THE HOSOYA INDEX OF GRAPHS FORMED BY A FRACTAL GRAPH

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950135 ◽  
Author(s):  
JIA-BAO LIU ◽  
JING ZHAO ◽  
JIE MIN ◽  
JINDE CAO
Keyword(s):  
Computational Complexity ◽  
Hosoya Index ◽  
Initial Graph ◽  
Np Complete

The computational complexity of the Hosoya index of a given graph is NP-Complete. Let [Formula: see text] be the graph constructed from [Formula: see text] by a triangle instead of all vertices of the initial graph [Formula: see text]. In this paper, we characterize the Hosoya index of the graph [Formula: see text]. To our surprise, it shows that the Hosoya index of [Formula: see text] is thoroughly given by the order and degrees of all the vertices of the initial graph [Formula: see text].

scholarly journals Popular Matching in Roommates Setting Is NP-hard

2021 ◽  
Vol 13 (2) ◽  
pp. 1-20
Author(s):  
Sushmita Gupta ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
Meirav Zehavi
Keyword(s):  
Computational Complexity ◽  
Np Hard ◽  
Np Complete ◽  
Open Question ◽  
Popular Matching

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

scholarly journals Stable roommates with narcissistic, single-peaked, and single-crossing preferences

2020 ◽  
Vol 34 (2) ◽  
Author(s):  
Robert Bredereck ◽  
Jiehua Chen ◽  
Ugo Paavo Finnendahl ◽  
Rolf Niedermeier
Keyword(s):  
Computational Complexity ◽  
Structural Property ◽  
Incomplete Preferences ◽  
Two Agents ◽  
Single Crossing ◽  
Np Complete

Abstract The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity—Stable Roommates remains NP-complete.

scholarly journals Computing the Interleaving Distance is NP-Hard

2019 ◽  
Vol 20 (5) ◽  
pp. 1237-1271 ◽  
Author(s):  
Håvard Bakke Bjerkevik ◽  
Magnus Bakke Botnan ◽  
Michael Kerber
Keyword(s):  
Computational Complexity ◽  
Complete Characterization ◽  
Np Hard ◽  
Slight Improvement ◽  
Approximation Factor ◽  
Distance Computation ◽  
Persistence Modules ◽  
Np Complete ◽  
Interleaving Distance

Abstract We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are 1-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement in the above reduction, showing that also the approximation of the interleaving distance is NP-hard for any approximation factor smaller than 3. Additionally, we obtain corresponding hardness results for the case that the modules are indecomposable, and in the setting of one-sided stability. Furthermore, we show that checking for injections (resp. surjections) between persistence modules is NP-hard. In conjunction with earlier results from computational algebra this gives a complete characterization of the computational complexity of one-sided stability. Lastly, we show that it is in general NP-hard to approximate distances induced by noise systems within a factor of 2.

scholarly journals LOGICS AND ALGEBRAS FOR MULTIPLE PLAYERS

2010 ◽  
Vol 3 (3) ◽  
pp. 485-519 ◽  
Author(s):  
LOES OLDE LOOHUIS ◽  
YDE VENEMA
Keyword(s):  
Computational Complexity ◽  
Winning Strategy ◽  
Boolean Algebras ◽  
Game Theoretic ◽  
Np Complete ◽  
Game Theoretic Semantics ◽  
Stone’S Theorem

We study a generalization of the standard syntax and game-theoretic semantics of logic, which is based on a duality between two players, to a multiplayer setting. We define propositional and modal languages of multiplayer formulas, and provide them with a semantics involving a multiplayer game. Our focus is on the notion of equivalence between two formulas, which is defined by saying that two formulas are equivalent if under each valuation, the set of players with a winning strategy is the same in the two respective associated games. We provide a derivation system which enumerates the pairs of equivalent formulas, both in the propositional case and in the modal case. Our approach is algebraic: We introduce multiplayer algebras as the analogue of Boolean algebras, and show, as the corresponding analog to Stone’s theorem, that these abstract multiplayer algebras can be represented as concrete ones which capture the game-theoretic semantics. For the modal case we prove a similar result. We also address the computational complexity of the problem whether two given multiplayer formulas are equivalent. In the propositional case, we show that this problem is co-NP-complete, whereas in the modal case, it is PSPACE-hard.

scholarly journals On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics

2008 ◽  
Vol 14 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Ian Pratt-Hartmann
Keyword(s):  
Computational Complexity ◽  
Related Problem ◽  
Satisfiability Problem ◽  
Propositional Satisfiability ◽  
Proof Systems ◽  
Transitive Verbs ◽  
Np Complete

AbstractThe numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic.

scholarly journals The Complexity of Checking Consistency of Pedigree Information and Related Problems

2003 ◽  
Vol 10 (17) ◽  
Author(s):  
Luca Aceto ◽  
Jens Alsted Hansen ◽  
Anna Ingólfsdóttir ◽  
Jacob Johnsen ◽  
John Knudsen
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Input Data ◽  
Single Gene ◽  
Pedigree Information ◽  
Consistency Checking ◽  
Computational Problem ◽  
Np Complete ◽  
Mendelian Laws

