Detecting Points in Integer Cones of Polytopes is Double-Exponentially Hard

Let d be a positive integer. For a finite set X ⊆ ℝ^d, we define its integer cone as the set . Goemans and Rothvoss showed that, given two polytopes with Ƥ being bounded, one can decide whether intersects Q in time [J. ACM 2020], where enc(·) denotes the number of bits required to encode a polytope through a system of linear inequalities. This result is the cornerstone of their XP algorithm for Bin Packing parameterized by the number of different item sizes.

We complement their result by providing a conditional lower bound. In particular, we prove that, unless the ETH fails, there is no algorithm which, given a bounded polytope Ƥ ⊆ ℝ^d and a point q ∈ ℤ^d, decides whether in time . Note that this does not rule out the existence of a fixed-parameter tractable algorithm for the problem, but shows that dependence of the running time on the parameter d must be at least doubly-exponential.

^* This work is a part of project BOBR (ŁK, KM, MP, MS) that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 948057). A. Lassota was supported by the Swiss National Science Foundation within the project Complexity of Integer Programming (207365). The arXiv version of the paper can be accessed at https://arxiv.org/abs/2307.00406