An easy subexponential bound for online chain partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.Comment: 23 pages, 11 figure

Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II

Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). We define the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the graph having a vertex for each path in P, and an edge between every pair of vertices representing two paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G=ENPT(P), and we say that is a representation of G. Our goal is to characterize the representation of chordless ENPT cycles (holes). To achieve this goal, we first assume that the EPT graph induced by the vertices of an ENPT hole is given. In [2] we introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize the representations of ENPT holes that satisfy (P1), (P2), (P3). In this work, we continue our work by relaxing these three assumptions one by one. We characterize the representations of ENPT holes satisfying (P3) by providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also show that there does not exist a polynomial-time algorithm to solve HamiltonianPairRec, unless P=NP

Online Graph Coloring with Predictions

We introduce learning augmented algorithms to the online graph coloring problem. Although the simple greedy algorithm FirstFit is known to perform poorly in the worst case, we are able to establish a relationship between the structure of any input graph $G$ that is revealed online and the number of colors that FirstFit uses for $G$. Based on this relationship, we propose an online coloring algorithm FirstFitPredictions that extends FirstFit while making use of machine learned predictions. We show that FirstFitPredictions is both \emph{consistent} and \emph{smooth}. Moreover, we develop a novel framework for combining online algorithms at runtime specifically for the online graph coloring problem. Finally, we show how this framework can be used to robustify by combining it with any classical online coloring algorithm (that disregards the predictions)

On-line coloring between two lines

We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses $O(\omega^3)$ colors on graphs with maximum clique size $\omega$. In contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when $\omega=2$. The {\em left-of} relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class $\mathcal{C}$ of posets obtained from convex sets between two lines with some other classes of posets: all $2$-dimensional posets and all posets of height $2$ are in $\mathcal{C}$ but there is a $3$-dimensional poset of height $3$ that does not belong to $\mathcal{C}$. We also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.Comment: grant support adde

New Steiner 2-designs from old ones by paramodifications

Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals