# Semidefinite Programming in Timetabling and Mutual-Exclusion Scheduling

## TL;DR

This paper develops semidefinite programming relaxations for complex mutual-exclusion scheduling and timetabling problems, providing strong theoretical bounds and demonstrating promising computational results on various graph types and benchmarks.

## Contribution

It introduces SDP relaxations for mutual-exclusion scheduling with multiple constraints, extending known bounds and applying them to practical timetabling problems.

## Key findings

- Strongest known bounds for these problems in polynomial time
- Encouraging computational results on diverse graph types
- Effective SDP relaxations for complex constraints

## Abstract

In scheduling and timetabling applications, the mutual-exclusion constraint stipulates that certain pairs of tasks that cannot be executed at the same time. This corresponds to the vertex colouring problem in graph theory, for which there are well-known semidefinite programming (SDP) relaxations. In practice, however, the mutual-exclusion constraint is typically combined with many other constraints, whose SDP representability has not been studied. We present SDP relaxations for a variety of mutual-exclusion scheduling and timetabling problems, starting from a bound on the number of tasks executed within each period, which corresponds to graph colouring bounded in the number of uses of each colour. In theory, this provides the strongest known bounds for these problems that are computable to any precision in time polynomial in the dimensions. In practice, we report encouraging computational results on random graphs, Knesser graphs, ``forbidden intersection'' graphs, the Toronto benchmark, and the International Timetabling Competition.