A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,for a connected graph G of n vertices and m edges, whereand ﹛x﹜ denotes the smallest integer ≧ x.We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

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On subgraphs of C2k-free graphs and a problem of Kühn and Osthus

Combinatorics Probability Computing ◽

10.1017/s0963548319000452 ◽

2020 ◽

Vol 29 (3) ◽

pp. 436-454

Author(s):

Dániel Grósz ◽

Abhishek Methuku ◽

Casey Tompkins

Keyword(s):

Short Proof ◽

Uniform Hypergraph ◽

Free Graph ◽

Bipartite Subgraph ◽

Image Position

AbstractLet c denote the largest constant such that every C6-free graph G contains a bipartite and C4-free subgraph having a fraction c of edges of G. Győri, Kensell and Tompkins showed that 3/8 ⩽ c ⩽ 2/5. We prove that c = 38. More generally, we show that for any ε > 0, and any integer k ⩾ 2, there is a C2k-free graph $G'$ which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction $$\Bigl(1-\frac{1}{2^{2k-2}}\Bigr)\frac{2}{2k-1}(1+\varepsilon)$$ of the edges of $G'$ . There also exists a C2k-free graph $G''$ which does not contain a bipartite and C4-free subgraph with more than a fraction $$\Bigl(1-\frac{1}{2^{k-1}}\Bigr)\frac{1}{k-1}(1+\varepsilon)$$ of the edges of $G''$ .One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős. For any ε > 0, and any integers a, b, k ⩾ 2, there exists an a-uniform hypergraph H of girth greater than k which does not contain any b-colourable subhypergraph with more than a fraction $$\Bigl(1-\frac{1}{b^{a-1}}\Bigr)(1+\varepsilon)$$ of the hyperedges of H. We also prove further generalizations of this theorem.In addition, we give a new and very short proof of a result of Kühn and Osthus, which states that every bipartite C2k-free graph G contains a C4-free subgraph with at least a fraction 1/(k−1) of the edges of G. We also answer a question of Kühn and Osthus about C2k-free graphs obtained by pasting together C2l’s (with k >l ⩾ 3).

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A Note on the Ohya-Masuda Quantum Algorithm

Open Systems & Information Dynamics ◽

10.1023/a:1015644508998 ◽

2002 ◽

Vol 09 (02) ◽

pp. 115-123

Author(s):

Miroljub Dugić

Keyword(s):

Density Matrix ◽

Complexity Theory ◽

Quantum Measurement ◽

Pure State ◽

Quantum Algorithm ◽

Satisfiability Problem ◽

Second Step ◽

Coherent Superposition ◽

Complete Problem ◽

Np Complete

We analyze the Ohya-Masuda quantum algorithm that solves the so-called “satisfiability” problem, which is an NP-complete problem of the complexity theory. We distinguish three steps in the algorithm, and analyze the second step, in which a coherent superposition of states (a “pure” state) transforms into an “incoherent” mixture presented by a density matrix. We show that, if “nonideal” (in analogy with “nonideal” quantum measurement), this transformation can make the algorithm to fail in some cases. On this basis we give some general notions on the physical implementation of the Ohya-Masuda algorithm.

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AN ANNEALED CHAOTIC MAXIMUM NEURAL NETWORK FOR BIPARTITE SUBGRAPH PROBLEM

International Journal of Neural Systems ◽

10.1142/s0129065704001917 ◽

2004 ◽

Vol 14 (02) ◽

pp. 107-116 ◽

Cited By ~ 1

Author(s):

JIAHAI WANG ◽

ZHENG TANG ◽

RONGLONG WANG

Keyword(s):

