Detecting Points in Integer Cones of Polytopes is Double-Exponentially Hard
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Let d be a positive integer. For a finite set X ⊆ℝ^d, we define its integer cone as the set 𝖨𝗇𝗍𝖢𝗈𝗇𝖾(X) := {∑_x ∈ Xλ_x · x |λ_x ∈ℤ_≥ 0}⊆ℝ^d. Goemans and Rothvoss showed that, given two polytopes 𝒫, 𝒬⊆ℝ^d with 𝒫 being bounded, one can decide whether 𝖨𝗇𝗍𝖢𝗈𝗇𝖾(𝒫∩ℤ^d) intersects 𝒬 in time 𝖾𝗇𝖼(𝒫)^2^𝒪(d)·𝖾𝗇𝖼(𝒬)^𝒪(1) [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 𝒫⊆ℝ^d and a point q ∈ℤ^d, decides whether q ∈𝖨𝗇𝗍𝖢𝗈𝗇𝖾(𝒫∩ℤ^d) in time 𝖾𝗇𝖼(𝒫, q)^2^o(d). 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.
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