{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:5VJFSG74PRYZU6BED2U6USJSRQ","short_pith_number":"pith:5VJFSG74","schema_version":"1.0","canonical_sha256":"ed52591bfc7c719a78241ea9ea49328c202584566dea2ab1d4a9e0ce7a621bbe","source":{"kind":"arxiv","id":"1210.8284","version":1},"attestation_state":"computed","paper":{"title":"Hardness and Approximation Results for $L_p$-Ball Constrained Homogeneous Polynomial Optimization Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Anthony Man-Cho So, Ke Hou","submitted_at":"2012-10-31T10:16:52Z","abstract_excerpt":"In this paper, we establish hardness and approximation results for various $L_p$-ball constrained homogeneous polynomial optimization problems, where $p \\in [2,\\infty]$. Specifically, we prove that for any given $d \\ge 3$ and $p \\in [2,\\infty]$, both the problem of optimizing a degree-$d$ homogeneous polynomial over the $L_p$-ball and the problem of optimizing a degree-$d$ multilinear form (regardless of its super-symmetry) over $L_p$-balls are NP-hard. On the other hand, we show that these problems can be approximated to within a factor of $\\Omega((\\log n)^{(d-2)/p} \\big/ n^{d/2-1})$ in deter"},"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":"1210.8284","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-10-31T10:16:52Z","cross_cats_sorted":[],"title_canon_sha256":"24009aa90484c042565d54ee2a5171b57f4610cad34e2baba90d818a9bbe84ef","abstract_canon_sha256":"a9f55e728f39e902c1ad2831c923bd943d43d5b8fc9ea2867f352021d3de59db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:55.453060Z","signature_b64":"LM5wyPeMLwCc+UQ0bK5bDwnl/3otKO0BzpLIKwToK7KYni4aKaIugWq8si7yzga8FGLK4b2df28ccdUP9lGaCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed52591bfc7c719a78241ea9ea49328c202584566dea2ab1d4a9e0ce7a621bbe","last_reissued_at":"2026-05-18T03:41:55.452283Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:55.452283Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hardness and Approximation Results for $L_p$-Ball Constrained Homogeneous Polynomial Optimization Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Anthony Man-Cho So, Ke Hou","submitted_at":"2012-10-31T10:16:52Z","abstract_excerpt":"In this paper, we establish hardness and approximation results for various $L_p$-ball constrained homogeneous polynomial optimization problems, where $p \\in [2,\\infty]$. Specifically, we prove that for any given $d \\ge 3$ and $p \\in [2,\\infty]$, both the problem of optimizing a degree-$d$ homogeneous polynomial over the $L_p$-ball and the problem of optimizing a degree-$d$ multilinear form (regardless of its super-symmetry) over $L_p$-balls are NP-hard. On the other hand, we show that these problems can be approximated to within a factor of $\\Omega((\\log n)^{(d-2)/p} \\big/ n^{d/2-1})$ in deter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.8284","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1210.8284","created_at":"2026-05-18T03:41:55.452420+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.8284v1","created_at":"2026-05-18T03:41:55.452420+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.8284","created_at":"2026-05-18T03:41:55.452420+00:00"},{"alias_kind":"pith_short_12","alias_value":"5VJFSG74PRYZ","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_16","alias_value":"5VJFSG74PRYZU6BE","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_8","alias_value":"5VJFSG74","created_at":"2026-05-18T12:26:56.085431+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ","json":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ.json","graph_json":"https://pith.science/api/pith-number/5VJFSG74PRYZU6BED2U6USJSRQ/graph.json","events_json":"https://pith.science/api/pith-number/5VJFSG74PRYZU6BED2U6USJSRQ/events.json","paper":"https://pith.science/paper/5VJFSG74"},"agent_actions":{"view_html":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ","download_json":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ.json","view_paper":"https://pith.science/paper/5VJFSG74","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.8284&json=true","fetch_graph":"https://pith.science/api/pith-number/5VJFSG74PRYZU6BED2U6USJSRQ/graph.json","fetch_events":"https://pith.science/api/pith-number/5VJFSG74PRYZU6BED2U6USJSRQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ/action/storage_attestation","attest_author":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ/action/author_attestation","sign_citation":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ/action/citation_signature","submit_replication":"https://pith.science/pith/5VJFSG74PRYZU6BED2U6USJSRQ/action/replication_record"}},"created_at":"2026-05-18T03:41:55.452420+00:00","updated_at":"2026-05-18T03:41:55.452420+00:00"}