{"paper":{"title":"Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Integer Linear Arithmetic","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG","cs.LO"],"primary_cat":"cs.CC","authors_text":"Geri Gokaj, Marvin K\\\"unnemann","submitted_at":"2025-02-07T00:29:05Z","abstract_excerpt":"In the last three decades, the $k$-SUM hypothesis has emerged as a satisfying explanation of long-standing time barriers for a variety of algorithmic problems. Yet to this day, the literature knows of only few proven consequences of a refutation of this hypothesis. Taking a descriptive complexity viewpoint, we ask: What is the largest logically defined class of problems \\emph{captured} by the $k$-SUM problem?\n  To this end, we introduce a class $\\mathsf{FOP}_{\\mathbb{Z}}$ of problems corresponding to deciding sentences in Presburger arithmetic/linear integer arithmetic over finite subsets of i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.04581","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.04581/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}