The Traveling Tournament Problem: Improved Algorithms Based on Cycle Packing

TL;DR

The paper tackles the Traveling Tournament Problem with a bound k on consecutive home/away games (TTP-k) by developing a novel k-cycle packing construction and integrating it with a refined Hamiltonian-cycle approach. By deriving stronger independent lower bounds and carefully analyzing two complementary constructions, the authors achieve improved approximation ratios across variants: TTP-3 is tightened to 139/87 + ε, TTP-4 to 17/10 + ε, and general TTP-k (k ≥ 5) to (5k^2−4k+3)/(2k(k+1)) + ε; LDTTP-k also benefits with a cycle-packing bound of (3k−3)/(2k−1) + ε and LDTTP-3 improving to 6/5 + ε. The methods hinge on cycle packing, random labeling with derandomization, and a careful combination of two schedule constructions to exploit structural trade-offs. These results advance the state of the art in approximation guarantees for TTP variants and highlight how cycle packing can yield practical improvements in sports-timetabling problems. The work also provides a pathway to tighter lower bounds and potential further gains via improved packing algorithms.

Abstract

The Traveling Tournament Problem (TTP) is a well-known benchmark problem in the field of tournament timetabling, which asks us to design a double round-robin schedule such that each pair of teams plays one game in each other's home venue, minimizing the total distance traveled by all $n$ teams ($n$ is even). TTP-$k$ is the problem with one more constraint that each team can have at most $k$-consecutive home games or away games. In this paper, we investigate schedules for TTP-$k$ and analyze the approximation ratio of the solutions. Most previous schedules were constructed based on a Hamiltonian cycle of the graph. We will propose a novel construction based on a $k$-cycle packing. Then, combining our $k$-cycle packing schedule with the Hamiltonian cycle schedule, we obtain improved approximation ratios for TTP-$k$ with deep analysis. The case where $k=3$, TTP-3, is one of the most investigated cases. We improve the approximation ratio of TTP-3 from $(1.667+\varepsilon)$ to $(1.598+\varepsilon)$, for any $\varepsilon>0$. For TTP-$4$, we improve the approximation ratio from $(1.750+\varepsilon)$ to $(1.700+\varepsilon)$. By a refined analysis of the Hamiltonian cycle construction, we also improve the approximation ratio of TTP-$k$ from $(\frac{5k-7}{2k}+\varepsilon)$ to $(\frac{5k^2-4k+3}{2k(k+1)}+\varepsilon)$ for any constant $k\geq 5$. Our methods can be extended to solve a variant called LDTTP-$k$ (TTP-$k$ where all teams are allocated on a straight line). We show that the $k$-cycle packing construction can achieve an approximation ratio of $(\frac{3k-3}{2k-1}+\varepsilon)$, which improves the approximation ratio of LDTTP-3 from $4/3$ to $(6/5+\varepsilon)$.

Figures (6)

• Figure 1: The schedule in the first time slot for an instance with $m=10$ and $r=2$
• Figure 2: The schedule in the second time slot for an instance with $m=10$ and $r=2$
• Figure 3: The schedule in the last time slot for an instance with $m=10$ and $r=2$
• Figure 4: The unarranged normal games between super-teams $U_i=\{x^i_1,x^i_2,x^i_3\}\cup\{x^i_4,x^i_5,x^i_6\}$, $U_j=\{x^j_1,x^j_2,x^j_3\}\cup\{x^j_4,x^j_5,x^j_6\}$
• Figure 5: The unarranged normal games on the cycle $C=(U_{i'_1},U_{i'_2},\dots, U_{i'_p}, U_{i'_1})_{i'}$, where we only consider two normal teams $\{x^i_1,x^i_{2k}\}$ for each super-team $U_{i'_i}$, $p=9$, and $k=3$

Theorems & Definitions (71)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3: christofides1976worstserdyukov1978some
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6