A Constructive FPT Algorithm for Integer Cone Optimization with Convex Costs
Klaus Jansen, Felix Ohnesorge, Björn Schumacher
原文摘要
We give a deterministic fixed-parameter algorithm for integer cone optimization with convex costs. Given $τ$ rational polytopes in $\R^{d+k}$, we combine generators whose first $d$ coordinates are integral and whose remaining $k$ coordinates may be fractional. Each generator has a convex cost depending only on its integer coordinates. We minimize the total cost of nonnegative integer combinations subject to arbitrary rational linear constraints on the full 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. The number $k$ of continuous coordinates contributes only to the polynomial input factor. For a single generator type, the running time becomes $2^{2^{O(d)}}\cdot |I|^{O(1)}$. Applications include constructive FPT algorithms for Cone and Polytope Intersection, high-multiplicity bin packing, vector bin packing, and makespan minimization on uniform and unrelated machines. The double-exponential dependence on $d$ is tight for Point-in-Cone and both bin-packing problems assuming $\mathrm{NP}\nsubseteq\mathrm{SUBEXP}$. The mixed-integer formulation also yields additive EPTASs for makespan minimization, machine covering, and envy minimization on identical machines, and additive schemes fixed-parameter tractable in $1/\varepsilon+τ$ on uniform machines.
cs.DS