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The Traveling Salesman Problem for Lines, Balls, and Planes
"We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in Rd, for d ⩾ 3) or improvements over previous approximations achievable in comparable times (for unit disks in the plane). (I) Given a set of n hyperplanes in Rd, a traveling salesman problem (TSP) tour whose length is at most O(1) times the optimal can be computed in O(n) time when d is constant. (II) Given a set of n lines in Rd, a TSP tour whose length is at most O(log 3n) times the optimal can be computed in polynomial time for all d. (III) Given a set of n unit balls in Rd, a TSP tour whose length is at most O(1) times the optimal can be computed in polynomial time when d is constant."