Consider the following simple coloring algorithm for a graph on $n$ vertices. Each vertex chooses a color from $\{1, \dotsc, Δ(G) + 1\}$ uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC'16] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected $O(n \log Δ)$ steps, which is optimal and proves a conjecture of Chakrabarty and Supinski [SOSA'20].
We consider a decentralized graph coloring model where each vertex only knows its own color and whether some neighbor has the same color as it. The networking community has studied this model extensively due to its applications to channel selection, rate adaptation, etc. Here, we analyze variants of a simple algorithm of Bhartia et al. [Proc., ACM MOBIHOC, 2016]. In particular, we introduce a variant which requires only $O(n\logΔ)$ expected recolorings that generalizes the coupon collector problem. Finally, we show that the $O(nΔ)$ bound Bhartia et al. achieve for their algorithm still holds a…
We develop a decentralized coloring approach to diversify the nodes in a complex network. The key is the introduction of a local conflict index that measures the color conflicts arising at each node which can be efficiently computed using only local information. We demonstrate via both synthetic and real-world networks that the proposed approach significantly outperforms random coloring as measured by the size of the largest color-induced connected component. Interestingly, for scale-free networks further improvement of diversity can be achieved by tuning a degree-biasing weighting parameter i…
We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg-Seymour and List-Coloring conjectures for (list-)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non-constructive. Our key insight is to show that the main result of Achlioptas, Iliopoulos and Kolmogorov for analyzing local search algorithms can be used to make constructive applications of a powerful version of the so-called Lopsided Lovasz Local Lemma. In particular, we use it to design algorithms that exploit the fact that correlations…
In this paper, we consider distributed coloring for planar graphs with a small number of colors. We present an optimal (up to a constant factor) $O(\log{n})$ time algorithm for 6-coloring planar graphs. Our algorithm is based on a novel technique that in a nutshell detects small structures that can be easily colored given a proper coloring of the rest of the vertices and removes them from the graph until the graph contains a small enough number of edges. We believe this technique might be of independent interest. In addition, we present a lower bound for 4-coloring planar graphs that essential…
An optimal algorithm is presented about Conflict-Free Coloring for connected subgraphs of tree of rings. Suppose the number of the rings in the tree is |T| and the maximum length of rings is |R|. A presented algorithm in [1] for a Tree of rings used O(log|T|.log|R|) colors but this algorithm uses O(log|T|+log|R|) colors. The coloring earned by this algorithm has the unique-min property, that is, the unique color is also minimum.
Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of $2$ of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree $Δ= O(\log n)$, which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or…
This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on coloring graphs of maximum degree $Δ$ with at most $Δ+1$ colors (or $Δ$ colors when some simple obstructions are forbidden). When $Δ$ is sufficiently large and $c\ge Δ-k_Δ+1$, for some integer $k_Δ\approx \sqrtΔ-2$, we give a distributed algorithm that given a $c$-colorable graph $G$ of maximum degree $Δ$, finds a $c$-coloring of $G$ in $\mi…
The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we obtain a kernel of bitsize $O(k^{q-1}\log{k})$ for q-Coloring parameterized by Vertex Cover, for any q >= 3. This size bound is optimal up to $k^{o(1)}$ factors assuming NP is not a subset of coNP/poly, and improve…
Given a large social or information network, how can we partition the vertices into sets (i.e., colors) such that no two vertices linked by an edge are in the same set while minimizing the number of sets used. Despite the obvious practical importance of graph coloring, existing works have not systematically investigated or designed methods for large complex networks. In this work, we develop a unified framework for coloring large complex networks that consists of two main coloring variants that effectively balances the tradeoff between accuracy and efficiency. Using this framework as a fundame…
Vizing's celebrated theorem asserts that any graph of maximum degree $Δ$ admits an edge coloring using at most $Δ+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses $2Δ-1$ colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with $Δ=O(\log n)$, and they conjectured the existence of online algorithms using $Δ(1+o(1))$ colors for $Δ=ω(\log n)$. Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS'03 a…
Graph coloring is one of the central problems in distributed graph algorithms. Much of the research on this topic has focused on coloring with $Δ+1$ colors, where $Δ$ denotes the maximum degree. Using $Δ+1$ colors may be unsatisfactory in sparse graphs, where not all nodes have such a high degree; it would be more desirable to use a number of colors that improves with sparsity. A standard measure that captures sparsity is arboricity, which is the smallest number of forests into which the edges of the graph can be partitioned. We present simple randomized distributed algorithms that, with high …
This paper introduces a variant of the classical edge coloring problem in graphs that can be applied to an offline scheduling problem for crossbar switches. We show that the problem is NP-complete, develop three lower bounds bounds on the optimal solution value and evaluate the performance of several approximation algorithms, both analytically and experimentally. We show how to approximate an optimal solution with a worst-case performance ratio of $3/2$ and our experimental results demonstrate that the best algorithms produce results that very closely track a lower bound.
Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for $(Δ+1)$-list coloring in the randomized ${\sf LOCAL}$ model running in $O(\mathsf{Det}_{\scriptscriptstyle d}(\text{poly} \log n))$ time, where $\mathsf{Det}_{\scriptscriptstyle d}(n')$ is the deterministic complexity of $(\text{deg}+1)$-list coloring on $n'$-vertex graphs. (In this problem, each $v$ has a palette of size $\text{deg}(v)+1$.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC'16, JACM'18] with complexi…
We present a new approach to randomized distributed graph coloring that is simpler and more efficient than previous ones. In particular, it allows us to tackle the $(\operatorname{deg}+1)$-list-coloring (D1LC) problem, where each node $v$ of degree $d_v$ is assigned a palette of $d_v+1$ colors, and the objective is to find a proper coloring using these palettes. While for $(Δ+1)$-coloring (where $Δ$ is the maximum degree), there is a fast randomized distributed $O(\log^3\log n)$-round algorithm (Chang, Li, and Pettie [SIAM J. Comp. 2020]), no $o(\log n)$-round algorithms are known for the D1LC…
This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger, Motwani, and Sudan \cite{KMS98} give a vector program for which a coloring of the graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has sufficiently high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have sufficiently high rank. In the case of the original Karger, Motwani…
Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree $Δ$, it follows from Vizing's Theorem that $O(m\log Δ)$ bits of advice suffice to achieve optimality, where $m$ is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $Δ$ is. On the other h…
Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for the online interval coloring problem. In this paper, for online unit interval coloring, we show that the number of colors used by the Kierstead-Trotter algorithm is at most $3 ω(G) - 3$, where $ω(G)$ is the size of the maximum clique in a given graph $G$, and it is the best possible.
In this paper we introduce and study a new problem named \emph{min-max edge $q$-coloring} which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer $q$. The goal is to color the edges of the graph with as many colors as possible such that: (a) any vertex is incident to at most $q$ different colors, and (b) the maximum size of a color group (i.e. set of edges identically colored) is minimized. We show the following results: 1. Min-max edge $q$-coloring is NP-hard, for any $q \ge 2$. 2. A polynomial time exact algorithm …
We examine the problem of almost-uniform sampling proper $q$-colorings of a graph whose maximum degree is $Δ$. A famous result, discovered independently by Jerrum(1995) and Salas and Sokal(1997), is that, assuming $q > (2+δ) Δ$, the Glauber dynamics (a.k.a. single-site dynamics) for this problem has mixing time $O(n \log n)$, where $n$ is the number of vertices, and thus provides a nearly linear time sampling algorithm for this problem. A natural question is the extent to which this algorithm can be parallelized. Previous work Feng, Sun and Yin [PODC'17] has shown that a $O(Δ\log n)$ time para…
One way to state the Load Coloring Problem (LCP) is as follows. Let $G=(V,E)$ be graph and let $f:V\rightarrow \{{\rm red}, {\rm blue}\}$ be a 2-coloring. An edge $e\in E$ is called red (blue) if both end-vertices of $e$ are red (blue). For a 2-coloring $f$, let $r'_f$ and $b'_f$ be the number of red and blue edges and let $μ_f(G)=\min\{r'_f,b'_f\}$. Let $μ(G)$ be the maximum of $μ_f(G)$ over all 2-colorings. We introduce the parameterized problem $k$-LCP of deciding whether $μ(G)\ge k$, where $k$ is the parameter. We prove that this problem admits a kernel with at most $7k$. Ahuja et al. (200…
In this paper, we propose a high-speed greedy sequential algorithm for the vertex coloring problem (VCP), based on the Wave Function Collapse algorithm, called Wave Function Collapse Coloring (WFC-C). An iteration of this algorithm goes through three distinct phases: vertex selection, color restriction through wave function collapsing, and domain propagation. In effect, WFC-C propagates color choices or "domain" restrictions beyond immediate neighbourhoods. This heuristic, combined with a series of other greedy optimizations, allows for a fast algorithm that prevents certain color conflicts. T…
We study a version of online edge coloring, where the goal is to color as many edges as possible using only a given number, $k$, of available colors. All of our results are with regard to competitive analysis. Previous attempts to identify optimal algorithms for this problem have failed, even for bipartite graphs. Thus, in this paper, we analyze even more restricted graph classes, paths and trees. For paths, we consider $k=2$, and for trees, we consider any $k \geq 2$. We prove that a natural greedy algorithm called First-Fit is optimal among deterministic algorithms, on paths as well as trees…
Register allocation, which is a crucial phase of a good optimizing compiler, relies on graph coloring. Hence, an efficient graph coloring algorithm is of paramount importance. In this work we try to learn a good heuristic for coloring interference graphs that are used in the register allocation phase. We aim to handle moderate sized interference graphs which have 100 nodes or less. For such graphs we can get the optimal allocation of colors to the nodes. Such optimal coloring is then used to train our Deep Learning network which is based on several layers of LSTM that output a color for each n…
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal sol…
