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Geometry-Dependent Approximation for Non-Monotone $k$-Submodular Maximization — arXiv:2610.04049

Source: arXiv
Original: https://arxiv.org/abs/2610.04049
License (as archived): CC0-1.0
Archived: 2026-10-06T05:31:58.269Z
SHA-256 of the archived text: 85480b95b28683b2a9df904ce7cde84c9fe9b1d40652da68b8d9cc0ff40ce9ba
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How to cite

Geometry-Dependent Approximation for Non-Monotone $k$-Submodular Maximization — arXiv:2610.04049 — arXiv, https://arxiv.org/abs/2610.04049 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2d83cd776136 (Metraipolis archive dc_2d83cd776136, sha256 85480b95b28683b2a9df904ce7cde84c9fe9b1d40652da68b8d9cc0ff40ce9ba)

BibTeX

@misc{dc2d83cd776136,
  title = {Geometry-Dependent Approximation for Non-Monotone $k$-Submodular Maximization — arXiv:2610.04049},
  howpublished = {arXiv},
  url = {https://arxiv.org/abs/2610.04049},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2d83cd776136; sha256 85480b95b28683b2a9df904ce7cde84c9fe9b1d40652da68b8d9cc0ff40ce9ba},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv, https://arxiv.org/abs/2610.04049. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study nonnegative, non-monotone $k$-submodular maximization with $k\ge2$ labels under support constraints, and show how the certified approximation coefficient improves as the support region permits more uniform selection. For a compact convex down-closed support region $P\subseteq[0,1]^n$, the diagonal level $\zeta(P)=\max\{t\in[0,1]:t {\bf 1} \in P\}$ ranges from $\zeta=0$, which carries no geometric promise, to $\zeta=1$, which is unrestricted support. Our main structural result is a comparator-uniform linearization of the multilinear extension, built from an objective-independent action and a comparator-independent update field. For $k\ge3$, its validity reduces, independently of the number of elements, to four polynomial inequalities of degree at most three in one or two variables, only one of which depends on $k$. Explicit parameter choices give a nondecreasing certified profile $\underline{\alpha}_k(\zeta)$, in closed form on all of $[0,1]$ when $k=2$. At $\zeta=0$ we certify $0.4456\ldots$ for $k=2$ and $0.4541\ldots$ for every $k\ge3$, improving the recent $\sqrt2-1$ guarantee for one matroid or one knapsack, as well as the $1/3$-type guarantees for a fixed number of budgets; at $\zeta=1$ we certify $1/2$ for $k=2$, $(\sqrt{17}-3)/2$ for $k=3,4$, and $k/(2k-1)$ for $k\ge5$, whose excess over $1/2$ is of order $1/k$ rather than the previous $1/k^2$. Value-retaining rounding transfers these guarantees to matroid and knapsack constraints, and the same field yields $O(\sqrt T)$ approximate regret online under gradient or post-decision value feedback.

Tema degli esploratori: agenti e intelligenza artificiale (elenco arXiv cs.LG, nuovi lavori del giorno).
Categorie arXiv: Data Structures and Algorithms (cs.DS) ; Machine Learning (cs.LG); Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2610.04049 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T05:27:05.939Z dalla pagina https://arxiv.org/list/cs.LG/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).