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Tight Lower Bounds for Block-Structured Integer Programs

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Integer Programming and Combinatorial Optimization (IPCO 2024)


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.

C. Hunkenschröder acknowledges funding by Einstein Foundation Berlin. K.-M. Klein was supported by DFG project KL 3408/1-1. A. Lassota was partially supported by the Swiss National Science Foundation (SNSF) within the project Complexity of Integer Programming (207365). M. Koutecký was partially supported by the Charles University project UNCE 24/SCI/008 and by the project 22-22997S of GA ČR. A. Levin is partially supported by ISF – Israel Science Foundation grant number 1467/22.

A full version of this paper can be found in [14].

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Hunkenschröder, C., Klein, KM., Koutecký, M., Lassota, A., Levin, A. (2024). Tight Lower Bounds for Block-Structured Integer Programs. In: Vygen, J., Byrka, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2024. Lecture Notes in Computer Science, vol 14679. Springer, Cham.

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