A Constructive FPT Algorithm for Integer Cone Optimization with Convex Costs

Abstract

We give a deterministic fixed-parameter algorithm for integer cone optimization with convex costs. Given $\tau$ rational polytopes in ${\mathbb{R}}^d$, each with a convex cost function, we minimize the total cost of nonnegative integer combinations of their lattice points subject to arbitrary rational linear constraints on the typewise aggregates and generator counts. Our algorithm runs in time \[ 2^{τ\log(τ+1)\,2^{O(d)}}\cdot |I|^{O(1)}, \] where $|I|$ is the total binary input length. It constructs an optimal representation using at most $2^{2d+1}$ distinct generators per type. For a single generator type, the running time becomes $2^{2^{O(d)}}\cdot |I{|}^{O(1)}$. This yields constructive FPT algorithms for Cone and Polytope Intersection and high-multiplicity bin packing, improving the $2^{2^{O(d\log d)}}\cdot |I{|}^{O(1)}$ bound of Koana and Kumabe (2026) for bin-packing decision. The dependence on $d$ is ETH-tight for point-in-cone, high-multiplicity bin packing, and vector bin packing. For vector bin packing, this establishes fixed-parameter tractability in the number of item types. We also obtain new FPT parameterizations for high-multiplicity scheduling on uniform and unrelated machines: makespan minimization and ${\ell }_q$-load minimization are FPT in the number of job and machine types.