Semidefinite Programming Relaxations in Timetabling (Abstract)

Definitions In linear programming (LP), the task is to optimise a linear combination cT x subject to linear constraints Ax = b, together with the constraint that each in vector x of n variables is nonnegative. The non-negativity of x, x ∈ (R+)n, can be seen be seen as a restriction of the variables to lie in the convex cone of the positive orthant. Using interior point methods, linear programming can be solved to any fixed precision in polynomial time. These methods also work for other symmetric convex cones. Semidefinite programming (SDP, Bellman & Fan, 1963; Alizadeh, 1995; Wolkowicz, Saigal, & Vandenberghe, 2000) is a generalisation of linear programming, replacing the vector variable with a square symmetric matrix variable and the polyhedral symmetric convex cone of the positive orthant with the non-polyhedral symmetric convex cone of positive semidefinite matrices. Note that an n× n matrix, M, is positive semidefinite if and only if yT My ≥ 0 for all y ∈ Rn. As all scalar multiplies of positive semidefinite matrices and convex combinations of pairs of positive semidefinite matrices are positive semidefinite, positive semidefinite matrices do form a convex cone in Rn 2 . We denote Aર B whenever A−B is positive semidefinite, and use ⟨A,B⟩ for the inner product of matrices, which is ∑i, j Ai, jB j,i. Formally, semidefinite programming is the minimisation of ⟨C,X⟩ such that ⟨Ai,X⟩= bi ∀i = 1 . . .m and X ર 0, where X is a (primal) square symmetric matrix variable, C and Ai are compatible symmetric matrices, m is the number of constraints, and b ∈Rm. Let us now consider a simple model of timetabling, underlying integer programming decompositions (Burke, Mareček, Parkes, & Rudová, 2010), for instance. The input consists of identifiers of events V, distinct enrolments U (“curricula”), rooms R, and periods P, plus mapping F : U→ 2V ∖ / 0 from “curricula” to non-empty sets of events. Conflict graph G = (V,E) is given