scholarly journals Matroids, Cyclic Flats, and Polyhedra

2021 ◽  
Author(s):  
◽  
Kadin Prideaux
Keyword(s):  
Polynomial Time ◽  
Solution Space ◽  
Axiom System ◽  
Linear Program ◽  
Integral Points ◽  
The Matrix ◽  
Axiom Systems

<p>Matroids have a wide variety of distinct, cryptomorphic axiom systems that are capable of defining them. A common feature of these is that they are able to be efficiently tested, certifying whether a given input complies with such an axiom system in polynomial time. Joseph Bonin and Anna de Mier, rediscovering a theorem first proved by Julie Sims, developed an axiom system for matroids in terms of their cyclic flats and the ranks of those cyclic flats. As with other matroid axiom systems, this is able to be tested in polynomial time. Distinct, non-isomorphic matroids may each have the same lattice of cyclic flats, and so matroids cannot be defined solely in terms of their cyclic flats. We do not have a clean characterisation of families of sets that are cyclic flats of matroids. However, it may be possible to tell in polynomial time whether there is any matroid that has a given lattice of subsets as its cyclic flats. We use Bonin and de Mier’s cyclic flat axioms to reduce the problem to a linear program, and show that determining whether a given lattice is the lattice of cyclic flats of any matroid corresponds to finding integral points in the solution space of this program, these points representing the possible ranks that may be assigned to the cyclic flats. We distinguish several classes of lattice for which solutions may be efficiently found, based upon the nature of the matrix of coefficients of the linear program, and of the polyhedron it defines, and then identify families of lattice that belong to those classes. We define operations and transformations on lattices of sets by examining matroid operations, and examine how these operations affect membership in the aforementioned classes. We conjecture that it is always possible to determine, in polynomial time, whether a given collection of subsets makes up the lattice of cyclic flats of any matroid.</p>

scholarly journals Matroids, Cyclic Flats, and Polyhedra

2021 ◽  
Author(s):  
◽  
Kadin Prideaux
Keyword(s):  
Polynomial Time ◽  
Solution Space ◽  
Axiom System ◽  
Linear Program ◽  
Integral Points ◽  
The Matrix ◽  
Axiom Systems

<p>Matroids have a wide variety of distinct, cryptomorphic axiom systems that are capable of defining them. A common feature of these is that they are able to be efficiently tested, certifying whether a given input complies with such an axiom system in polynomial time. Joseph Bonin and Anna de Mier, rediscovering a theorem first proved by Julie Sims, developed an axiom system for matroids in terms of their cyclic flats and the ranks of those cyclic flats. As with other matroid axiom systems, this is able to be tested in polynomial time. Distinct, non-isomorphic matroids may each have the same lattice of cyclic flats, and so matroids cannot be defined solely in terms of their cyclic flats. We do not have a clean characterisation of families of sets that are cyclic flats of matroids. However, it may be possible to tell in polynomial time whether there is any matroid that has a given lattice of subsets as its cyclic flats. We use Bonin and de Mier’s cyclic flat axioms to reduce the problem to a linear program, and show that determining whether a given lattice is the lattice of cyclic flats of any matroid corresponds to finding integral points in the solution space of this program, these points representing the possible ranks that may be assigned to the cyclic flats. We distinguish several classes of lattice for which solutions may be efficiently found, based upon the nature of the matrix of coefficients of the linear program, and of the polyhedron it defines, and then identify families of lattice that belong to those classes. We define operations and transformations on lattices of sets by examining matroid operations, and examine how these operations affect membership in the aforementioned classes. We conjecture that it is always possible to determine, in polynomial time, whether a given collection of subsets makes up the lattice of cyclic flats of any matroid.</p>

scholarly journals On A Quantum Proof Of The Equivalence of P And NP

2019 ◽  
Author(s):  
Adewale Oluwasanmi
Keyword(s):  
Polynomial Time ◽  
Solution Space ◽  
Quantum Systems ◽  
Satisfiability Problem ◽  
Quantum Mechanical ◽  
Correctness Proof ◽  
Classical Systems ◽  
Classical Architecture ◽  
General Quantum

