The incomplete Traveling Tournament Problem

Abstract

We present a new problem called the incomplete Traveling Tournament problem, which introduces the well known Traveling Tournament Problem into the realm of incomplete round-robin tournaments. We focus on the case where teams can face each opponent at most once. We give a formal description of this problem and show that it is NP-hard. We first discuss how we can obtain lower bounds and how to strengthen them. Then, we propose two integer programming formulations and compare their LP-relaxations. We also propose a third formulation that assumes that home-away patterns of teams are fixed. We discuss how a recently proposed metaheuristic for incomplete round-robin scheduling can be tailored to our problem. In doing so, we present a novel neighborhood structure and show it fully connects the home-away pattern solution space. Finally, problem instances are proposed, for which we derive lower and upper bounds. We show that these instances are challenging, making the development of efficient algorithms for the incomplete Traveling Tournament problem an interesting direction for future research.

Figures (2)

• Figure 1: Example of the ILB for an iTTP with 6 teams and 4 rounds. The optimal travel distance of dotted teams is when they mutually travel to each other. The optimal travel distance of teams 4 and 5 is when they travel to each other and to team 3. Team 6 is not visited by any team according to the ILB.
• Figure 2: \ref{['example1']}, which shows that constructive greedy matching does not allow a bounded approximation ratio. Only the matches with a travel distance of 1 or 2 are depicted.

Theorems & Definitions (11)

• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• Theorem 4.2
• proof
• Example 1
• Theorem 4.3
• proof