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        "description": "The approximation scheme concerns Euclidean TSP in fixed dimension with fixed requested accuracy. Its randomized structural theorem succeeds with a stated probability over shifts, not for every shift. Portals, crossing bounds, preprocessing and rounding have costs; the method is neither an exact solver nor a uniform efficient scheme in dimension/accuracy. General metric TSP and later practical improvements are separate. Theoretical parallelism is not a measured hardware speedup.",
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      "sources": [
        {
          "citation": "Arora: Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems (1998 article family)",
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      "statement": "# Arora’s geometric approximation scheme for the traveling salesman\n\nA randomly shifted geometric dissection has a near-optimal Euclidean tour with few boundary crossings; portal-based dynamic programming gives an approximation scheme in any fixed dimension.\n\nA delivery route can have far too many possible visiting orders to check them all. Arora showed that, when locations lie in a fixed-dimensional geometric space, one can search a much smaller structured family and still obtain a route arbitrarily close to the optimum. Randomly shifted squares and a limited set of boundary portals make the smaller problems fit together. For fixed accuracy, the planar algorithm has nearly linear asymptotic running time, with substantial dependence on the requested accuracy. The guarantee applies to approximation, not exact optimality, and does not extend to every abstract distance table. It succeeds with probability at least one half for a random dissection; repeating independent trials raises the success probability. The paper acknowledges that straightforward implementations are slow. Mitchell independently developed a related planar approximation scheme, while earlier work supplied partitioning, patching and approximation ideas. WRS has not independently implemented the full algorithm or certified the proof.\n\n## Scientific scope\n\nThe approximation scheme concerns Euclidean TSP in fixed dimension with fixed requested accuracy. Its randomized structural theorem succeeds with a stated probability over shifts, not for every shift. Portals, crossing bounds, preprocessing and rounding have costs; the method is neither an exact solver nor a uniform efficient scheme in dimension/accuracy. General metric TSP and later practical improvements are separate. Theoretical parallelism is not a measured hardware speedup.\n\n## People and systems\n\nSanjeev Arora\n\n\n\n## Original sources\n\n- [Arora: Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems (1998 article family)](https://graphics.stanford.edu/courses/cs468-06-winter/Papers/arora-tsp.pdf)\n",
      "summary": "A randomly shifted geometric dissection has a near-optimal Euclidean tour with few boundary crossings; portal-based dynamic programming gives an approximation scheme in any fixed dimension.\n\nA delivery route can have far too many possible visiting orders to check them all. Arora showed that, when locations lie in a fixed-dimensional geometric space, one can search a much smaller structured family and still obtain a route arbitrarily close to the optimum. Randomly shifted squares and a limited set of boundary portals make the smaller problems fit together. For fixed accuracy, the planar algorithm has nearly linear asymptotic running time, with substantial dependence on the requested accuracy. The guarantee applies to approximation, not exact optimality, and does not extend to every abstract distance table. It succeeds with probability at least one half for a random dissection; repeating independent trials raises the success probability. The paper acknowledges that straightforward implementations are slow. Mitchell independently developed a related planar approximation scheme, while earlier work supplied partitioning, patching and approximation ideas. WRS has not independently implemented the full algorithm or certified the proof.",
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