Konstanty Junosza-Szaniawski

On improved exact algorithms for L(2, 1)-labeling of graphs

L(2, 1)-labeling is graph labeling model where adjacent vertices get labels that differ by at least 2 and vertices in distance 2 get different labels. In this paper we present an algorithm for finding an optimal L(2, 1)-labeling (i.e. an L(2, 1)-labeling in which largest label is the least possible) of a graph with time complexity O*(3.5616n), which improves a previous best result: O*(3.8739n).

On the complexity of exact algorithm for L (2, 1)-labeling of graphs

L(2,1)-labeling is a graph coloring model inspired by a frequency assignment in telecommunication. It asks for such a labeling of vertices with nonnegative integers that adjacent vertices get labels that differ by at least 2 and vertices in distance 2 get different labels. It is known that for any k >= 4 it is NP-complete to determine if a graph has a L(2,1)-labeling with no label greater than k. In this paper we present a new bound on complexity of an algorithm for finding an optimal L(2,1)-labeling (i.e. an L(2,1)-labeling in which the largest label is the least possible). We improve the upper complexity bound of the algorithm from O^@?(3.5616^n) to O^@?(3.2361^n). Moreover, we establish a lower complexity bound of the presented algorithm, which is @W^@?(3.0739^n).

Fast exact algorithm for L(2,1)-labeling of graphs

An L(2,1)-labeling of a graph is a mapping from its vertex set into nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. The span of such a labeling is the maximum label used, and the L(2,1)-span of a graph is the minimum possible span of its L(2,1)-labelings. We show how to compute the L(2,1)-span of a connected graph in time O^*(2.6488^n). Previously published exact exponential time algorithms were gradually improving the base of the exponential function from 4 to the so far best known 3, with 3 itself seemingly having been the Holy Grail for quite a while. As concerns special graph classes, we are able to solve the problem in time O^*(2.5944^n) for claw-free graphs, and in time O^*(2^n^-^r(2+nr)^r) for graphs having a dominating set of size r.

Fast exact algorithm for L(2, 1)-labeling of graphs

An L(2, 1)-labeling of a graph is a mapping from its vertex set into nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. The span of such a labeling is the maximum label used, and the L(2, 1)-span of a graph is the minimum possible span of its L(2, 1)- labelings. We show how to compute the L(2, 1)-span of a connected graph in time O*(2.6488n). Previously published exact exponential time algorithms were gradually improving the base of the exponential function from 4 to the so far best known 3.2361, with 3 seemingly having been the Holy Grail.

Exact and approximation algorithms for a soft rectangle packing problem

We consider the problem of finding an arrangement of rectangles with given areas that minimizes the total length of all inner and outer border lines. We present a polynomial time approximation algorithm and derive an upper bound estimation on its approximation ratio. Furthermore, we give a formulation of the problem as mixed-integer nonlinear program and show that it can be approximatively reformulated as linear mixed-integer program. On a test set of problem instances, we compare our approximation algorithm with another one from the literature. Using a standard numerical mixed-integer linear solver, we show that adding the solutions from the approximation algorithm as advanced starter helps to reduce the overall solution time for proven global optimality, or gives better primal and dual bounds if a certain time-limit is reached before.

Beyond Homothetic Polygons: Recognition and Maximum Clique

We study the Clique problem in classes of intersection graphs of convex sets in the plane. The problem is known to be NP-complete in convex-sets intersection graphs and straight-line-segments intersection graphs, but solvable in polynomial time in intersection graphs of homothetic triangles. We extend the latter result by showing that for every convex polygon P with k sides, every n-vertex graph which is an intersection graph of homothetic copies of P contains at most n 2k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, so called k DIR -CONV, which are intersection graphs of convex polygons whose all sides are parallel to at most k directions. We further provide lower bounds on the numbers of maximal cliques, discuss the complexity of recognizing these classes of graphs and present relationship with other classes of convex-sets intersection graphs.

Fixing Improper Colorings of Graphs

In this paper we consider a variation of a recoloring problem, called the r-Color-Fixing. Let us have some non-proper r-coloring ϕ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is “the most similar” to ϕ, i.e. the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any r ≥ 3, but is Fixed Parameter Tractable (FPT), when parametrized by the number of allowed transformations k. We provide an \(\mathcal{O}^*(2^n)\) algorithm for the problem (for any fixed r) and a linear algorithm for graphs with bounded treewidth. Finally, we investigate the fixing number of a graph G. It is the maximum possible distance (in the number of transformations) between some non-proper coloring of G and a proper one.

Determining the L(2,1)-span in polynomial space

A

Counting maximal independent sets in subcubic graphs

k

Travel Time Prediction for Trams in Warsaw

-L(2,1)-labeling of a graph is a function from its vertex set into the set