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

Tim Randolph, Karol Węgrzycki

TL;DR

This work introduces a parameterization of problems on integer sets by the doubling constant $\mathcal{C}$ and develops a suite of algorithms that exploit additive structure. Central to the approach is a constructive Freiman's theorem that yields explicit generalized arithmetic progressions containing the input, enabling reductions to structured integer programs and tight reductions among Subset Sum, HBILP, and ILP. The paper delivers: (i) a near-linear FPT algorithm for constructing Freiman progressions, (ii) polynomial-time DP-based feasibility for Binary ILP under constant doubling, (iii) XP-time solvability for Subset Sum and related problems with precise doubling-based bounds, (iv) reductions showing equivalences between Subset Sum and HBILP, (v) near-XP or subexponential-time results for Unbounded Subset Sum, and (vi) a k-SUM algorithm with constant doubling using sparse convolution, including tight results for the 4-SUM case under the k-SUM conjecture. The results bridge additive combinatorics and algorithm design, offering new tractable regimes for classical hard problems when the input exhibits additive structure, and they provide near-optimal algorithmic behavior under widely believed complexity conjectures.

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.

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

TL;DR

This work introduces a parameterization of problems on integer sets by the doubling constant and develops a suite of algorithms that exploit additive structure. Central to the approach is a constructive Freiman's theorem that yields explicit generalized arithmetic progressions containing the input, enabling reductions to structured integer programs and tight reductions among Subset Sum, HBILP, and ILP. The paper delivers: (i) a near-linear FPT algorithm for constructing Freiman progressions, (ii) polynomial-time DP-based feasibility for Binary ILP under constant doubling, (iii) XP-time solvability for Subset Sum and related problems with precise doubling-based bounds, (iv) reductions showing equivalences between Subset Sum and HBILP, (v) near-XP or subexponential-time results for Unbounded Subset Sum, and (vi) a k-SUM algorithm with constant doubling using sparse convolution, including tight results for the 4-SUM case under the k-SUM conjecture. The results bridge additive combinatorics and algorithm design, offering new tractable regimes for classical hard problems when the input exhibits additive structure, and they provide near-optimal algorithmic behavior under widely believed complexity conjectures.

Abstract

We study the parameterized complexity of algorithmic problems whose input is an integer set in terms of the doubling constant , 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 with polynomially-bounded variables and constraints can be determined in time when the column set of the constraint matrix has doubling constant . Second, we show that the Subset Sum and Unbounded Subset Sum problems can be solved in time and , respectively, where the notation hides functions that depend only on the doubling constant . 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 -SUM as well as tight lower bounds for 4-SUM and nearly tight lower bounds for -SUM, under the -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.
Paper Structure (28 sections, 25 theorems, 60 equations)

This paper contains 28 sections, 25 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Results
  4. Contribution 1: A Constructive Freiman's Theorem in Near-Linear FPT Time.
  5. Organization
  6. Preliminaries
  7. Freiman's Theorem Made Constructive in FPT Time
  8. Integer Programming with Constant Doubling
  9. -Binary ILP Feasibility
  10. -Bounded ILP Feasibility
  11. Subset Sum with Constant Doubling
  12. Reduction from -Subset Sum to Hyperplane-Constrained Binary ILP Feasibility
  13. Equivalence Between HBILP Feasibility and Subset Sum
  14. Reduction from BILP Feasibility to HBILP Feasibility
  15. Unbounded Subset Sum with Constant Doubling
  16. ...and 13 more sections

Key Result

Theorem 1.1

An instance $\mathcal{I}$ of $\mathcal{C}$-Binary ILP Feasibility on $n$ variables can be solved in time $n^{O_\mathcal{C}(1)} \cdot \poly(|\mathcal{I}|)$.We write $|\mathcal{I}|$ to denote the size of the ILP instance $\mathcal{I}$. In the word RAM model (see sec:prelims), this is $\poly(m, n)$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1: Freiman's Theorem, freiman1964addition, see zhao2022graph for a modern presentation
  • Theorem 3.2: FPT Freiman's Theorem
  • Remark 4.1
  • Theorem 4.1
  • proof
  • Corollary 4.1
  • proof
  • Corollary 5.1: $\mathcal{C}$-Subset Sum is in XP
  • ...and 42 more