We present an algorithm, along with a correctness proof, for solving the 3 Satisfiability problem that is inspired by quantum mechanical principles and that runs in polynomial time with respect to the size of the input problem. Even though we term both our algorithm and its associated proof as quantum (for reasons which we will demonstrate), it is intended to be run on standard classical architecture. In the article, we posit that the 3 Satisfiability problem has an intrinsic complex quantum form that can be programmed in order to build a model of the solution space for satisfiable instances or show that such a model cannot be constructed. This yields surprising results on the ability for classical systems to abstractly simulate general quantum systems.

scholarly journals A factor graph based genetic algorithm

2014 ◽  
Vol 24 (3) ◽  
pp. 621-633 ◽  
Author(s):  
B. Hoda Helmi ◽  
Adel T. Rahmani ◽  
Martin Pelikan
Keyword(s):  
Genetic Algorithm ◽  
Polynomial Time ◽  
Mathematical Analysis ◽  
Matrix Factorization ◽  
Optimization Problems ◽  
Factor Graph ◽  
The Matrix ◽  
Separable Problems ◽  
Linkage Learning ◽  
Non Negative Matrix Factorization

Abstract We propose a new linkage learning genetic algorithm called the Factor Graph based Genetic Algorithm (FGGA). In the FGGA, a factor graph is used to encode the underlying dependencies between variables of the problem. In order to learn the factor graph from a population of potential solutions, a symmetric non-negative matrix factorization is employed to factorize the matrix of pair-wise dependencies. To show the performance of the FGGA, encouraging experimental results on different separable problems are provided as support for the mathematical analysis of the approach. The experiments show that FGGA is capable of learning linkages and solving the optimization problems in polynomial time with a polynomial number of evaluations.

scholarly journals GRAVITY ON A FUZZY SPHERE

2004 ◽  
Vol 19 (03) ◽  
pp. 361-370 ◽  
Author(s):  
P. VALTANCOLI
Keyword(s):  
Matrix Model ◽  
Solution Space ◽  
Two Dimensional ◽  
Fuzzy Sphere ◽  
Gauge Transformations ◽  
The Matrix ◽  
Analogous Model ◽  
Dimensional Gravity ◽  
Noncommutative Gravity ◽  
Noncommutative Plane

We propose an action for gravity on a fuzzy sphere, based on a matrix model. We find striking similarities with an analogous model of two-dimensional gravity on a noncommutative plane, i.e. the solution space of both models is spanned by pure U(2) gauge transformations acting on the background solution of the matrix model, and there exist deformations of the classical diffeomorphisms which preserve the two-dimensional noncommutative gravity actions.

scholarly journals Safe Approximation—An Efficient Solution for a Hard Routing Problem

Algorithms ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 48
Author(s):  
András Faragó ◽  
Zohre R. Mojaveri
Keyword(s):  
Polynomial Time ◽  
Flow Problem ◽  
Large Deviation ◽  
Directed Graphs ◽  
Solution Space ◽  
Polynomial Time Algorithm ◽  
Np Hard ◽  
Routing Problem ◽  
Practical Applications ◽  
Unsplittable Flow

The Disjoint Connecting Paths problem and its capacitated generalization, called Unsplittable Flow problem, play an important role in practical applications such as communication network design and routing. These tasks are NP-hard in general, but various polynomial-time approximations are known. Nevertheless, the approximations tend to be either too loose (allowing large deviation from the optimum), or too complicated, often rendering them impractical in large, complex networks. Therefore, our goal is to present a solution that provides a relatively simple, efficient algorithm for the unsplittable flow problem in large directed graphs, where the task is NP-hard, and is known to remain NP-hard even to approximate up to a large factor. The efficiency of our algorithm is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm for approximation. Rather than giving up the search for an exact solution, we restrict the solution space to a subset that is the most important for applications, and excludes only a small part that is marginal in some well-defined sense. Specifically, the sacrificed part only contains scenarios where some edges are very close to saturation. Since nearly saturated links are undesirable in practical applications, therefore, excluding near saturation is quite reasonable from the practical point of view. We refer the solutions that contain no nearly saturated edges as safe solutions, and call the approach safe approximation. We prove that this safe approximation can be carried out efficiently. That is, once we restrict ourselves to safe solutions, we can find the exact optimum by a randomized polynomial time algorithm.

