The computational complexity of deciding if a given set of points in $\mathcal{Z}^d $ has an integral Radon partition is discussed, and it is shown that if d is fixed, then this problem is in P, while ifd is part of the input, it is NP-complete.Expand

It is shown that ILP is FPT parameterized by the largest coefficient $\|A\|_\infty$ and the primal or dual treedepth of $A$, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP.Expand

An efficient iterative approximation algorithm is described, that finds a colourful T whose convex hull contains a point e-close to b, and its real arithmetic and Turing time complexities are analyzed.Expand

An algorithm which runs in time O (n3L) having cubic dependency on n regardless of the bimatrix A is established and can be used to define a hierarchy of approximations for any integer programming problem.Expand

It is shown that (computable) certificates exist for any algorithm that is capable of producing solutions of guaranteed accuracy, and how the implementation of the ellipsoid method and other cutting plane algorithms can be augmented with the computation of such certificates.Expand

This work considers the problem of maximizing convex functionals over matroid bases and shows that it is efficiently solvable when a suitable parameter is restricted.Expand

It is shown that integer programming can be solved in time, and a strongly-polynomial algorithm is derived, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data.Expand