{"paper":{"title":"Hardness of A/E-Design under Partition Constraints","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Nikhil Bansal, Yuze Xu","submitted_at":"2026-08-05T23:33:19Z","abstract_excerpt":"We consider the A/E-design problem under partition constraints: Given vectors $v_1,\\ldots,v_N\\in \\R^d$ and a partition matroid on $[N]$, find a base $S$ of the matroid that minimizes $\\tr(M(S)^{-1})$ or $\\lambda_{\\max}(M(S)^{-1})$ where $M(S)=\\sum_{i \\in S} v_i v_i^\\top$.\n  In contrast to D-design, where good estimation and approximation guarantees are known as a function of $d$, we show that no reasonable approximation exists for A/E-design. This answers a question of Brown, Laddha and Singh. The proof is based on an elementary reduction from three-dimensional matching."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05468","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/2608.05468/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"}