Acyclic and frugal colourings of graphs

Given a graph G = (V, E), a proper vertex colouring of V is t-frugal if no colour appears more than t times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind, Molloy and Reed [14] studied proper t-frugal colourings and Yuster [19] studied acyclic proper 2-frugal colourings. In this paper, we expand and generalise this study. In particular, we consider vertex colourings that are not necessarily proper, and in this case, we find qualitative connections with colourings that are t-improper -colourings in which the colour classes induce subgraphs of maximum degree at most t -for choices of t near to d

Distributed (Δ+1)(\Delta+1)-Coloring in Sublogarithmic Rounds

We give a new randomized distributed algorithm for $(\Delta+1)$-coloring in the LOCAL model, running in $O(\sqrt{\log \Delta})+ 2^{O(\sqrt{\log \log n})}$ rounds in a graph of maximum degree~$\Delta$. This implies that the $(\Delta+1)$-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of $\Omega \left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log \Delta}{\log \log \Delta} \right) \right)$ by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains $\Delta+1$ colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts

Proper Conflict-free Coloring of Graphs with Large Maximum Degree

A proper coloring of a graph is conflict-free if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every graph $G$ has a proper conflict-free coloring with at most $5\Delta(G)/2$ colors and conjectured that $\Delta(G)+1$ colors suffice for every connected graph $G$ with $\Delta(G)\ge 3$. Our first main result is that even for list-coloring, $\left\lceil 1.6550826\Delta(G)+\sqrt{\Delta(G)}\right\rceil$ colors suffice for every graph $G$ with $\Delta(G)\ge 10^{9}$; we also prove slightly weaker bounds for all graphs with $\Delta(G)\ge 750$. These results follow from our more general framework on proper conflict-free list-coloring of a pair consisting of a graph $G$ and a ``conflict'' hypergraph ${\mathcal H}$. As another corollary of our results in this general framework, every graph has a proper $(\sqrt{30}+o(1))\Delta(G)^{1.5}$-list-coloring such that every bi-chromatic component is a path on at most three vertices, where the number of colors is optimal up to a constant factor. Our proof uses a fairly new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lov\'{a}sz Local Lemma or entropy compression. We also prove an asymptotically optimal result for a fractional analogue of our general framework for proper conflict-free coloring for pairs of a graph and a conflict hypergraph. A corollary states that every graph $G$ has a fractional $(1+o(1))\Delta(G)$-coloring such that every fractionally bi-chromatic component has at most two vertices. In particular, it implies that the fractional analogue of the conjecture of Caro et al. holds asymptotically in a strong sense

New bounds for odd colourings of graphs

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $\sigma$. The odd chromatic number of a graph $G$, denoted by $\chi_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $\sigma \colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $\Delta\ge 3$ has odd-chromatic number at most $\Delta+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $\Delta$, $\chi_o(G)\le\Delta+O(\ln\Delta)$ as $\Delta \to \infty$. We also prove that $\chi_o(G)\le\lfloor3\Delta/2\rfloor+2$ for every $\Delta$. If moreover the minimum degree $\delta$ of $G$ is sufficiently large, we have $\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta)$ and $\chi_o(G) = O(\chi(G)\ln \Delta)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{\deg(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant

Sublinear Time and Space Algorithms for Correlation Clustering via Sparse-Dense Decompositions

We present a new approach for solving (minimum disagreement) correlation clustering that results in sublinear algorithms with highly efficient time and space complexity for this problem. In particular, we obtain the following algorithms for $n$-vertex $(+/-)$-labeled graphs $G$: -- A sublinear-time algorithm that with high probability returns a constant approximation clustering of $G$ in $O(n\log^2{n})$ time assuming access to the adjacency list of the $(+)$-labeled edges of $G$ (this is almost quadratically faster than even reading the input once). Previously, no sublinear-time algorithm was known for this problem with any multiplicative approximation guarantee. -- A semi-streaming algorithm that with high probability returns a constant approximation clustering of $G$ in $O(n\log{n})$ space and a single pass over the edges of the graph $G$ (this memory is almost quadratically smaller than input size). Previously, no single-pass algorithm with $o(n^2)$ space was known for this problem with any approximation guarantee. The main ingredient of our approach is a novel connection to sparse-dense graph decompositions that are used extensively in the graph coloring literature. To our knowledge, this connection is the first application of these decompositions beyond graph coloring, and in particular for the correlation clustering problem, and can be of independent interest

Algorithms for Fundamental Problems in Computer Networks.

Traditional studies of algorithms consider the sequential setting, where the whole input data is fed into a single device that computes the solution. Today, the network, such as the Internet, contains of a vast amount of information. The overhead of aggregating all the information into a single device is too expensive, so a distributed approach to solve the problem is often preferable. In this thesis, we aim to develop efficient algorithms for the following fundamental graph problems that arise in networks, in both sequential and distributed settings. Graph coloring is a basic symmetry breaking problem in distributed computing. Each node is to be assigned a color such that adjacent nodes are assigned different colors. Both the efficiency and the quality of coloring are important measures of an algorithm. One of our main contributions is providing tools for obtaining colorings of good quality whose existence are non-trivial. We also consider other optimization problems in the distributed setting. For example, we investigate efficient methods for identifying the connectivity as well as the bottleneck edges in a distributed network. Our approximation algorithm is almost-tight in the sense that the running time matches the known lower bound up to a poly-logarithmic factor. For another example, we model how the task allocation can be done in ant colonies, when the ants may have different capabilities in doing different tasks. The matching problems are one of the classic combinatorial optimization problems. We study the weighted matching problems in the sequential setting. We give a new scaling algorithm for finding the maximum weight perfect matching in general graphs, which improves the long-standing Gabow-Tarjan's algorithm (1991) and matches the running time of the best weighted bipartite perfect matching algorithm (Gabow and Tarjan, 1989). Furthermore, for the maximum weight matching problem in bipartite graphs, we give a faster scaling algorithm whose running time is faster than Gabow and Tarjan's weighted bipartite {it perfect} matching algorithm.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113540/1/hsinhao_1.pd

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM