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(It is well known that the same problem over arbitrary integers is equivalent to the above definition, by linear-time randomized reductions.) We prove that this definition of k-SUM remains W[1]-hard, and is in fact W[1]-complete: k-SUM can be reduced to f(k) * n^o(1) instances of k-Clique.\n  - The maximum node-weighted k-Clique and node-weighted k-dominating set problems can be re"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1311.3054","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-11-13T09:26:57Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"13dac0dfb412303b7a53d286499a7836e551919e58ebb57e3d211eb6799d0169","abstract_canon_sha256":"932ad33b691df059a760a6bc90844f04147e8251096bc6cfca00b4fab944f098"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:03.139578Z","signature_b64":"K7oX+J1gnd2zxRxmRwNC8Mw0UxBeA7YaAvNB6palBtwP6CTVV7NMrZfGVvgQ+FctkUuXqGMyOxtDyVq0xYfqDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1bd9ba8f34906384efe74f4b18c7842d045015cdeec2c022e8b192b07ba07b0","last_reissued_at":"2026-05-18T01:26:03.139152Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:03.139152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Losing Weight by Gaining Edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CC","authors_text":"Amir Abboud, Kevin Lewi, Ryan Williams","submitted_at":"2013-11-13T09:26:57Z","abstract_excerpt":"We present a new way to encode weighted sums into unweighted pairwise constraints, obtaining the following results.\n  - Define the k-SUM problem to be: given n integers in [-n^2k, n^2k] are there k which sum to zero? 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