On complexity of minimization and compression problems for models of sequential choice

AbstractA model of sequential choice of depth k for binary relations r1,r2,…,rk on a set A of alternatives relates each X⊆A to its subset Crk(…Cr2(Cr1(X))…), where Cr(Y)={y∈Y|(∀z∈Y)zr̄y}, Y⊆A. The minimization problem is of building an equivalent model of minimal depth; the compression problem is posed similarly, but the model built must satisfy some “insertion” condition. We prove that for k⩾3, the minimization and compression problems for models of depth k are NP-hard (for k=2, they are polynomial). Parameters of local algorithms solving these problems are investigated, and it is shown that the compression problem is decidable by algorithms working with neighbourhoods of size 3, whereas the minimization problem is not decidable for any finite neighbourhood size. For an arbitrary k, a model of depth k is built such that its minimization problem is not decidable with the use of neighbourhoods smaller than the whole model