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

Semidefinite Programming in Timetabling and Mutual-Exclusion Scheduling