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.

翻译：我们研究一类算法问题的参数化复杂度，其输入为整数集$A$，并以倍加常数$C := |A + A|/|A|$作为参数——该常数是加性结构的基本度量。我们通过针对两个困难且被深入研究的经典问题——整数规划与子集和问题——的新结果，证明这一新参数化方式具有算法实用性。首先，我们证明在倍加常数参数化下，判定有界整数规划的可行性是一个可处理问题。具体而言，我们证明当约束矩阵的列集具有倍加常数$C$时，包含$n$个多项式有界变量与$m$个约束的整数规划$I$的可行性可在$n^{O_C(1)} poly(|I|)$时间内判定。其次，我们证明子集和问题与无界子集和问题可分别在$n^{O_C(1)}$时间与$n^{O_C(\log \log \log n)}$时间内求解，其中$O_C$记号隐藏了仅依赖于倍加常数$C$的函数。我们还证明了在有界倍加条件下为子集和问题实现FPT算法，与为Box ILP的参数化复杂度取得里程碑式结果具有等价性。最后，基于k-SUM猜想，我们为k-SUM问题设计了近线性时间算法，并为4-SUM问题建立了紧下界，为k-SUM问题建立了近紧下界。我们的多项结果依赖于一项新证明：加性组合学核心定理——Freiman定理——可实现高效构造性证明。该结果可能具有独立的研究价值。