Generating Approximate Solutions to the TTP using a Linear Distance Relaxation

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.

💡 Research Summary

The paper tackles the Traveling Tournament Problem (TTP), a classic NP‑hard scheduling problem in which n sports teams must play a double round‑robin tournament while minimizing the total travel distance of all teams. Recognizing that many real‑world leagues have stadiums aligned along a single train line, the authors introduce a linear‑distance relaxation, the Linear Distance Traveling Tournament Problem (LD‑TTP), in which each team’s venue is placed on a one‑dimensional line and the distance between teams i and j is simply |x_i – x_j|. This metric collapses the triangle inequality to an equality, dramatically simplifying the combinatorial structure of the problem.

The first contribution is an exhaustive enumeration of optimal schedules for the smallest non‑trivial instances, n = 4 and n = 6. By generating all possible round‑robin matchings, imposing the usual home‑away balance and no‑repeat‑venue constraints, and evaluating the total linear travel cost, the authors identify the complete set of feasible schedules (2 for n = 4, 12 for n = 6) and compute the exact optimal distances. These tiny instances serve as a benchmark for later approximation results.

The core of the paper is a novel “expander construction” that yields a feasible double round‑robin schedule for any n congruent to 4 modulo 6. The construction proceeds by splitting the teams into two equal halves, arranging intra‑half matchings as simple cyclic permutations, and then inserting cross‑half matchings so that each team meets every opponent exactly once in each direction. Home‑away assignments are alternated round by round, guaranteeing the standard TTP constraints. The authors prove that the total travel distance D_expander produced by this scheme never exceeds 4/3 times the optimal linear‑distance cost D_opt, regardless of the actual placement of the teams on the line. This 4/3 approximation ratio improves on the best known bound for the general TTP (5/3 + ε) and holds uniformly for all possible linear configurations.

Having established a strong theoretical guarantee for the linear case, the authors explore how the LD‑TTP solution can be leveraged for ordinary, non‑linear TTP instances. Their heuristic simply projects the actual two‑ or three‑dimensional team locations onto a line (using sorting or a least‑squares regression line), solves the LD‑TTP exactly on the projected coordinates, and then maps the resulting schedule back to the original geometry, recomputing travel costs with the true Euclidean distances. This “assume‑a‑line” approach does not alter the schedule structure but adapts the distance evaluation to the real layout.

Experimental evaluation on standard six‑team benchmark sets—including circular, random planar, and random three‑dimensional placements—shows that the line‑based schedule coincides with the true optimal schedule in every case. For larger instances (n = 8, 10, 12) the method consistently yields solutions within 2–5 % of the best known distances, and even on a perfect circle the observed factor is only about 1.12, far below the theoretical 4/3 bound. These results suggest that the linear relaxation captures much of the essential structure of the TTP, even when the underlying geography is far from linear.

The paper concludes by outlining future research directions: extending the expander construction to the other residue classes (n ≡ 0 or 2 mod 6), handling multiple venues or neutral sites, and applying the methodology to real league data such as Major League Baseball or the Korean Baseball Organization. Overall, the work provides a fresh theoretical lens—linear distance relaxation—through which the notoriously hard TTP can be approximated more tightly, and demonstrates that the resulting schedules are practically competitive across a wide range of realistic settings.