scholarly journals Abstract Tropical Linear Programming

10.37236/7718 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Georg Loho
Keyword(s):  
Linear Programming ◽  
Upper Bound ◽  
Axiomatic Approach ◽  
Bipartite Graphs ◽  
Complexity Measure ◽  
Integer Vector ◽  
Feasible Point ◽  
Additional Sign

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. We obtain an algorithm based on an axiomatic approach to this generalization.  It builds on the introduction of signed tropical matroids based on the polyhedral properties of triangulations of the product of two simplices and the combinatorics of the associated set of bipartite graphs with an additional sign information. Finally, we establish an upper bound for our feasibility algorithm applied to a system of min-plus-inequalities in terms of the secondary fan of a product of two simplices. The appropriate complexity measure is a shortest integer vector in a cone of the secondary fan associated to the system.

scholarly journals On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1778
Author(s):  
Fangyun Tao ◽  
Ting Jin ◽  
Yiyou Tu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Equitable Partition ◽  
Special Cases ◽  
Vertex Set ◽  
Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

scholarly journals Matrix Games with Interval-Valued 2-Tuple Linguistic Information

Games ◽  
2018 ◽  
Vol 9 (3) ◽  
pp. 62 ◽  
Author(s):  
Anjali Singh ◽  
Anjana Gupta
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Real World ◽  
Upper Bound ◽  
Duality Theorem ◽  
Matrix Game ◽  
Matrix Games ◽  
Linguistic Information ◽  
The Real ◽  
Interval Valued

In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.

scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

Linear Programming Approach for Power System State Estimation Using Upper Bound Optimization Techniques

2010 ◽  
Vol 11 (3) ◽  
Author(s):  
Thukaram Dhadbanjan ◽  
Seshadri Sravan Kumar Vanjari
Keyword(s):  
Linear Programming ◽  
State Estimation ◽  
Upper Bound ◽  
Measurement Errors ◽  
Large Scale ◽  
Optimization Technique ◽  
Optimization Techniques ◽  
Programming Approach ◽  
Linear Programming Approach ◽  
Bound Optimization

State estimation plays an important role in real time security monitoring and control of power systems. There are many problems in the implementation of state estimator for large scale networks due to measurement errors, weights given and the numerical ill-conditioning associated with the solution techniques. In this paper a new formulation using linear programming approach is presented. The formulation is devoid of weights and errors associated with the measurements are taken care of in constraints. The non linear problem is linearized at previous operating state and constraints are set up using flow mismatches. The implementation of the formulation exploits sparse features of the network matrices and avoids matrix inversions. Upper bound optimization technique is employed to solve the linear programming problem. Illustration of the proposed approach on sample 3-bus and 6-bus systems and a practical Indian Southern grid 72 bus equivalent system are presented.

scholarly journals New Upper Bound for Sums of Dilates

10.37236/6576 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Albert Bush ◽  
Yi Zhao
Keyword(s):  
Upper Bound ◽  
Bipartite Graphs ◽  
Binary Expansion ◽  
Main Technique

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A = \{ \lambda a : a \in A\}$. Suppose $r, h\in \mathbb{Z}$ are sufficiently large and comparable to each other. We prove that if $|A+A| \le K |A|$ and $\lambda_1, \ldots, \lambda_h \le 2^r$, then \[ |\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{ 7 rh /\ln (r+h) } |A|. \]This improves upon a result of Bukh who shows that\[ |\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{O(rh)} |A|. \]Our main technique is to combine Bukh's idea of considering the binary expansion of $\lambda_i$ with a result on biclique decompositions of bipartite graphs.

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