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PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Simpler and More General Distributed Coloring Based on Si...
Marc Fuchs, Fabian Kuhn · 2024-05-08 · via cs.DS updates on arXiv.org

In this paper, we give list coloring variants of simple iterative defective coloring algorithms. Formally, in a list defective coloring instance, each node $v$ of a graph is given a list $L_v$ of colors and a list of allowed defects $d_v(x)$ for the colors. Each node $v$ needs to be colored with a color $x\in L_v$ such that at most $d_v(x)$ neighbors of $v$ also pick the same color $x$. For a defect parameter $d$, it is known that by making two sweeps in opposite order over the nodes of an edge-oriented graph with maximum outdegree $β$, one can compute a coloring with $O(β^2/d^2)$ colors such that every node has at most $d$ outneighbors of the same color. We generalize this and show that if all nodes have lists of size $p^2$ and $\forall v:\sum_{x\in L_v}(d_v(x)+1)>p\cdotβ$, we can make two sweeps of the nodes such that at the end, each node $v$ has chosen a color $x\in L_v$ for which at most $d_v(x)$ outneighbors of $v$ are colored with color $x$. Our algorithm is simpler and computationally significantly more efficient than existing algorithms for similar list defective coloring problems. We show that the above result can in particular be used to obtain an alternative $\tilde{O}(\sqrtΔ)+O(\log^* n)$-round algorithm for the $(Δ+1)$-coloring problem in the CONGEST model. The neighborhood independence $θ$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. It is known that by doing a single sweep over the nodes of a graph of neighborhood independence $θ$, one can compute a $d$-defective coloring with $O(θ\cdot Δ/d)$ colors. We extend this approach to the list defective coloring setting and use it to obtain an efficient recursive coloring algorithm for graphs of neighborhood independence $θ$. In particular, if $θ=O(1)$, we get an $(\logΔ)^{O(\log\logΔ)}+O(\log^* n)$-round algorithm.