Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917733477646336 |
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| author | Randolph, Tim Węgrzycki, Karol |
| author_facet | Randolph, Tim Węgrzycki, Karol |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM Randolph, Tim Węgrzycki, Karol Data Structures and Algorithms F.2.1; F.2.2 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. |
| title | Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM |
| topic | Data Structures and Algorithms F.2.1; F.2.2 |
| url | https://arxiv.org/abs/2407.18228 |