The complexity of the T-coloring problem for graphs with small degree

AbstractIn the paper we consider a generalized vertex coloring model, namely T-coloring. For a given finite set T of nonnegative integers including 0, a proper vertex coloring is called a T-coloring if the distance of the colors of adjacent vertices is not an element of T. This problem is a generalization of the classic vertex coloring and appeared as a model of the frequency assignment problem. We present new results concerning the complexity of T-coloring with the smallest span on graphs with small degree Δ. We distinguish between the cases that appear to be polynomial or NP-complete. More specifically, we show that our problem is polynomial on graphs with Δ⩽2 and in the case of k-regular graphs it becomes NP-hard even for every fixed T and every k>3. Also, the case of graphs with Δ=3 is under consideration. Our results are based on the complexity properties of the homomorphism of graphs

Szeregowanie rozrzedzonych systemów zadań jednostkowych 1- i 2-procesowych w oknach czasowych

Tyt. z nagł.STRESZCZENIE: W artykule rozważono rozrzedzone systemy niepodzielnych zadań 1- i 2-procesorowych o jednostkowych czasach wykonywania. Przedstawiono wielomianowe algorytmy wykorzystujące programowanie dynamiczne, pozwalające na znalezienie optymalnego uszeregowania względem szerokiej rodziny funkcji kryterialnych. Stopień rozrzedzenia systemu zdefiniowano, posługując się jego modelem grafowym - w zakresie naszego zainteresowania leżą jedynie takie instancje problemów szeregowania, których modelami są grafy o ograniczonej liczbie cyklomatycznej. Istotnym elementem opracowanych procedur są algorytmy rozwiązujące pewne zagadnienia związane z wyszukiwaniem skojarzeń w grafach. SŁOWA KLUCZOWE: algorytm wielomianowy, kolorowanie grafów, np-zupełność, okna czasowe, szeregowanie zadań, zadania wieloprocesorowe. ABSTRACT: In the paper sparse systems of dedicated 1- and 2-processor tasks with unit execution times are considered. Polynomial-time algorithms based on dynamic programming are given. These algorithms allow finding optimal solutions with respect to broad range of criterion functions. The sparsity of a system is measured in terms of the number of edges in the corresponding scheduling graph. More precisely, we are focused on graphs whose cyclomatic number is bounded by a constant. Our algorithms invoke procedures for finding maximal matching in graphs. KEYWORDS: polynomial algorithm, graph coloring, time windows, task scheduling, multiprocessor tasks, np-completeness

Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs

We consider a list cost coloring of vertices and edges in the model of vertex, edge, total and pseudototal coloring of graphs. We use a dynamic programming approach to derive polynomial-time algorithms for solving the above problems for trees. Then we generalize this approach to arbitrary graphs with bounded cyclomatic numbers and to their multicolorings

Introduction to Quantum Computing

info:eu-repo/semantics/publishe

On a matching distance between rooted phylogenetic trees

The Robinson-Foulds (RF) distance is the most popular method of evaluating the dissimilarity between phylogenetic trees. In this paper, we define and explore in detail properties of the Matching Cluster (MC) distance, which can be regarded as a refinement of the RF metric for rooted trees. Similarly to RF, MC operates on clusters of compared trees, but the distance evaluation is more complex. Using the graph theoretic approach based on a minimum-weight perfect matching in bipartite graphs, the values of similarity between clusters are transformed to the final MC-score of the dissimilarity of trees. The analyzed properties give insight into the structure of the metric space generated by MC, its relations with the Matching Split (MS) distance of unrooted trees and asymptotic behavior of the expected distance between binary n-leaf trees selected uniformly in both MC and MS (Θ(n 3/2 ))

A polynomial algorithm for finding T-span of generalized cacti

AbstractIt has been known for years that the problem of computing the T-span is NP-hard in general. Recently, Giaro et al. (Discrete Appl. Math., to appear) showed that the problem remains NP-hard even for graphs of degree Δ⩽3 and it is polynomially solvable for graphs with degree Δ⩽2. Herein, we extend the latter result. We introduce a new class of graphs which is large enough to contain paths, cycles, trees, cacti, polygon trees and connected outerplanar graphs. Next, we study the properties of graphs from this class and prove that the problem of computing the T-span for these graphs is polynomially solvable