The nonidealness index of circulant matrices

Ideal matrices are precisely those matrices M where the set covering polyhedron Q ∗ ( M ) equals the polyhedron Q ( M ) = { x : M x ≥ 1 , x ≥ 0 } . In a previous work (2006) we defined a nonidealness index equivalent to max { t : Q ( M ) ⊂ t Q ∗ ( M ) } . Given an arbitrary matrix M the nonideal index is NP-hard to compute and for most matrices it remains unknown. known family of minimally nonideal matrices is the one of the incidence matrices of chordless odd cycles. A natural generalization of them is given by circulant matrices. Circulant ideal matrices have been completely identified by Cornuéjols and Novick (1994). In this work we obtain a bound for the nonidealness index of circulant matrices and determine it for some particular cases.