Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM

TL;DR

This paper explores the parameterized complexity of problems involving integer sets with small doubling, providing new algorithms and bounds for Integer Programming, Subset Sum, and k-SUM based on this structural property.

Contribution

It introduces the doubling constant as a parameter to develop efficient algorithms for Integer Programming, Subset Sum, and k-SUM, and proves the constructive version of Freiman's Theorem.

Findings

Integer Program feasibility is tractable with small doubling constant

Subset Sum can be solved in time depending on the doubling constant

Near-linear algorithms and bounds for k-SUM and related problems

Abstract

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.