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

Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways.In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present only when it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to "efficiently" solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter $\alpha\in\mathbb{N}$. Nevertheless, here it is proved that the probability of requiring a value of $\alpha>k$ to obtain a solution for a random graph decreases exponentially: $P(\alpha>k) \leq 2^{-(k+1)}$, making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.Comment: Working pape

Improved Distributed Fractional Coloring Algorithms

We prove new bounds on the distributed fractional coloring problem in the LOCAL model. Fractional $c$-colorings can be understood as multicolorings as follows. For some natural numbers $p$ and $q$ such that $p/q\leq c$, each node $v$ is assigned a set of at least $q$ colors from $\{1,\dots,p\}$ such that adjacent nodes are assigned disjoint sets of colors. The minimum $c$ for which a fractional $c$-coloring of a graph $G$ exists is called the fractional chromatic number $\chi_f(G)$ of $G$. Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any constant $\epsilon>0$, a fractional $(\Delta+\epsilon)$-coloring can be computed in $\Delta^{O(\Delta)} + O(\Delta\cdot\log^* n)$ rounds. We show that such a coloring can be computed in only $O(\log^2 \Delta)$ rounds, without any dependency on $n$. We further show that in $O\big(\frac{\log n}{\epsilon}\big)$ rounds, it is possible to compute a fractional $(1+\epsilon)\chi_f(G)$-coloring, even if the fractional chromatic number $\chi_f(G)$ is not known. That is, this problem can be approximated arbitrarily well by an efficient algorithm in the LOCAL model. For the standard coloring problem, it is only known that an $O\big(\frac{\log n}{\log\log n}\big)$-approximation can be computed in polylogarithmic time in the LOCAL model. We also show that our distributed fractional coloring approximation algorithm is best possible. We show that in trees, which have fractional chromatic number $2$, computing a fractional $(2+\epsilon)$-coloring requires at least $\Omega\big(\frac{\log n}{\epsilon}\big)$ rounds. We finally study fractional colorings of regular grids. In [Bousquet, Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded dimension, a fractional $(2+\epsilon)$-coloring can be computed in time $O(\log^* n)$. We show that such a coloring can even be computed in $O(1)$ rounds in the LOCAL model

Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time

The described multicoloring problem has direct applications in the context of wireless ad hoc and sensor networks. In order to coordinate the access to the shared wireless medium, the nodes of such a network need to employ some medium access control (MAC) protocol. Typical MAC protocols control the access to the shared channel by time (TDMA), frequency (FDMA), or code division multiple access (CDMA) schemes. Many channel access schemes assign a fixed set of time slots, frequencies, or (orthogonal) codes to the nodes of a network such that nodes that interfere with each other receive disjoint sets of time slots, frequencies, or code sets. Finding a valid assignment of time slots, frequencies, or codes hence directly corresponds to computing a multicoloring of a graph $G$. The scarcity of bandwidth, energy, and computing resources in ad hoc and sensor networks, as well as the often highly dynamic nature of these networks require that the multicoloring can be computed based on as little and as local information as possible