The metric polytope is a \(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array}} \right) \) -dimensional convex polytope defined by its 4 \(\left( {\begin{array}{*{20}c} n \\ 3 \\ \end{array}} \right)… Expand

The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of S i for 1≤i<j≤d+1 and new colourful Caratheodory theorems are obtained.Expand

A colourful generalization of Liu's simplicial depth is introduced and a parity property and conjecture are proved that the minimum colourful simplicial Depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1.Expand

The Oberwolfach problem (OP) asks whether Kn (for n odd) or Kn minus a 1-factor (for n even) admits a 2-factorization where each 2-factor is isomorphic to a given 2-factor F . The order n and the… Expand

It is shown that deciding if a given sequence is the degree sequence of a 3-hypergraph is NP-complete, thereby solving a 30 year long open problem and implying that optimization over hypergraphs is hard already for simple concave functions.Expand

We consider a family of linear optimization problems over the n-dimensional Klee—Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This… Expand