{"paper":{"title":"Finding Submodularity Hidden in Symmetric Difference","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Junpei Nakashima, Masafumi Yamashita, Shuji Kijima, Yukiko Yamauchi","submitted_at":"2017-12-23T06:25:43Z","abstract_excerpt":"A set function $f$ on a finite set $V$ is submodular if $f(X) + f(Y) \\geq f(X \\cup Y) + f(X \\cap Y)$ for any pair $X, Y \\subseteq V$. The symmetric difference transformation (SD-transformation) of $f$ by a canonical set $S \\subseteq V$ is a set function $g$ given by $g(X) = f(X \\vartriangle S)$ for $X \\subseteq V$,where $X \\vartriangle S = (X \\setminus S) \\cup (S \\setminus X)$ denotes the symmetric difference between $X$ and $S$. Submodularity and SD-transformations are regarded as the counterparts of convexity and affine transformations in a discrete space, respectively. However, submodularit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08721","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"}