Matrix Power ◽  
Necessary Constraints ◽  
One Way Function ◽  
Polynomial Time Reduction

This paper is a continuation of our previous publication of enhanced matrix power function (MPF) as a conjectured one-way function. We are considering a problem introduced in our previous paper and prove that tis problem is NP-Complete. The proof is based on the dual interpretation of well known multivariate quadratic (MQ) problem defined over the binary field as a system of MQ equations, and as a general satisfiability (GSAT) problem. Due to this interpretation the necessary constraints to MPF function for cryptographic protocols construction can be added to initial GSAT problem. Then it is proved that obtained GSAT problem is NP-Complete using Schaefer dichotomy theorem. Referencing to this result, GSAT problem by polynomial-time reduction is reduced to the sub-problem of enhanced MPF, hence the latter is NP-Complete as well.

scholarly journals On paths, trails and closed trails in edge-colored graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Laurent Gourvès ◽  
Adria Lyra ◽  
Carlos A. Martinhon ◽  
Jérôme Monnot
Keyword(s):  
Polynomial Time ◽  
Optimal Solution ◽  
Colored Graph ◽  
Edge Disjoint ◽  
Colored Graphs ◽  
International Audience ◽  
Np Complete ◽  
T Path ◽  
Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

scholarly journals The Complexity Landscape of Resource-Constrained Scheduling

2020 ◽  
Author(s):  
Robert Ganian ◽  
Thekla Hamm ◽  
Guillaume Mescoff
Keyword(s):  
Artificial Intelligence ◽  
Polynomial Time ◽  
Project Scheduling ◽  
Scheduling Problem ◽  
Comprehensive Understanding ◽  
Resource Constrained ◽  
Special Cases ◽  
Project Scheduling Problem ◽  
Np Complete ◽  
New Algorithms

The Resource-Constrained Project Scheduling Problem (RCPSP) and its extension via activity modes (MRCPSP) are well-established scheduling frameworks that have found numerous applications in a broad range of settings related to artificial intelligence. Unsurprisingly, the problem of finding a suitable schedule in these frameworks is known to be NP-complete; however, aside from a few results for special cases, we have lacked an in-depth and comprehensive understanding of the complexity of the problems from the viewpoint of natural restrictions of the considered instances. In the first part of our paper, we develop new algorithms and give hardness-proofs in order to obtain a detailed complexity map of (M)RCPSP that settles the complexity of all 1024 considered variants of the problem defined in terms of explicit restrictions of natural parameters of instances. In the second part, we turn to implicit structural restrictions defined in terms of the complexity of interactions between individual activities. In particular, we show that if the treewidth of a graph which captures such interactions is bounded by a constant, then we can solve MRCPSP in polynomial time.

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