• Corpus ID: 245836914

A Knapsack Intersection Hierarchy Applied to All-or-Nothing Flow in Trees

@inproceedings{Jozefiak2022AKI,
  title={A Knapsack Intersection Hierarchy Applied to All-or-Nothing Flow in Trees},
  author={Adam Jozefiak and F. Bruce Shepherd and Noah Weninger},
  year={2022}
}

• Adam Jozefiak, F. B. Shepherd, Noah Weninger
• Published 9 January 2022
• Computer Science, Mathematics

We introduce a natural knapsack intersection hierarchy for strengthening linear programming relaxations of packing integer programs, i.e., max{wx : x ∈ P ∩{0, 1}} where P = {x ∈ [0, 1] : Ax ≤ b} and A, b, w ≥ 0. The t level P t corresponds to adding cuts associated with the integer hull of the intersection of any t knapsack constraints (rows of the constraint matrix). This model captures the maximum possible strength of “t-row cuts”, an approach often used by solvers for small t. If A is m × n, then P is the integer hull of P and P 1 corresponds to adding cuts for each associated single-row knapsack problem. Thus, even separating over P 1 is NP-hard. However, for fixed t and any ǫ > 0, results of Pritchard imply there is a polytime (1 + ǫ)-approximation for P . We then investigate the hierarchy’s strength in the context of the well-studied all-or-nothing flow problem in trees (also called unsplittable flow on trees). For this problem, we show that the integrality gap of P t is O(n/t) and give examples where the gap is Ω(n/t). We then examine the stronger formulation Prank where all rank constraints are added. For P t rank, our best lower bound drops to Ω(1/c) at level t = n c for any c > 0. Moreover, on a well-known class of “bad instances” due to Friggstad and Gao, we show that we can achieve this gap; hence a constant integrality gap for these instances is obtained at level n.