TY - GEN
T1 - A Simple Algorithm for Combinatorial n-Fold ILPs Using the Steinitz Lemma
AU - Gupta, Sushmita
AU - Jain, Pallavi
AU - Seetharaman, Sanjay
AU - Zehavi, Meirav
N1 - Publisher Copyright:
© Sushmita Gupta, Pallavi Jain, Sanjay Seetharaman, and Meirav Zehavi.
PY - 2025/1/1
Y1 - 2025/1/1
N2 - We present an algorithm for a class of n-fold ILPs whose existing algorithms in literature are often either (1) based on the augmentation framework where one starts with an arbitrary solution and then iteratively moves towards an optimal solution by solving appropriate programs; or (2) require solving a linear relaxation of the program; or (3) are based on decomposition/proximity based arguments. Combinatorial n-fold ILPs is a class of n-fold ILPs introduced and studied by Knop et al. [MP2020] that captures several other problems in a variety of domains. We present a simple and direct algorithm that solves combinatorial n-fold ILPs with unbounded non-negative variables via an application of the Steinitz lemma. Depending on the structure of the input ILP, we also improve upon the existing algorithms in the literature in terms of the running time, thereby showing an improvement that mirrors the one shown by Rohwedder [ICALP2025] contemporaneously and independently.
AB - We present an algorithm for a class of n-fold ILPs whose existing algorithms in literature are often either (1) based on the augmentation framework where one starts with an arbitrary solution and then iteratively moves towards an optimal solution by solving appropriate programs; or (2) require solving a linear relaxation of the program; or (3) are based on decomposition/proximity based arguments. Combinatorial n-fold ILPs is a class of n-fold ILPs introduced and studied by Knop et al. [MP2020] that captures several other problems in a variety of domains. We present a simple and direct algorithm that solves combinatorial n-fold ILPs with unbounded non-negative variables via an application of the Steinitz lemma. Depending on the structure of the input ILP, we also improve upon the existing algorithms in the literature in terms of the running time, thereby showing an improvement that mirrors the one shown by Rohwedder [ICALP2025] contemporaneously and independently.
KW - n-fold integer linear program
KW - parameterized algorithms
UR - https://www.scopus.com/pages/publications/105031361532
U2 - 10.4230/LIPIcs.IPEC.2025.14
DO - 10.4230/LIPIcs.IPEC.2025.14
M3 - Conference contribution
AN - SCOPUS:105031361532
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 20th International Symposium on Parameterized and Exact Computation, IPEC 2025
A2 - Agrawal, Akanksha
A2 - Leeuwen , Erik Jan van
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 20th International Symposium on Parameterized and Exact Computation, IPEC 2025
Y2 - 17 September 2025 through 19 September 2025
ER -