The Traveling Salesman Problem for Lines, Balls, and Planes
Adrian Dumitrescu, Csaba D. Tóth · ACM Transactions on Algorithms · 2016
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 R d , 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 R d , 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 R d , a TSP tour whose length is at most O (log 3 n ) times the optimal can be computed in polynomial time for all d . (III) Given a set of n unit balls in R d , a TSP tour whose length is at most O (1) times the optimal can be computed in polynomial time when d is constant.