{"paper":{"title":"Near-optimal small-depth lower bounds for small distance connectivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Igor C. Oliveira, Li-Yang Tan, Rocco A. Servedio, Xi Chen","submitted_at":"2015-09-24T19:01:34Z","abstract_excerpt":"We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{\\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were $n^{k^{\\exp(-O(d))}}$ (due to Beame, Impagliazzo, and Pitassi [BIP98]) and $n^{\\Omega((\\log k)/d)}$ (following from a recent formula size lower bound of Rossman [Ros14]). Our lower bound is quite close to optimal, since a simple construction gives depth-$d$ circuits of size $n^{O(k^{2/d})}$ for this problem (and strengthening our bound even to $n^{k^{\\Omega(1/d)}}$ woul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07476","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"}