Parametric nonlinear discrete optimization over well-described sets and matroid intersections

Computation in multicriteria matroid optimization

“…Moreover the author showed that the vertices of W P M can be found in polynomial time. A similar result can be found in [Berstein et al 2009]. Algorithm 4 can (and does in practice) pick up non-vertex integral boundary points.…”

“…. , a p ) is a p-tuple of distinct positive integers a k that are binary encoded, and the integers W The following lemma is a variation on facts pointed out in [Berstein et al 2009]. With W having a generalized unary encoding, it is simple to bound the size of #W e(B M ).…”

“…In order to present a general situation for which some of our heuristics are efficient, we consider the "generalized unary encoding" of the weight matrix W ∈ R d×n introduced in [Berstein et al 2009]. We consider weights W i,j of the form…”

“…Likewise for optimizing f or with a min-max objective, we simply evaluate over all projected bases found as we go through them. This is the same methodology as [Berstein et al 2009] where the · 9 authors prove efficient deterministic algorithms, given sufficient conditions on the input, to list all projected bases. Efficiency follows due to assumptions on the input, i.e.…”

“…In [Berstein et al 2009], the authors used matroid intersections to solve the problem: Given x ∈ R d find B ∈ B M such that W e(B) = x if such a B exists. Guided by the success of our previous heuristics, we had the idea of using either Heuristic 2 or Heuristic 1 to find a B ∈ B M such that W e(B) = x.…”