scholarly journals Generalized Max-Cut and the Approximation Ratio

2021 ◽  
Author(s):  
Junsi Zhang
Keyword(s):  
Approximation Ratio ◽  
Quadratic Program ◽  
Input Graph ◽  
Real Numbers ◽  
Rounding Algorithm

In this thesis, we formulate a new problem based on Max-Cut called Generalized Max-Cut. This problem requires a graph as input and two real numbers (a, b) where a > 0 and −a < b < a and outputs a number. The restriction on the pair (a, b) is to avoid trivializing the problem. We formulate a quadratic program for Generalized Max-Cut and relax it to a semi-definite program. Most algorithms in this thesis will require solving this semi-definite program. The main algorithm in this thesis is the 2-Dimensional Rounding algorithm, designed by Avidor and Zwick, with the restriction that the semi-definite program of the input graph must have 2-Dimensional solutions. This algorithm uses a factor of randomness, β ∈ [0, 1], that is dependent on the integer input to Generalized Max-Cut. We improve the performance of this algorithm by numerically finding better β.

scholarly journals Generalized Max-Cut and the Approximation Ratio

2021 ◽  
Author(s):  
Junsi Zhang
Keyword(s):  
Approximation Ratio ◽  
Quadratic Program ◽  
Input Graph ◽  
Real Numbers ◽  
Rounding Algorithm

In this thesis, we formulate a new problem based on Max-Cut called Generalized Max-Cut. This problem requires a graph as input and two real numbers (a, b) where a > 0 and −a < b < a and outputs a number. The restriction on the pair (a, b) is to avoid trivializing the problem. We formulate a quadratic program for Generalized Max-Cut and relax it to a semi-definite program. Most algorithms in this thesis will require solving this semi-definite program. The main algorithm in this thesis is the 2-Dimensional Rounding algorithm, designed by Avidor and Zwick, with the restriction that the semi-definite program of the input graph must have 2-Dimensional solutions. This algorithm uses a factor of randomness, β ∈ [0, 1], that is dependent on the integer input to Generalized Max-Cut. We improve the performance of this algorithm by numerically finding better β.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Dynamics of a family of orbitally continuous mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3507-3517
Author(s):  
Abhijit Pant ◽  
R.P. Pant ◽  
Kuldeep Prakash
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Real Numbers ◽  
Continuous Mappings ◽  
Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

Hermite-Hadamard and simpson-like type inequalities for differentiable p-quasi-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 5945-5953 ◽  
Author(s):  
İmdat İsçan ◽  
Sercan Turhan ◽  
Selahattin Maden
Keyword(s):  
Convex Functions ◽  
Real Numbers ◽  
Absolute Value ◽  
Special Means ◽  
Quasi Convexity

In this paper, we give a new concept which is a generalization of the concepts quasi-convexity and harmonically quasi-convexity and establish a new identity. A consequence of the identity is that we obtain some new general inequalities containing all of the Hermite-Hadamard and Simpson-like type for functions whose derivatives in absolute value at certain power are p-quasi-convex. Some applications to special means of real numbers are also given.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

Approximation Algorithms for Submodular Data Summarization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-31
Author(s):  
Kai Han ◽  
Shuang Cui ◽  
Tianshuai Zhu ◽  
Enpei Zhang ◽  
Benwei Wu ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Fundamental Problem ◽  
Randomized Algorithm ◽  
Deterministic Algorithm ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Performance Bounds ◽  
Data Summarization ◽  
Submodular Optimization ◽  
Knapsack Constraint

Data summarization, i.e., selecting representative subsets of manageable size out of massive data, is often modeled as a submodular optimization problem. Although there exist extensive algorithms for submodular optimization, many of them incur large computational overheads and hence are not suitable for mining big data. In this work, we consider the fundamental problem of (non-monotone) submodular function maximization with a knapsack constraint, and propose simple yet effective and efficient algorithms for it. Specifically, we propose a deterministic algorithm with approximation ratio 6 and a randomized algorithm with approximation ratio 4, and show that both of them can be accelerated to achieve nearly linear running time at the cost of weakening the approximation ratio by an additive factor of ε. We then consider a more restrictive setting without full access to the whole dataset, and propose streaming algorithms with approximation ratios of 8+ε and 6+ε that make one pass and two passes over the data stream, respectively. As a by-product, we also propose a two-pass streaming algorithm with an approximation ratio of 2+ε when the considered submodular function is monotone. To the best of our knowledge, our algorithms achieve the best performance bounds compared to the state-of-the-art approximation algorithms with efficient implementation for the same problem. Finally, we evaluate our algorithms in two concrete submodular data summarization applications for revenue maximization in social networks and image summarization, and the empirical results show that our algorithms outperform the existing ones in terms of both effectiveness and efficiency.

scholarly journals On monotonicity and search strategies in face-based copositivity detection algorithms

2021 ◽  
Author(s):  
B. G.-Tóth ◽  
E. M. T. Hendrix ◽  
L. G. Casado
Keyword(s):  
Research Question ◽  
Quadratic Function ◽  
Mixed Integer ◽  
Quadratic Program ◽  
Matrix Methods ◽  
Detection Algorithms ◽  
Standard Simplex ◽  
Mathematical Properties ◽  
The Face ◽  
Spatial Branch And Bound

AbstractOver the last decades, algorithms have been developed for checking copositivity of a matrix. Methods are based on several principles, such as spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the relative interior minima of the standard quadratic program (StQP) and monotonicity. We derive several theoretical properties related to convexity and monotonicity of the standard quadratic function over faces of the standard simplex. We illustrate with numerical instances up to 28 dimensions the use of monotonicity in face-based algorithms. The question is what traversal through the face graph of the standard simplex is more appropriate for which matrix instance; top down or bottom up approaches. This depends on the level of the face graph where the minimum of StQP can be found, which is related to the density of the so-called convexity graph.

Optimization studies of low-density polyethylene process: effect of different interval numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ashraf Azmi ◽  
Suhairi Abdul Sata ◽  
Fakhrony Sholahudin Rohman ◽  
Norashid Aziz
Keyword(s):  
Dynamic Optimization ◽  
Monomer Conversion ◽  
Polymerization Process ◽  
Low Density Polyethylene ◽  
Quadratic Program ◽  
Flow Index ◽  
Low Density ◽  
Orthogonal Collocation ◽  
Interval Numbers ◽  
Optimization Study

AbstractThe highly exothermic nature of the low-density polyethylene (LDPE) polymerization process and the heating-cooling prerequisite in tubular reactor can lead to various problems particularly safety and economic. These issues complicate the monomer conversion maximization approaches. Consequently, the dynamic optimization study to obtain maximum conversion of the LDPE is carried out. A mathematical model has been developed and validated using industrial data. In the dynamic optimization study, maximum monomer conversion (XM) is considered as the objective function, whereas the constraint and bound consists of maximum reaction temperature and product melt flow index (MFI). The orthogonal collocation (OC) on finite elements is used to convert the original optimization problems into Nonlinear Programming (NLP) problems, which are then solved using sequential quadratic program (SQP) methods. The result shows that five interval numbers produce better optimization result compared to one and two intervals.

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