The convex hull of transposition matrices

Abstract

Many combinatorial optimization proble ms can be expressed as requiring the ex tre mization of a linear fun ction over some finite set S of points in a real N-dime nsion al space. To bring the theoretical and computational resources of linear programming to bear, it is necessary to cha racterize the convex hull K (S) of S as the solutio n-set of a " ni cely structured" family of linear inequalities and equations. The outs tanding example , arising in connec ti on with the assignment prob le m of operations research , has S = S", the se t of n by n permutation matrices (regarded as points in n2 dim ension rll s pace). Here a we ll-known theore m I (Birkhoff-Hoffman-vo n Ne umann -W ielan dt et al.) identifies K (5,,) as the set of all n by n doubly stochasti c matrices X = (Xi), i.e. matrices with nonnegative e ntries and with e ach row and column summing to 1. It is expected th at a si milar cl 'aracte r ization of K ( Cn ), where Cll is the set of all cyclic permutation matrices, would be va lua ble in connection with the travelin g saleman proble m, but no such c harac teri zation has been given as ye t. For a given n le t TIc(l ~ c ~ n) de note the se t of n by n permutation matrices for which the decomposition of the associated perm utation into di sjoin t cycles conta in s exactly c cycles (i ncl udin g cycles of le ngth o ne). Sin ce Cn = TIl , the re mark endi ng the last paragraph suggests lookin g at th e "other e nd" of th e seque nce {nc}~= I' The situ ations for TIn and TInI are simp le, a nd form the s ubject of thi s note. Clearly TIll consists of the identity matrix In, so that K (TI,,) = {JII}' We go on to characterize K CTIn l ) as well as K(TIn1 U TI,, ) = K(TIn 1 U { In}) . Note that llll I consists of the n(n -1)/2 transposition matrices Tpq(l ~ p < q ~ n) defined by