scholarly journals The Computational Complexity of Some Problems of Linear Algebra

1996 ◽  
Vol 3 (33) ◽  
Author(s):  
Jonathan F. Buss ◽  
Gudmund Skovbjerg Frandsen ◽  
Jeffery O. Shallit
Keyword(s):  
Computational Complexity ◽  
Commutative Ring ◽  
Linear Algebra ◽  
Polynomial Time ◽  
Matrix Rank ◽  
Maximum Rank ◽  
The Matrix ◽  
Np Complete

We consider the computational complexity of some problems dealing with matrix rank.<br /> Let E, S be subsets of a commutative ring R.<br />Let x1, x2, ..., xt be variables. Given a matrix M = M(x1, x2, ..., xt)<br />with entries chosen from E union {x1, x2, ..., xt}, we want to determine<br />maxrankS(M) = max rank M(a1, a2, ... , at)<br />and<br />minrankS(M) = min rank M(a1, a2, ..., at). <br />There are also variants of these problems that specify more about the<br />structure of M, or instead of asking for the minimum or maximum rank, <br />ask if there is some substitution of the variables that makes the matrix<br /> invertible or noninvertible.<br />Depending on E, S, and on which variant is studied, the complexity<br />of these problems can range from polynomial-time solvable to random<br />polynomial-time solvable to NP-complete to PSPACE-solvable to<br />unsolvable.

scholarly journals On some aspects of the matrix data perturbation in linear program

2003 ◽  
Vol 13 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Margita Kon-Popovska
Keyword(s):  
Linear Program ◽  
Linear Functions ◽  
Basic Solution ◽  
System Matrix ◽  
Data Perturbation ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Base Matrix ◽  
Technology Resources ◽  
Optimal Basic Solution

Linear program under changes in the system matrix coefficients has proved to be more complex than changes of the coefficients in objective functions and right hand sides. The most of the previous studies deals with problems where only one coefficient, a row (column), or few rows (columns) are linear functions of a parameter. This work considers a more general case, where all the coefficients are polynomial (in the particular case linear) functions of the parameter tT??R. For such problems, assuming that some non-singularity conditions hold and an optimal base matrix is known for some particular value t of the parameter, corresponding explicit optimal basic solution in the neighborhood of t is determined by solving an augmented LP problem with real system matrix coefficients. Parametric LP can be utilized for example to model the production problem where, technology, resources, costs and similar categories vary with time. .

Computing the variance of tour costs over the solution space of the TSP in polynomial time

2012 ◽  
Vol 53 (3) ◽  
pp. 711-728
Author(s):  
Paul J. Sutcliffe ◽  
Andrew Solomon ◽  
Jenny Edwards
Keyword(s):  
Polynomial Time ◽  
Solution Space

scholarly journals RANK LOGIC IS DEAD, LONG LIVE RANK LOGIC!

2019 ◽  
Vol 84 (1) ◽  
pp. 54-87
Author(s):  
ERICH GRÄDEL ◽  
WIED PAKUSA
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Expressive Power ◽  
Matrix Rank ◽  
Original Definition ◽  
The Matrix ◽  
Prime Fields ◽  
Solvability Problem ◽  
Image Position ◽  
Open Question

AbstractMotivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. WhileFPRcan express most of the known queries that separateFPCfromPtime, almost nothing was known about the limitations of its expressive power.In our first main result we show that the extensions ofFPCby rank operators over different prime fields are incomparable. This solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic${\text{FPR}}^{\text{*}}$with an operator that uniformly expresses the matrix rank over finite fields is more expressive thanFPR.One important step in our proof is to consider solvability logicFPSwhich is the analogous extension ofFPCby quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers.

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