Krzysztof Giaro

Consecutive colorings of the edges of general graphs

Abstract Given an n -vertex graph G , an edge-coloring of G with natural numbers is a consecutive (or interval) coloring if the colors of edges incident with each vertex are distinct and form an interval of integers. In this paper we prove that if G has a consecutive coloring and n⩾3 then S(G)⩽2n−4 , where S(G) is the maximum number of colors allowing a consecutive coloring. Next, we investigate the so-called deficiency of G , a natural measure of how far it falls of being consecutively colorable. Informally, we define the deficiency def (G) of G as the minimum number of pendant edges which would need to be attached in order that the resulting supergraph has such a coloring, and compute this number in the case of cycles, wheels and complete graphs.

Matching Split Distance for Unrooted Binary Phylogenetic Trees

The reconstruction of evolutionary trees is one of the primary objectives in phylogenetics. Such a tree represents the historical evolutionary relationship between different species or organisms. Tree comparisons are used for multiple purposes, from unveiling the history of species to deciphering evolutionary associations among organisms and geographical areas. In this paper, we propose a new method of defining distances between unrooted binary phylogenetic trees that is especially useful for relatively large phylogenetic trees. Next, we investigate in detail the properties of one example of these metrics, called the Matching Split distance, and describe how the general method can be extended to nonbinary trees.

Compact scheduling of zero-one time operations in multi-stage systems

A problem of no-wait scheduling of zero-one time operations without allowing inserted idle times is considered in the case of open, flow and mixed shop. We show that in the case of open shop this problem is equivalent to the problem of consecutive coloring the edges of a bipartite graph G. In the cases of flow shop and mixed shop this problem is equivalent to the problem of consecutive coloring the edges of G with some additional restrictions. Moreover, in all shops under consideration the problem is shown to be strongly NP-hard. Since such colorings are not always possible when the number of processors m > 3 for open shop (m > 2 for flow shop), we concentrate on special families of scheduling graphs, e.g. paths and cycles, trees, complete bipartite graphs, which can be optimally colored in polynomial time.

TreeCmp: Comparison of Trees in Polynomial Time

When a phylogenetic reconstruction does not result in one tree but in several, tree metrics permit finding out how far the reconstructed trees are from one another. They also permit to assess the accuracy of a reconstruction if a true tree is known. TreeCmp implements eight metrics that can be calculated in polynomial time for arbitrary (not only bifurcating) trees: four for unrooted (Matching Split metric, which we have recently proposed, Robinson-Foulds, Path Difference, Quartet) and four for rooted trees (Matching Cluster, Robinson-Foulds cluster, Nodal Splitted and Triple). TreeCmp is the first implementation of Matching Split/Cluster metrics and the first efficient and convenient implementation of Nodal Splitted. It allows to compare relatively large trees. We provide an example of the application of TreeCmp to compare the accuracy of ten approaches to phylogenetic reconstruction with trees up to 5000 external nodes, using a measure of accuracy based on normalized similarity between trees.

Edge-chromatic sum of trees and bounded cyclicity graphs

Abstract In the minimum edge-sum coloring problem the aim is to color the edges of a graph with positive integers such that the sum of all colors is as small as possible. In this note we study the complexity of this problem on some sparse graphs. In particular, we give polynomial-time algorithms for trees and k -cyclic graphs.

On the deficiency of bipartite graphs

Abstract Given a graph G, an edge-coloring of G with colors 1,2,3,… is consecutive if the colors of edges incident to each vertex form an interval of integers. This paper is devoted to bipartite graphs which do not have such a coloring of edges. We investigate their consecutive coloring deficiency, or shortly the deficiency d(G) of G, i.e. the minimum number of pendant edges whose attachment to G makes it consecutively colorable. In particular, we show that there are bipartite graphs whose deficiency approaches the number of vertices.

A graph coloring approach to scheduling of multiprocessor tasks on dedicated machines with availability constraints

We address a generalization of the classical 1- and 2-processor unit execution time scheduling problem on dedicated machines. In our chromatic model of scheduling machines have non-simultaneous availability times and tasks have arbitrary release times and due dates. Also, the versatility of our approach makes it possible to generalize all known classical criteria of optimality. Under these stipulations we show that the problem of optimal scheduling of sparse tree-like instances can be solved in polynomial time. However, if we admit dense instances then the problem becomes NP-hard, even if there are only two machines.

NP-hardness of compact scheduling in simplified open and flow shops

Abstract In practical task scheduling it is sometimes required that the components of a system perform consecutively. Such a scheduling is called scheduling without waiting periods or no-wait and/or no-idle. In this article we study the complexity of some simplified scheduling problems of this kind in open shop and flow shop settings. In particular, we show that many trivial questions about the existence of schedule become NP-hard, even if there are only two machines or if the scheduling graph of a system is a path or a cycle.

A 27/26-Approximation Algorithm for the Chromatic Sum Coloring of Bipartite Graphs

We consider the CHROMATIC SUM PROBLEM on bipartite graphs which appears to be much harder than the classical CHROMATIC NUMBER PROBLEM. We prove that the CHROMATIC SUM PROBLEM is NP-complete on planar bipartite graphs with ? ? 5, but polynomial on bipartite graphs with ? ? 3, for which we construct an O(n2)-time algorithm. Hence, we tighten the borderline of intractability for this problem on bipartite graphs with bounded degree, namely: the case ? = 3 is easy, ? = 5 is hard. Moreover, we construct a 27/26-approximation algorithm for this problem thus improving the best known approximation ratio of 10/9.

On a matching distance between rooted phylogenetic trees

Abstract 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(n3/2)).