The Minimum-weight Ideal Problem for Signed Posets

The concept of signed poset has recently been introduced by V. Reiner as a generalization of that of ordinary poset (partially ordered set). We consider the problem of finding a minimum-weight ideal of a signed poset. We show a representation theorem that there exists a bijection between the set of all the ideals of a signed poset and the set of all the "reduced ideals" (defined here) of the associated ordinary poset, which was earlier proved by S. D. Fischer in his Ph.D. thesis. I t follows from this representation theorem that the minimum-weight ideal problem for a signed poset can be reduced to a problem of finding a minimum-weight (reduced) ideal of the associated ordinary poset and hence to a minimum-cut problem. We also consider the case when the weight of an ideal is defined in terms of two weight functions. The problem is also reduced to a minimum-cut problem by the same reduction technique as above. Furt,hermore, the relationship between the minimum-weight ideal problem and a certain bisubmodular function minimization problem is revealed.