{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/190409354","title":"Submodular Maximization Beyond Non-negativity: Guarantees, Fast Algorithms, and Applications","arxiv_id":"1904.09354","date":"2019-04-19","proceeding":null,"authors":["Christopher Harshaw","Moran Feldman","Justin Ward","Amin Karbasi"],"abstract":"It is generally believed that submodular functions -- and the more general\nclass of $\\gamma$-weakly submodular functions -- may only be optimized under\nthe non-negativity assumption $f(S) \\geq 0$. In this paper, we show that once\nthe function is expressed as the difference $f = g - c$, where $g$ is monotone,\nnon-negative, and $\\gamma$-weakly submodular and $c$ is non-negative modular,\nthen strong approximation guarantees may be obtained. We present an algorithm\nfor maximizing $g - c$ under a $k$-cardinality constraint which produces a\nrandom feasible set $S$ such that $\\mathbb{E} \\left[ g(S) - c(S) \\right] \\geq\n(1 - e^{-\\gamma} - \\epsilon) g(OPT) - c(OPT)$, whose running time is $O\n(\\frac{n}{\\epsilon} \\log^2 \\frac{1}{\\epsilon})$, i.e., independent of $k$. We\nextend these results to the unconstrained setting by describing an algorithm\nwith the same approximation guarantees and faster $O(\\frac{n}{\\epsilon}\n\\log\\frac{1}{\\epsilon})$ runtime. The main techniques underlying our algorithms\nare two-fold: the use of a surrogate objective which varies the relative\nimportance between $g$ and $c$ throughout the algorithm, and a geometric sweep\nover possible $\\gamma$ values. Our algorithmic guarantees are complemented by a\nhardness result showing that no polynomial-time algorithm which accesses $g$\nthrough a value oracle can do better. We empirically demonstrate the success of\nour algorithms by applying them to experimental design on the Boston Housing\ndataset and directed vertex cover on the Email EU dataset.","url_abs":"http://arxiv.org/abs/1904.09354v1","url_pdf":"http://arxiv.org/pdf/1904.09354v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"190409354","repo_url":"https://github.com/crharshaw/submodular-minus-linear","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"experimental-design","task_name":"Experimental Design"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1904.09354","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}