An Improved Algorithm for a Bipartite Traveling Tournament in Interleague Sports Scheduling

TL;DR

The bipartite traveling tournament problem (BTTP) seeks a distance-minimizing inter-league schedule between two $n$-team leagues under No-repeat, direct-traveling, and bounded-by-3 constraints across $2n$ days. The authors introduce a novel 3-path construction that reduces BTTP to finding high-quality $3$-path packings, together with a new lower bound based on minimum weight cycle packings, enabling a randomized (derandomizable) $O(n^3)$-time algorithm with a theoretical approximation ratio of $\tfrac{3}{2}+\varepsilon$ for all $n$ and fixed $\varepsilon>0$. The core analysis connects the packing quality to the overall itinerary weight, and shows that with suitable packings the schedule achieves the claimed bound; a small-instance adaptation further ensures feasibility for nearly all $n$. Experimental results on an NBA-inspired 32-team instance (n=16 per league) demonstrate notable practical performance, with the 3-cycle variant achieving gaps to a strong lower bound well below the theoretical target, validating the method's applicability beyond asymptotics.

Abstract

The bipartite traveling tournament problem (BTTP) addresses inter-league sports scheduling, which aims to design a feasible bipartite tournament between two $n$-team leagues under some constraints such that the total traveling distance of all participating teams is minimized. Since its introduction, several methods have been developed to design feasible schedules for NBA, NPB and so on. In terms of solution quality with a theoretical guarantee, previously only a $(2+\varepsilon)$-approximation is known for the case that $n\equiv 0 \pmod 3$. Whether there are similar results for the cases that $n\equiv 1 \pmod 3$ and $n\equiv 2 \pmod 3$ was asked in the literature. In this paper, we answer this question positively by proposing a $(3/2+\varepsilon)$-approximation algorithm for any $n$ and any constant $\varepsilon>0$, which also improves the previous approximation ratio for the case that $n\equiv 0 \pmod 3$.

Figures (4)

• Figure 1: The super-game schedule in the first time slot, where $m=5$ and $l=2$.
• Figure 2: The super-game schedule in the second slot, where $m=5$ and $l=2$.
• Figure 3: An illustration of patching paths into a cycle: there are three paths, denoted by solid lines, and they are patched into a cycle using three edges $e_1,e_2,e_3$, denoted by dashed lines.
• Figure 4: An illustration of the locations of the 32 teams in our instance.

Theorems & Definitions (24)

• Claim 1
• proof
• Claim 2
• proof
• Claim 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof