eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
96:1
96:19
10.4230/LIPIcs.ESA.2024.96
article
Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM
Randolph, Tim
1
https://orcid.org/0000-0002-1263-1024
Węgrzycki, Karol
2
https://orcid.org/0000-0001-9746-5733
Harvey Mudd College, Claremont, CA, USA
Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany
We study the parameterized complexity of algorithmic problems whose input is an integer set A in terms of the doubling constant 𝒞 := |A+A| / |A|, a fundamental measure of additive structure. We present evidence that this new parameterization is algorithmically useful in the form of new results for two difficult, well-studied problems: Integer Programming and Subset Sum.
First, we show that determining the feasibility of bounded Integer Programs is a tractable problem when parameterized in the doubling constant. Specifically, we prove that the feasibility of an integer program ℐ with n polynomially-bounded variables and m constraints can be determined in time n^{O_𝒞(1)} ⋅ poly(|ℐ|) when the column set of the constraint matrix has doubling constant 𝒞.
Second, we show that the Subset Sum and Unbounded Subset Sum problems can be solved in time n^{O_C(1)} and n^{O_𝒞(log log log n)}, respectively, where the O_C notation hides functions that depend only on the doubling constant 𝒞. We also show the equivalence of achieving an FPT algorithm for Subset Sum with bounded doubling and achieving a milestone result for the parameterized complexity of Box ILP. Finally, we design near-linear time algorithms for k-SUM as well as tight lower bounds for 4-SUM and nearly tight lower bounds for k-SUM, under the k-SUM conjecture.
Several of our results rely on a new proof that Freiman’s Theorem, a central result in additive combinatorics, can be made efficiently constructive. This result may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.96/LIPIcs.ESA.2024.96.pdf
Parameterized algorithms
parameterized complexity
additive combinatorics
Subset Sum
integer programming
doubling constant