Piotr Borowiecki

Dynamic Coloring of Graphs

Dynamics is an inherent feature of many real life systems so it is natural to define and investigate the properties of models that reflect their dynamic nature. Dynamic graph colorings can be naturally applied in system modeling, e.g. for scheduling threads of parallel programs, time sharing in wireless networks, session scheduling in high-speed LANs, channel assignment in WDM optical networks as well as traffic scheduling. In the dynamic setting of the problem, a graph we color is not given in advance and new vertices together with adjacent edges are revealed one after another at algorithms input during the coloring process. Moreover, independently of the algorithm, some vertices may lose their colors and the algorithm may be asked to color them again. We formally define a dynamic graph coloring problem, the dynamic chromatic number and prove various bounds on its value. We also analyze the effectiveness of the dynamic coloring algorithm Dynamic-Fit for selected classes of graphs. In particular, we deal with trees, products of graphs and classes of graphs for which Dynamic-Fit is competitive. Motivated by applications, we state the problem of dynamic coloring with discoloring constraints for which the performance of the dynamic algorithm Time-Fit is analyzed and give a characterization of graphs k-critical for Time-Fit. Since for any fixed k > 0 the number of such graphs is finite, it is possible to decide in polynomial time whether Time-Fit will always color a given graph with at most k colors.

GreedyMAX-type algorithms for the maximum independent set problem

A maximum independent set problem for a simple graph G = (V, E) is to find the largest subset of pairwise nonadjacent vertices. The problem is known to be NP-hard and it is also hard to approximate. Within this article we introduce a non-negative integer valued function p defined on the vertex set V (G) and called a potential function of a graph G, while P(G) = maxv∈V(G) p(v) is called a potential of G. For any graph P(G) ≤ Δ(G), where Δ(G) is the maximum degree of G. Moreover, Δ(G) - P(G) may be arbitrarily large. A potential of a vertex lets us get a closer insight into the properties of its neighborhood which leads to the definition of the family of GreedyMAX-type algorithms having the classical GreedyMAX algorithm as their origin. We establish a lower bound 1/(P + 1) for the performance ratio of GreedyMAX-type algorithms which favorably compares with the bound 1/(Δ + 1) known to hold for GreedyMAX. The cardinality of an independent set generated by any GreedyMAX-type algorithm is at least Σv∈V(G)(p(v)+1)-1, which strengthens the bounds of Turaan and Caro-Wei stated in terms of vertex degrees.

New potential functions for greedy independence and coloring

A potential function f G of a finite, simple and undirected graph G = ( V , E ) is an arbitrary function f G : V ( G ) ? N 0 that assigns a nonnegative integer to every vertex of a graph G . In this paper we define the iterative process of computing the step potential function q G such that q G ( v ) ? d G ( v ) for all v ? V ( G ) . We use this function in the development of new Caro-Wei-type and Brooks-type bounds for the independence number α ( G ) and the Grundy number ? ( G ) . In particular, we prove that ? ( G ) ? Q ( G ) + 1 , where Q ( G ) = max { q G ( v ) | v ? V ( G ) } and α ( G ) ? ? v ? V ( G ) ( q G ( v ) + 1 ) - 1 . This also establishes new bounds for the number of colors used by the algorithm Greedy and the size of an independent set generated by a suitably modified version of the classical algorithm GreedyMAX.

Distributed graph searching with a sense of direction

In this work we consider the edge searching problem for vertex-weighted graphs with arbitrarily fast and invisible fugitive. The weight function

Computational aspects of greedy partitioning of graphs

{\omega }

Distributed Evacuation in Graphs with Multiple Exits

ω provides for each vertex

Independence in uniform linear triangle-free hypergraphs

v