Multi-index planar transportation polytopes with maximum number of vertices

Asymptotic behaviour of multi-index axial transportation polyhedra

“…In the case where ρ -2 and the number of restrictions t < c min{m/ In ra, n/ In n}, where c is some positive constant that does not depend on m and n, this assertion was first established in [6,7].…”

“…The following theorem gives a positive answer to the conjecture concerning the asymptotic behaviour of the class of CTPs of maximal diameter formulated in [6]. …”

“…The negative answer was given in [2][3][4], whereas in [2,5] it was shown that almost all CTPs possess the maximal numbers of facets (faces of maximal dimensionality). Recently, in [6][7][8] In this paper, the latter result is extended to the case of p-index, p > 2, axial transportation polyhedra (p-ATPs). Besides, it is proved that almost all p-ATPs are of diameter no less than Σ£=Ι n s -p + 1, where n\ χ HI χ ... χ η ρ is the order of the p-ATP.…”

“…In [6,7], a criterion was given for belonging a polyhedron of order m χ η to the class &k(m,ri) of the non-degenerate CTPs with the maximal number of &-faces for all k 6 {0, 1, ... ,d -1}, where d = (in -\}(n -1). It is interesting to clear up the asymptotic behaviour of these classes as m and n tend to infinity.…”

“…Then, by virtue of Theorem 2 and Theorem 1 of [6], in order to prove Theorem 3, we have to choose a pair (/,/) € K (a',b',a*,b*) satisfying the inequality …”