Improved approximation guarantees for multi-label optimization problems
Geometry-Dependent Approximation for Non-Monotone $k$-Submodular Maximization
Data Structures and AlgorithmsMachine Learning
Summary
This paper studies a difficult kind of optimization problem where you assign one of several labels to items to get the best possible total value, but the value can be tricky and not always increasing as you add labels. The authors found a new mathematical way to analyze and improve the quality of solutions depending on how much freedom you have to pick labels. They give explicit formulas for how good their method is, which improve on previous guarantees, and also show how to use their approach for related practical problems like resource allocation with constraints.
What this means in practice
- •For operations research teams: Maximize resource allocation efficiency in complex multi-label assignment problems under budget or matroid constraints using improved approximation guarantees.
- •For online optimization engineers: Implement better online decision algorithms with provable near-optimal regret bounds when feedback is limited to gradients or post-decision values.
Authors
Vaneet Aggarwal
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 $ζ(P)=\max\{t\in[0,1]:t {\bf 1} \in P\}$ ranges from $ζ=0$, which carries no geometric promise, to $ζ=1$, 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 $\underlineα_k(ζ)$, in closed form on all of $[0,1]$ when $k=2$. At $ζ=0$ 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 $ζ=1$ 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.