Geometry-Dependent Approximation for Non-Monotone $k$-Submodular Maximization

2026-10-02

AI Summary

This study addresses non-monotone $k$-submodular maximization under cardinality constraints, aiming to overcome existing approximation ratio bottlenecks. Methodologically, it proposes a geometry-dependent linearization framework that exploits diagonal-level optimization over the support region to enhance approximation guarantees. By constructing a linearized structure with uniform comparators, the validity verification is reduced to a small set of polynomial inequalities, enabling closed-form certification. The approach further integrates multilinear extensions, value-preserving rounding, and online gradient feedback techniques. This work significantly improves the approximation ratio beyond 0.45 and surpasses the classical $1/2$ theoretical barrier in the unconstrained setting. Moreover, the proposed framework naturally generalizes to online scenarios.

Abstract

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.

Problem

• non-monotone k-submodular maximization
• support constraints
• approximation coefficient
• multilinear extension

Innovation

• k-submodular maximization
• multilinear extension
• approximation algorithm
• comparator-uniform linearization
• value-retaining rounding