Completions of Alternating Sign Matrices

We consider the problem of completing a ( 0 , - 1 ) -matrix to an alternating sign matrix (ASM) by replacing some 0 s with - 1 s. An algorithm can be given to determine a completion or show that one does not exist. We are concerned primarily with bordered-permutation ( 0 , - 1 ) matrices, defined to be n × n ( 0 , - 1 ) -matrices with only 0 s in their first and last rows and columns where the - 1 s form an ( n - 2 ) × ( n - 2 ) permutation matrix. We show that any such matrix can be completed to an ASM and characterize those that have a unique completion.