The Weighted Linear Ordering Problem - Springer
...Abstract In this work, we introduce and analyze an extension of the Linear Ordering Problem (LOP). The LOP aims to find a simultaneous permutation of rows and columns of a given weight matrix such that the sum of the weights in the upper triangle is maximized. We propose the weighted Linear Ordering Problem (wLOP) that additionally considers individual node weights....
https://link.springer.com/content/pdf/10.1007/978-3-030-48439-2_27.pdf
The linear ordering problem: an algorithm for the optimal solution
...In this paper we describe and implement an algorithm for the exact solution of the Linear Ordering problem. Linear Ordering is the problem of finding a linear order of the nodes of a graph such ......
https://www.researchgate.net/publication/272456381_The_linear_ordering_problem_an_algorithm_for_the_optimal_solution
Multilevel Algorithms for Linear Ordering Problems
...Linear ordering problems are combinatorial optimization problems which deal with the minimization of different functionals in which the graph vertices are mapped onto (1, 2, ..., n). These problems are widely used and studied in many practical and theoretical applications. In this paper we present a variety of linear-time algorithms for these problems inspired by the Algebraic Multi-grid ......
https://www.wisdom.weizmann.ac.il/~achi/tr07-03.pdf
SCIP Doxygen Documentation: Linear Ordering - scipopt.org
...The linear ordering problem is the following: Given a positive integer \ ( n \) and an \ ( n \times n \) matrix \ ( W \) the goal is to find a linear order of \ ( \ {1, \dots, n\} \) such that the sum of weights \ ( w_ {ij}\) for all pairs in which \ ( x \) comes before \ ( j \) in the order is maximized....
https://scipopt.org/doc-7.0.0/html/LOP_MAIN.php
Transitions from P to NP-hardness: the case of the Linear Ordering Problem
...This is done by means of the linear ordering problem. More specifically, for this problem we prove that the objective function can be expressed as the sum of two objective functions, one of which is associated with a P problem (an exact polynomial time algorithm is proposed to solve it), while the other is associated with an NP-hard problem....
https://arxiv.org/pdf/2205.12943
Linear Ordering Problem - GRAFO Research Group
...Obviously, by setting arc weights wij = cij for the complete digraph Dn, the triangulation problem for C can be solved as linear ordering problem Dn. Conversely,a linear odering problem for Dn can be transformed to a triangulation problem for an (n, n) matrix C by setting cij = wij and the diagonal entries cii = 0 (or to arbitrary values)....
https://grafo.etsii.urjc.es/optsicom/lolib.html
Linear Ordering Problem I (lop) - MatMod
...Problem , ? = {1, ... , } > 0 : Given a numerical matrix , with and , the linear ordering problem consists of finding a simultaneous permutation of the rows and columns of the matrix , such that the sum of the entries above the main diagonal is maximized (or equivalently, the sum of the entries below the main diagonal is minimized). 0?1...
https://matmod.ch/lpl/PDF/lop.pdf
Block-insertion-based algorithms for the linear ordering problem
...Highlights Four properties of the Linear Ordering Problem. Three new local search schemes. The block-insertion-based iterated local search and memetic algorithm that frequently produce better solutions than the state-of-the-art meta-heuristics....
https://www.sciencedirect.com/science/article/pii/S030505481930303X
The Weighted Linear Ordering Problem | Springer Nature Link
...In this work, we introduce and analyze an extension of the Linear Ordering Problem ( LOP) . The LOP aims to find a simultaneous permutation of rows and columns of a given weight matrix such that the sum of the weights in the upper triangle is maximized. We propose the weighted Linear Ordering Problem ( wLOP) that additionally considers individual node weights. First, we argue that in several ......
https://link.springer.com/chapter/10.1007/978-3-030-48439-2_27
A Cutting Plane Algorithm for the Linear Ordering Problem
...The linear ordering problem is an NP-hard combinatorial optimization problem with a large number of applications (including triangulation of input-output matrices, archaeological seriation, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences). In a former paper, we have investigated the facet structure of the 0/1 -polytope associated ......
https://www.jstor.org/stable/170944