Abstract
Given a compact E⊂Rn and s>0, the maximum distance problem seeks a compact and connected subset of Rn of smallest one dimensional Hausdorff measure whose s-neighborhood covers E. For E⊂R2, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius s, which cover E, solves the maximum distance problem. The main difficulty in proving this result is overcome by the proof of a key lemma which states that one is able to cover the s-neighborhood of a Lipschitz curve Γ in R2 with a finite number of balls of radius s, and connect their centers with another Lipschitz curve Γ*, where H1 (Γ*) is arbitrarily close to H1(Γ). We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at github.com/mtdaydream/MDP_MST.
Highlights
There are many variants of the traveling salesman problem in R2
For E ⊂ R2, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius s, which cover E, solves the maximum distance problem
The main difficulty in proving this result is overcome by the proof of a key lemma which states that one is able to cover the s-neighborhood of a Lipschitz curve Γ in R2 with a finite number of balls of radius s, and connect their centers with another Lipschitz curve Γ∗, where H1(Γ∗) is arbitrarily close to H1(Γ)
Summary
There are many variants of the traveling salesman problem in R2. The classic problem seeks the shortest connected tour through a finite collection of points E = {xi}Ni=1 ⊂ R2, where the points represent cities a salesman has to visit. The proof of Theorem 3.4 will follow from Lemma 3.3, which constitutes the heart of our paper This lemma says that given any > 0, the s-neighborhood of any Lipschitz curve Γ is contained in a finite number of balls of radius s, whose centers are connected by another finite continuum Γ∗ such that H1(Γ∗) is within of H1(Γ). In Lemma 3.2 we assume that E is the s-neighborhood of a line segment, and in Proposition 3.1, we assume that the s-maximum distance minimizer of E is a C1 curve, rather than merely a finite continuum as we do so in Theorem 3.4. The equivalence follows from a classic geometric measure theory result which states that any compact, connected set Γ with H1(Γ) < +∞ is 1-rectifiable, and a slightly more refined result, Theorem 2.1, stated next.
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