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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
Pebble Guided Near Optimal Treasure Hunt in Anonymous Graphs
Barun Gorain, Kaushik Mondal, Himadri Nayak, Supantha Pandit · 2021-03-10 · via cs.DS updates on arXiv.org

We study the problem of treasure hunt in a graph by a mobile agent. The nodes in the graph are anonymous and the edges at any node $v$ of degree $deg(v)$ are labeled arbitrarily as $0,1,\ldots, deg(v)-1$. A mobile agent, starting from a node, must find a stationary object, called {\it treasure} that is located on an unknown node at a distance $D$ from its initial position. The agent finds the treasure when it reaches the node where the treasure is present. The {\it time} of treasure hunt is defined as the number of edges the agent visits before it finds the treasure. The agent does not have any prior knowledge about the graph or the position of the treasure. An Oracle, that knows the graph, the initial position of the agent, and the position of the treasure, places some pebbles on the nodes, at most one per node, of the graph to guide the agent towards the treasure. We target to answer the question: what is the fastest possible treasure hunt algorithm regardless of the number of pebbles are placed? We show an algorithm that uses $O(D \log Δ)$ pebbles to find the treasure in a graph $G$ in time $O(D \log Δ+ \log^3 Δ)$, where $Δ$ is the maximum degree of a node in $G$ and $D$ is the distance from the initial position of the agent to the treasure. We show an almost matching lower bound of $Ω(D \log Δ)$ on time of the treasure hunt using any number of pebbles.