{"paper":{"title":"Parametric Shortest Paths in Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Jaikumar Radhakrishnan, Kshitij Gajjar","submitted_at":"2018-11-13T05:35:47Z","abstract_excerpt":"We construct a family of planar graphs $\\{G_n\\}_{n\\geq 4}$, where $G_n$ has $n$ vertices including a source vertex $s$ and a sink vertex $t$, and edge weights that change linearly with a parameter $\\lambda$ such that, as $\\lambda$ varies in $(-\\infty,+\\infty)$, the piece-wise linear cost of the shortest path from $s$ to $t$ has $n^{\\Omega(\\log n)}$ pieces. This shows that lower bounds obtained earlier by Carstensen (1983) and Mulmuley \\& Shah (2001) for general graphs also hold for planar graphs, thereby refuting a conjecture of Nikolova (2009). Gusfield (1980) and Dean (2009) showed that the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05115","kind":"arxiv","version":3},"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"}