Tight Lower Bounds for Block-Structured Integer Programs

TL;DR

The paper analyzes block-structured integer programs, focusing on tree-fold and multi-stage IPs, and shows ETH-based lower bounds that nearly close the gap with the best known algorithms. It unveils a bi-/tri-diagonal matrix-based reduction framework to derive both triple-exponential and double-exponential lower bounds in the number of levels and block sizes, respectively, while also proving unconditional lower bounds on Graver-basis norms that underpin algorithmic complexity. A refined perspective using tree-depth parameters further clarifies the intrinsic hardness of these problems, indicating no substantial improvements over current approaches. The results substantially advance our understanding of the limits of block-structured IP solvers and provide a new toolkit for translating simple matrix structures into hard instances.

Abstract

We study fundamental block-structured integer programs called tree-fold and multi-stage IPs. Tree-fold IPs admit a constraint matrix with independent blocks linked together by few constraints in a recursive pattern; and transposing their constraint matrix yields multi-stage IPs. The state-of-the-art algorithms to solve these IPs have an exponential gap in their running times, making it natural to ask whether this gap is inherent. We answer this question affirmative. Assuming the Exponential Time Hypothesis, we prove lower bounds showing that the exponential difference is necessary, and that the known algorithms are near optimal. Moreover, we prove unconditional lower bounds on the norms of the Graver basis, a fundamental building block of all known algorithms to solve these IPs. This shows that none of the current approaches can be improved beyond this bound.

Figures (2)

• Figure 1: On the left, a schematic multi-stage with three levels is presented. On the right, a schematic tree-fold with 4 layers is pictured. All entries within a rectangle can be non-zero, all entries outside of the rectangles must be zero.
• Figure 2: On the left, a $2$-stage stochastic matrix with blocks $A_i$ and $D_i$, $i \in [n]$ is presented. On the right, an $n$-fold matrix with blocks $C_i$ and $D_i$, $i \in [n]$ is pictured. All entries not belonging to a block are zero.

Theorems & Definitions (20)

• definition thmcounterdefinition: Tree-fold and multi-stage matrices
• definition thmcounterdefinition: $n$-fold and $2$-stage stochastic matrices
• definition thmcounterdefinition: Bi-diagonal, Tri-diagonal Matrix
• lemma thmcounterlemma
• proof
• corollary thmcountercorollary
• theorem thmcountertheorem: A lower bound for multi-stage IPs
• corollary thmcountercorollary
• proof : of Theorem \ref{['thm:multi-stage']}
• lemma thmcounterlemma: knop2020tight