In some domestic professional sports leagues, the home stadiums are located in cities connected by a common train line running in one direction. For these instances, we can incorporate this geographical information to determine optimal or nearly-optimal solutions to the n-team Traveling Tournament Problem (TTP), an NP-hard sports scheduling problem whose solution is a double round-robin tournament schedule that minimizes the sum total of distances traveled by all n teams. We introduce the Linear Distance Traveling Tournament Problem (LD-TTP), and solve it for n=4 and n=6, generating the complete set of possible solutions through elementary combinatorial techniques. For larger n, we propose a novel "expander construction" that generates an approximate solution to the LD-TTP. For n congruent to 4 modulo 6, we show that our expander construction produces a feasible double round-robin tournament schedule whose total distance is guaranteed to be no worse than 4/3 times the optimal solution, regardless of where the n teams are located. This 4/3-approximation for the LD-TTP is stronger than the currently best-known ratio of 5/3 + epsilon for the general TTP. We conclude the paper by applying this linear distance relaxation to general (non-linear) n-team TTP instances, where we develop fast approximate solutions by simply "assuming" the n teams lie on a straight line and solving the modified problem. We show that this technique surprisingly generates the distance-optimal tournament on all benchmark sets on 6 teams, as well as close-to-optimal schedules for larger n, even when the teams are located around a circle or positioned in three-dimensional space.

We introduce a linear distance relaxation of the n-team Traveling Tournament Problem (TTP), a simple yet powerful heuristic that temporarily "assumes"' the n teams are located on a straight line, thereby reducing the n ( n –1)/2 pairwise distance parameters to just n –1 variables. The modified problem then becomes easier to analyze, from which we determine an approximate solution for the actual instance on n teams. We present combinatorial techniques to solve the Linear Distance TTP (LD-TTP) for n = 4 and n = 6, without any use of computing, generating the complete set of optimal distances regardless of where the n teams are located. We show that there are only 295 non-isomorphic schedules that can be a solution to the 6-team LD-TTP, and demonstrate that in all previously-solved benchmark TTP instances on 6 teams, the distance-optimal schedule appears in this list of 295, even when the six teams are arranged in a circle or located in three-dimensional space. We then extend the LD-TTP to multiple rounds, and apply our theory to produce a nearly-optimal regular-season schedule for the Nippon Pro Baseball league in Japan. We conclude the paper by generalizing our theory to the n -team LD-TTP, producing a feasible schedule whose total distance is guaranteed to be no worse than 4/3 times the optimal solution.