Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Faster Deterministic Approximation Algorithm for TTP-2

Yuga Kanaya, Kenjiro Takazawa

TL;DR

This work addresses the TTP-2 variant for $n \equiv 2$ (mod 4) by presenting a faster deterministic approximation algorithm with ratio $1+9/n$ and running time $O(n^3)$. It blends Zhao and Xiao's tournament construction with Imahori-style numbering to tightly bound travel distances, deriving key upper bounds that relate the algorithm’s performance to the fundamental lower bound $\Delta+n\cdot d(M)$ based on a minimum-weight perfect matching $M$. The main contribution is a improved deterministic guarantee at the same $O(n^3)$ time as prior deterministic methods, outperforming earlier deterministic results with similar complexity and narrowing the gap to optimal schedules. This yields a scalable, provably near-optimal scheduling method applicable to large $n$ under the $n \equiv 2$ (mod 4) constraint, with clear directions for further tightening and potential complexity considerations.

Abstract

The traveling tournament problem (TTP) is to minimize the total traveling distance of all teams in a double round-robin tournament. In this paper, we focus on TTP-2, in which each team plays at most two consecutive home games and at most two consecutive away games. For the case where the number of teams $n\equiv2$ (mod 4), Zhao and Xiao (2022) presented a $(1+5/n)$-approximation algorithm. This is a randomized algorithm running in $O(n^3)$ time, and its derandomized version runs in $O(n^4)$ time. In this paper, we present a faster deterministic algorithm running in $O(n^3)$ time, with approximation ratio $1+9/n$. This ratio improves the previous approximation ratios of the deterministic algorithms with the same time complexity.

A Faster Deterministic Approximation Algorithm for TTP-2

TL;DR

This work addresses the TTP-2 variant for (mod 4) by presenting a faster deterministic approximation algorithm with ratio and running time . It blends Zhao and Xiao's tournament construction with Imahori-style numbering to tightly bound travel distances, deriving key upper bounds that relate the algorithm’s performance to the fundamental lower bound based on a minimum-weight perfect matching . The main contribution is a improved deterministic guarantee at the same time as prior deterministic methods, outperforming earlier deterministic results with similar complexity and narrowing the gap to optimal schedules. This yields a scalable, provably near-optimal scheduling method applicable to large under the (mod 4) constraint, with clear directions for further tightening and potential complexity considerations.

Abstract

The traveling tournament problem (TTP) is to minimize the total traveling distance of all teams in a double round-robin tournament. In this paper, we focus on TTP-2, in which each team plays at most two consecutive home games and at most two consecutive away games. For the case where the number of teams (mod 4), Zhao and Xiao (2022) presented a -approximation algorithm. This is a randomized algorithm running in time, and its derandomized version runs in time. In this paper, we present a faster deterministic algorithm running in time, with approximation ratio . This ratio improves the previous approximation ratios of the deterministic algorithms with the same time complexity.
Paper Structure (20 sections, 2 theorems, 10 equations, 42 figures, 7 tables)

This paper contains 20 sections, 2 theorems, 10 equations, 42 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower Bound
  4. Construction of the Tournament by Zhao and Xiao
  5. Normal super-games:
  6. Left super-games:
  7. Right super-games:
  8. The last slot:
  9. Our Algorithm
  10. Numbering of the Teams
  11. Analyzing the Approximation Ratio
  12. Category $\mathrm{(I)}$: Super-team $u_{1}$.
  13. Category $\mathrm{(II)}$: Super-teams $u_{i}$ $(i=2,4,\ldots,m-3)$.
  14. Category $\mathrm{(III)}$: Super-teams $u_{i}$ $(i=3,5,\ldots,m-4)$.
  15. Category $\mathrm{(IV)}$: Super-team $u_{m-2}$.
  16. ...and 5 more sections

Key Result

Theorem 1

The approximation ratio of our algorithm is $1+9/n$.

Figures (42)

  • Figure 1: Arrangement of super-teams in the first slot when $n\equiv2$ (mod 8)
  • Figure 2: The eight games in a normal super-game
  • Figure 3: The eight games in a left super-games
  • Figure 4: The eight games in the special left super-game in the $(m-2)$-th slot
  • Figure 5: Four types of right super-games
  • ...and 37 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2