🤖 AI Summary
This paper investigates the parameterized tractability of block-structured integer programming—specifically two-stage stochastic IPs and n-fold IPs—when the global constraint matrix contains arbitrarily large entries. Addressing the limitation of prior work requiring all input coefficients to be bounded, we establish, for the first time, that feasibility checking for two-stage stochastic IPs and linear optimization for *uniform* n-fold IPs are fixed-parameter tractable (FPT) when parameterized solely by the local matrix dimensions and the maximum absolute value of the right-hand sides (D_i). We further show that uniformity is necessary for n-fold IPs to admit such FPT algorithms. Our approach integrates Graver basis theory, potential function analysis, dynamic programming, integer conic decomposition, and distance-sensitive search to design a weakly polynomial-time FPT algorithm. This work establishes the first FPT framework for block-structured IPs tolerant of large matrix entries, tightening the parameter dependence from global coefficient magnitude to local magnitudes—thereby substantially broadening the applicability of parameterized algorithms for structured integer programming.
📝 Abstract
We study two classic variants of block-structured integer programming. Two-stage stochastic programs are integer programs of the form ${A_i mathbf{x} + D_i mathbf{y}_i = mathbf{b}_i extrm{ for all }i=1,ldots,n}$, where $A_i$ and $D_i$ are bounded-size matrices. On the other hand, $n$-fold programs are integer programs of the form ${{sum_{i=1}^n C_imathbf{y}_i=mathbf{a}} extrm{ and } D_imathbf{y}_i=mathbf{b}_i extrm{ for all }i=1,ldots,n}$, where again $C_i$ and $D_i$ are bounded-size matrices. It is known that solving these kind of programs is fixed-parameter tractable when parameterized by the maximum dimension among the relevant matrices $A_i,C_i,D_i$ and the maximum absolute value of any entry appearing in the constraint matrix. We show that the parameterized tractability results for two-stage stochastic and $n$-fold programs persist even when one allows large entries in the global part of the program. More precisely, we prove that: - The feasibility problem for two-stage stochastic programs is fixed-parameter tractable when parameterized by the dimensions of matrices $A_i,D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we allow matrices $A_i$ to have arbitrarily large entries. - The linear optimization problem for $n$-fold integer programs that are uniform -- all matrices $C_i$ are equal -- is fixed-parameter tractable when parameterized by the dimensions of matrices $C_i$ and $D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we require that $C_i=C$ for all $i=1,ldots,n$, but we allow $C$ to have arbitrarily large entries. In the second result, the uniformity assumption is necessary; otherwise the problem is $mathsf{NP}$-hard already when the parameters take constant values. Both our algorithms are weakly polynomial: the running time is measured in the total bitsize of the input.
