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cs.DS updates on arXiv.org

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
Closed form distance formula for the balanced multiple tr...
Wolfgang Garn · 2020-01-22 · via cs.DS updates on arXiv.org

As a first contribution the mTSP is solved using an exact method and two heuristics, where the number of nodes per route is balanced. The first heuristic uses a nearest node approach and the second assigns the closest vehicle (salesman). A comparison of heuristics with test-instances being in the Euclidean plane showed similar solution quality and runtime. On average, the nearest node solutions are approximately one percent better. The closest vehicle heuristic is especially important when the nodes (customers) are not known in advance, e.g. for online routing. Whilst the nearest node is preferable when one vehicle has to be used multiple times to service all customers. The second contribution is a closed form formula that describes the mTSP distance dependent on the number of vehicles and customers. Increasing the number of salesman results in an approximately linear distance growth for uniformly distributed nodes in a Euclidean grid plane. The distance growth is almost proportional to the square root of number of customers (nodes). These two insights are combined in a single formula. The minimum distance of a node to $n$ uniformly distributed random (real and integer) points was derived and expressed as functional relationship dependent on the number of vehicles. This gives theoretical underpinnings and is in agreement with the distances found via the previous mTSP heuristics. Hence, this allows to compute all expected mTSP distances without the need of running the heuristics.