About equivalent interval colorings of weighted graphs

AbstractGiven a graph G=(V,E) with strictly positive integer weights ωi on the vertices i∈V, a k-interval coloring of G is a function I that assigns an interval I(i)⊆{1,…,k} of ωi consecutive integers (called colors) to each vertex i∈V. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0̸ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χint(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ωi=1 for all vertices i∈V).Two k-interval colorings I1 and I2 are said equivalent if there is a permutation π of the integers 1,…,k such that ℓ∈I1(i) if and only if π(ℓ)∈I2(i) for all vertices i∈V. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions

A new lower bound for doubly metric dimension and related extremal differences

In this paper a new graph invariant based on the minimal hitting set problem is introduced. It is shown that it represents a tight lower bound for the doubly metric dimension of a graph. Exact values of new invariant for paths, stars, complete graphs and complete bipartite graph are obtained. The paper analyzes some tight bounds for the new invariant in general case. Also several extremal differences between some related invariants are determined

The traveling salesman problem: The spectral radius and the length of an optimal tour

We consider the symmetric traveling salesman problem (TSP) with instances represented by complete graphs with distances between cities as edge weights. Computational experiments with randomly generated instances on 50 and 100 vertices with the uniform distribution of integer edge weights in interval [1, 100] show that there exists a correlation between the sequences of the spectral radii of the distance matrices and the lengths of optimal tours obtained by the well known TSP solver Concorde. In this paper we give a partial theoretical explanation of this correlation.Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles. Sciences mathématiques. . - 43 , 151 (2018

Heuristic approach to train rescheduling

Starting from the defined network topology and the timetable assigned beforehand, the paper considers a train rescheduling in respond to disturbances that have occurred. Assuming that the train trips are jobs, which require the elements of infrastructure - resources, it was done by the mapping of the initial problem into a special case of job shop scheduling problem. In order to solve the given problem, a constraint programming approach has been used. A support to fast finding "enough good" schedules is offered by original separation, bound and search heuristic algorithms. In addition, to improve the time performance, instead of the actual objective function with a large domain, a surrogate objective function is used with a smaller domain, if there is such.

DOI: 10.2298/YUJOR0701009M HEURISTIC APPROACH TO TRAIN RESCHEDULING

Abstract: Starting from the defined network topology and the timetable assigned beforehand, the paper considers a train rescheduling in respond to disturbances that have occurred. Assuming that the train trips are jobs, which require the elements of infrastructure – resources, it was done by the mapping of the initial problem into a special case of job shop scheduling problem. In order to solve the given problem, a constraint programming approach has been used. A support to fast finding “enough good” schedules is offered by original separation, bound and search heuristic algorithms. In addition, to improve the time performance, instead of the actual objective function with a large domain, a surrogate objective function is used with a smaller domain, if there is such

Finding minimal branchings with a given number of arcs

We describe an algorithm for finding a minimal s-branching (where s is a given number of its arcs) in a weighted digraph with an a symetric weight matrix. The algorithm uses the basic principles of the method (previously developed by J. Edmonds) for determining a minimal branching in the case when the number of its arcs is not specified in advance. Here we give a proof of the correctness for the described algorithm

Computing strong metric dimension of some special classes of graphs by genetic algorithms

In this paper we consider the NP-hard problem of determining the strong metric dimension of graphs. The problem is solved by a genetic algorithm that uses binary encoding and standard genetic operators adapted to the problem. This represents the first attempt to solve this problem heuristically. We report experimental results for the two special classes of ORLIB test instances: crew scheduling and graph coloring

Computing the metric dimension of graphs by genetic algorithms

Graph theory, Metric dimension, Evolutionary approach,