k-SUM is complete for the existential fragment of linear integer arithmetic sentences, and 3-SUM is complete for sentences using at most three linear inequalities, unifying many fine-grained reductions.
Paul Chew, Dorit Dor, Alon Efrat, and Klara Kedem
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Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Integer Linear Arithmetic
k-SUM is complete for the existential fragment of linear integer arithmetic sentences, and 3-SUM is complete for sentences using at most three linear inequalities, unifying many fine-grained reductions.