We consider a special block-structured integer program with $n$ blocks. Every block has a prescribed number of vectors that need to be selected while minimizing the objective $c^{\intercal}x + g(Ax)$, where $c$ is a rational cost vector, $A$ is an integral matrix with $d$ rows, and $g \colon \mathbb{Z}^d \to \mathbb{Q}\cup \{+\infty\}$ is an arbitrary function. We access $g$ via an oracle that minimizes $c^{\intercal}x + g(Ax)$ when all but $O(d)$ variables are fixed. Our algorithm runs in time $(ndΔ)^{O(d)}(Q + \text{poly}(|I|))$ where $Δ$ is the largest absolute value of an entry in $A$, $Q$ is the necessary time to evaluate the oracle for $g$ and $|I|$ is the size of the input. This model is a variation of the model studied by Dadush et al.~[IPCO 23]. Their model has only a single block but individual upper and lower bounds on the variables instead of prescribed block sums. For their model they present an algorithm with running time $t^d(dΔ)^{O(d^2)}Q$ where $t$ is the number of variables. We use our algorithm to solve mixed-integer linear programs with no upper bounds on the variables, a linear objective, and arbitrarily many constraints involving only fractional variables in time $(mΔ)^{O(m)}\text{poly}(|I|)$, where $m$ is the number of constraints involving integral variables and $Δ$ is the largest absolute value of coefficients of integral variables. This improves the dependence on $m$ in the exponent to asymptotically match the dependency in the integer setting. Further, we use our algorithm to generalize the parameterized algorithm for scheduling on uniformly related machines by Rohwedder~[ICALP 25] to obtain an algorithm for multiple knapsack with running time $w_{\max}^{O(d)}\text{poly}(|I|)$ where $w_{\max}$ is the maximum weight of an item and $d$ the number of different weights.
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.
We give a fixed-parameter tractable (FPT) algorithm with running time $f(k,\barΔ)\cdot {|I|}^{O(1)}$ for integer linear programs with 4-block structure, parameterized by the maximum block dimension $k$ and the largest absolute entry $\barΔ$ in the local-column blocks. This result resolves a long-standing open question in parameterized complexity, and answers a conjecture by Eisenbrand and Rothvoss (2026) in the positive. This result has several implications for other block-structured integer programming models, including mixed fracture number and special cases of 4-block programs with large entries outside of the diagonal. Important tools for this algorithm are structural properties of generalized $n$-fold integer programs shown by Ligthart~(2026) and an algorithm by Veselov, Gribanov, Zolotykh and Chirkov~(2020) for optimizing discrete convic functions. Independently and shortly after the first version of our paper appeared, Koutecký, Lassota, and Ligthart (2026) obtained an FPT algorithm for 4-block integer programming.