Consistency checking is a fundamental computational problem in genetics. Given a pedigree and information on the genotypes (of some) of the individuals in it, the aim of consistency checking is to determine whether these data are consistent with the classic Mendelian laws of inheritance. This problem arose originally from the geneticists' need to filter their input data from erroneous information, and is well motivated from both a biological and a sociological viewpoint. This paper shows that consistency checking is NP-complete, even with focus on a single gene and in the presence of three alleles. Several other results on the computational complexity of problems from genetics that are related to consistency checking are also offered. In particular, it is shown that checking the consistency of pedigrees over two alleles, and of pedigrees without loops, can be done in polynomial time.

scholarly journals Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

2021 ◽  
Author(s):  
Thomas Bläsius ◽  
Philipp Fischbeck ◽  
Tobias Friedrich ◽  
Maximilian Katzmann
Keyword(s):  
Computational Complexity ◽  
Structural Properties ◽  
Random Graphs ◽  
Polynomial Time ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Vertex Cover ◽  
Optimal Solution ◽  
Np Complete

AbstractThe computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.

scholarly journals The Computational Complexity of Structure-Based Causality

2017 ◽  
Vol 58 ◽  
pp. 431-451 ◽  
Author(s):  
Gadi Aleksandrowicz ◽  
Hana Chockler ◽  
Joseph Y. Halpern ◽  
Alexander Ivrii
Keyword(s):  
Computational Complexity ◽  
Complexity Classes ◽  
Final Version ◽  
New Family ◽  
Original Definition ◽  
Np Complete ◽  
Definition Of

Halpern and Pearl introduced a definition of actual causality; Eiter and Lukasiewicz showed that computing whether X = x is a cause of Y = y is NP-complete in binary models (where all variables can take on only two values) and Σ^P_2 -complete in general models. In the final version of their paper, Halpern and Pearl slightly modified the definition of actual cause, in order to deal with problems pointed out by Hopkins and Pearl. As we show, this modification has a nontrivial impact on the complexity of computing whether {X} = {x} is a cause of Y = y. To characterize the complexity, a new family D_k^P , k = 1, 2, 3, . . ., of complexity classes is introduced, which generalises the class DP introduced by Papadimitriou and Yannakakis (DP is just D_1^P). We show that the complexity of computing causality under the updated definition is D_2^P -complete. Chockler and Halpern extended the definition of causality by introducing notions of responsibility and blame, and characterized the complexity of determining the degree of responsibility and blame using the original definition of causality. Here, we completely characterize the complexity using the updated definition of causality. In contrast to the results on causality, we show that moving to the updated definition does not result in a difference in the complexity of computing responsibility and blame.

scholarly journals Low complexity algorithms in knot theory

2019 ◽  
Vol 29 (02) ◽  
pp. 245-262
Author(s):  
Olga Kharlampovich ◽  
Alina Vdovina
Keyword(s):  
Computational Complexity ◽  
Time Complexity ◽  
Linear Time ◽  
Circuit Complexity ◽  
Equivalence Problem ◽  
Low Complexity ◽  
Combinatorial Structure ◽  
Complexity Classes ◽  
Np Complete ◽  
Almost All

Agol, Haas and Thurston showed that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. This shows that (unless P[Formula: see text]NP) the genus problem has high computational complexity even for knots in a 3-manifold. We initiate the study of classes of knots where the genus problem and even the equivalence problem have very low computational complexity. We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace[Formula: see text]. Alternating knots with some additional combinatorial structure will be referred to as standard. As expected, almost all alternating knots of a given genus are standard. We show that the genus problem for these knots belongs to [Formula: see text] circuit complexity class. We also show, that the equivalence problem for such knots with [Formula: see text] crossings has time complexity [Formula: see text] and is in Logspace[Formula: see text] and [Formula: see text] complexity classes.

scholarly journals Minimal and Hyper-Minimal Biautomata

2016 ◽  
Vol 27 (02) ◽  
pp. 161-185
Author(s):  
Markus Holzer ◽  
Sebastian Jakobi
Keyword(s):  
Computational Complexity ◽  
Minimization Problem ◽  
Finite Automata ◽  
Deterministic Finite Automata ◽  
Np Complete ◽  
Carry Over

We compare deterministic finite automata (DFAs) and biautomata under the following two aspects: structural similarities between minimal and hyper-minimal automata, and computational complexity of the minimization and hyper-minimization problem. Concerning classical minimality, the known results such as isomorphism between minimal DFAs, and NL-completeness of the DFA minimization problem carry over to the biautomaton case. But surprisingly this is not the case for hyper-minimization: the similarity between almost-equivalent hyper-minimal biautomata is not as strong as it is between almost-equivalent hyper-minimal DFAs. Moreover, while hyper-minimization is NL-complete for DFAs, we prove that this problem turns out to be computationally intractable, i.e., NP-complete, for biautomata.

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