Math Levels the Playing Field for Sports Scheduling

For college student-athletes, a sports schedule can be a major challenge to juggle with the rest of their priorities.

Traveling for extended periods, recovering from tough competition: These don’t always affect every team of a conference equally. Some teams might have to travel further, play more games in fewer days, or have fewer home events than other members of their conference.

Now, a Trinity University mathematics duo is implementing an innovative scheduling method in the DIII Southern Athletic Association (SAA) that aims to create the most level playing field possible. It’s called “atomic scheduling,” and it’s being pioneered by math professor Eduardo Cabral Balreira, Ph.D., and Jared Gallagher ’25, a double major in math and mathematical finance who runs track and field and cross country for Trinity.

“The traditional way to schedule, the way they do it in the NFL or the NBA, is called ‘integer programming,’ where you let the computer run for a long period of time, coming up with the best possible solution for a schedule with given constraints,” Balreira says. “But with atomic scheduling, which is our ‘graph theoretical’ or ‘combinatorial’ approach, we translate constraints into priorities to generate a finite set of viable options. This lowers the search time significantly and allows the commissioner to select the schedule that best satisfies the most important priorities for each sport.”

In a perfect world, Balreira says, members of Trinity’s SAA conference would simply randomize the matchups for every sport each year. But the real world doesn’t work that way: There are constraints of logistics, facility availability, and physical limitations, too. And then there’s the headache of scheduling each program around the other programs in its athletic department.

“Particular campuses can’t have too many events happening at the same time. Football needs to coordinate with volleyball or soccer,” Balreira says. “Then you have the issue that many teams want to minimize the number of travels to Texas because it’s always a flight and it is expensive; we look for ways that each team can play all its Texas matchups in the same trip. ”

However, Gallagher notes that the atomic model flourishes under these complications because each constraint narrows down the number of possibilities.

“Does every team need a bye (a break) before they play a big week? Do teams need a home game in the last couple of weeks of the season? You can't have three weekends in a row on the road … you have to have a home game in the last two weeks. Those are some priorities we've been given,” Gallagher says. “Now, we find schedules that meet all those required constraints, and we have this list of possible schedules to meet all priorities.”

There are still some inherent challenges that even math can’t fix, Balreira notes. However, the atomic formula can minimize these injustices.

“As we’re doing the schedule for soccer, there will be a team every year that's going to have to play four games over the span of eight days,” Balreira says. “There's no way mathematically that that can be avoided. So fairness here would be to make sure that this team is different every year.”

SAA commissioner Stacey LaDew notes that this research has proven “tremendously helpful” to the conference. 

"Cabral Balreira and his student team have been invaluable partners to the SAA. Thanks to their use of technology and advanced modeling, we can evaluate millions of scheduling possibilities in a fraction of the time,” Ladew says. “There’s no denying that the makeup and geographic spread of our conference presents unique challenges, but working with the Trinity team has allowed us to be both creative and innovative in meeting the priorities of the league. We’ve been able to craft schedules that are not only optimized for equity and fairness, but also aligned with the mission and values that define the SAA."

Gallagher says this is a project of paramount importance for all student-athletes.

“At the college level, we see it all the time that playing a bad schedule might affect your chances of making it to the postseason,” Gallagher says. “So I really liked the idea of making a fair schedule, where everyone starts on a level playing field. I think athletes should just be able to go compete and see what they can do, but it's difficult for teams that the schedule doesn't favor.”

And this is the type of project that reaffirms his decision to try DIII sports at a place like Trinity. Gallagher eventually wants to work in the sports industry, perhaps in sports analytics or logistics, and this research has been the latest in a long string of experiences that are propelling him toward a career in sports.

At Trinity, he’s gotten to do data analysis with the soccer team, while the scheduling project has delved into the logistics side of sports and graph theory. And Gallagher also had an internship in summer 2024 branching into the business side of sports. 

“It's cool to be able to see all the different sides of sports and how different fields of math go into each of those,” Gallagher says. “I think the liberal arts give you the chance to take courses in all sorts of different fields of math and business and sports, and it wouldn't be possible anywhere else to have that many types of different experiences.”