Neural Network ◽

Parallel Algorithm ◽

Chaotic Dynamics ◽

Neural Model ◽

Steepest Descent Method ◽

Stable Equilibrium Point ◽

Local Minima ◽

Complete Problem ◽

Bipartite Subgraph ◽

Np Complete

In this paper, based on maximum neural network, we propose a new parallel algorithm that can help the maximum neural network escape from local minima by including a transient chaotic neurodynamics for bipartite subgraph problem. The goal of the bipartite subgraph problem, which is an NP–complete problem, is to remove the minimum number of edges in a given graph such that the remaining graph is a bipartite graph. Lee et al. presented a parallel algorithm using the maximum neural model (winner-take-all neuron model) for this NP–complete problem. The maximum neural model always guarantees a valid solution and greatly reduces the search space without a burden on the parameter-tuning. However, the model has a tendency to converge to a local minimum easily because it is based on the steepest descent method. By adding a negative self-feedback to the maximum neural network, we proposed a new parallel algorithm that introduces richer and more flexible chaotic dynamics and can prevent the network from getting stuck at local minima. After the chaotic dynamics vanishes, the proposed algorithm is then fundamentally reined by the gradient descent dynamics and usually converges to a stable equilibrium point. The proposed algorithm has the advantages of both the maximum neural network and the chaotic neurodynamics. A large number of instances have been simulated to verify the proposed algorithm. The simulation results show that our algorithm finds the optimum or near-optimum solution for the bipartite subgraph problem superior to that of the best existing parallel algorithms.

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On Minimum Representations of Matched Formulas

Journal of Artificial Intelligence Research ◽

10.1613/jair.4517 ◽

2014 ◽

Vol 51 ◽

pp. 707-723

Author(s):

O. Cepek ◽

S. Gursky ◽

P. Kucera

Keyword(s):

Normal Form ◽

Boolean Function ◽

Minimization Problem ◽

Conjunctive Normal Form ◽

Boolean Formula ◽

Polynomial Hierarchy ◽

Partial Assignment ◽

Complete Problem ◽

Boolean Minimization ◽

Np Complete

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

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Geodetic Number of Powers of Cycles

Symmetry ◽

10.3390/sym10110592 ◽

2018 ◽

Vol 10 (11) ◽

pp. 592

Author(s):

Mohammad Abudayah ◽

Omar Alomari ◽

Hassan Ezeh

Keyword(s):

Distributed Computing ◽

Computer Networks ◽

Important Class ◽

Graph Invariant ◽

Complete Problem ◽

Geodetic Number ◽

Geodetic Set ◽

Np Complete

The geodetic number of a graph is an important graph invariant. In 2002, Atici showed the geodetic set determination of a graph is an NP-Complete problem. In this paper, we compute the geodetic set and geodetic number of an important class of graphs called the k-th power of a cycle. This class of graphs has various applications in Computer Networks design and Distributed computing. The k-th power of a cycle is the graph that has the same set of vertices as the cycle and two different vertices in the k-th power of this cycle are adjacent if the distance between them is at most k.

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Hamilton Connectivity of Convex Polytopes with Applications to Their Detour Index

Complexity ◽

10.1155/2021/6684784 ◽

2021 ◽

Vol 2021 ◽

pp. 1-23

Author(s):

Sakander Hayat ◽

Asad Khan ◽

Suliman Khan ◽

Jia-Bao Liu

Keyword(s):

Computer Science ◽

Connected Graph ◽

Electrical Engineering ◽

Convex Polytopes ◽

Hamiltonian Path ◽

Infinite Family ◽

Complete Problem ◽

Analytical Formulas ◽

Np Complete ◽

Hamilton Connected

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

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A Public-Key Cryptosystem Based on the Matrix Cover NP-Complete Problem

Advances in Cryptology ◽

10.1007/978-1-4757-0602-4_3 ◽

1983 ◽

pp. 21-37

Author(s):

Ravi Janardan ◽

K. B. Lakshmanan

Keyword(s):

Public Key ◽

Public Key Cryptosystem ◽

Complete Problem ◽

The Matrix ◽

Np Complete

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COMPUTING THE RUPTURE DEGREE IN COMPOSITE GRAPHS

International Journal of Foundations of Computer Science ◽

10.1142/s012905411000726x ◽

2010 ◽

Vol 21 (03) ◽

pp. 311-319 ◽

Cited By ~ 2

Author(s):

AYSUN AYTAC ◽

ZEYNEP NIHAN ODABAS

Keyword(s):

Complete Graph ◽

Connected Graph ◽

The Other ◽

Trade Off ◽

Number Of Components ◽

Np Complete ◽

Work Done ◽

Better Than

The rupture degree of an incomplete connected graph G is defined by [Formula: see text] where w(G - S) is the number of components of G - S and m(G - S) is the order of a largest component of G - S. For the complete graph Kn, rupture degree is defined as 1 - n. This parameter can be used to measure the vulnerability of a graph. Rupture degree can reflect the vulnerability of graphs better than or independent of the other parameters. To some extent, it represents a trade-off between the amount of work done to damage the network and how badly the network is damaged. Computing the rupture degree of a graph is NP-complete. In this paper, we give formulas for the rupture degree of composition of some special graphs and we consider the relationships between the rupture degree and other vulnerability parameters.

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