New Algorithm for Combinatorial $n$-folds and Applications

TL;DR

This paper introduces a specialized divide-and-conquer algorithm for combinatorial $n$-fold ILPs with a specific block structure, enabling efficient solutions for problems like scheduling, string proximity, and graph imbalance.

Contribution

The paper presents a novel algorithm tailored to a specific class of $n$-fold ILPs, improving solution times and extending applicability to various combinatorial problems.

Findings

Efficient feasibility and optimal solution algorithms with time $(n r \Delta)^{O(r)} \log(\|b\\|_ ext{max})$.

Matching lower bounds for scheduling problems on uniform machines.

Improved parameter dependency for graph imbalance problems.

Abstract

Block-structured integer linear programs (ILPs) play an important role in various application fields. We address $n$-fold ILPs where the matrix $\mathcal{A}$ has a specific structure, i.e., where the blocks in the lower part of $\mathcal{A}$ consist only of the row vectors $(1,\dots,1)$. In this paper, we propose an approach tailored to exactly these combinatorial $n$-folds. We utilize a divide and conquer approach to separate the original problem such that the right-hand side iteratively decreases in size. We show that this decrease in size can be calculated such that we only need to consider a bounded amount of possible right-hand sides. This, in turn, lets us efficiently combine solutions of the smaller right-hand sides to solve the original problem. We can decide the feasibility of, and also optimally solve, such problems in time $(n r \Delta)^{O(r)} \log(\|b\|_\infty),$ where $n$ is the number of blocks, $r$ the number of rows in the upper blocks and $\Delta=\|A\|_\infty$. We complement the algorithm by discussing applications of the $n$-fold ILPs with the specific structure we require. We consider the problems of (i) scheduling on uniform machines, (ii) closest string and (iii) (graph) imbalance. Regarding (i), our algorithm results in running times of $p_{\max}^{O(d)}|I|^{O(1)},$ matching a lower bound derived via ETH. For (ii) we achieve running times matching the current state-of-the-art in the general case. In contrast to the state-of-the-art, our result can leverage a bounded number of column-types to yield an improved running time. For (iii), we improve the parameter dependency on the size of the vertex cover.