2026-07-22T08:36:03Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:22711 2026-04-17T05:32:38Z ddc:004 open_access

Fine-Grained Equivalence for Problems Related to Integer Linear Programming Rohwedder, Lars Węgrzycki, Karol Integer Programming Fine-Grained Complexity Fixed-Parameter Tractable Algorithms Integer Linear Programming with n binary variables and m many 0/1-constraints can be solved in time 2^Õ(m²) poly(n) and it is open whether the dependence on m is optimal. Several seemingly unrelated problems, which include variants of Closest String, Discrepancy Minimization, Set Cover, and Set Packing, can be modelled as Integer Linear Programming with 0/1 constraints to obtain algorithms with the same running time for a natural parameter m in each of the problems. Our main result establishes through fine-grained reductions that these problems are equivalent, meaning that a 2^O(m^{2-ε}) poly(n) algorithm with ε > 0 for one of them implies such an algorithm for all of them. In the setting above, one can alternatively obtain an n^O(m) time algorithm for Integer Linear Programming using a straightforward dynamic programming approach, which can be more efficient if n is relatively small (e.g., subexponential in m). We show that this can be improved to {n'}^O(m) + O(nm), where n' is the number of distinct (i.e., non-symmetric) variables. This dominates both of the aforementioned running times. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Lars Rohwedder and Karol Węgrzycki 2025 Is Part Of LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2025.83 urn:nbn:de:0030-drops-227114 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.83 eng https://creativecommons.org/licenses/by/4.0/legalcode