## Discrete-Convex-Analysis-Based Framework for Warm-Starting Algorithms with Predictions

### Shinsaku Sakaue · Taihei Oki

##### Hall J #541

Keywords: [ discrete convex analysis ] [ algorithms with predictions ] [ Combinatorial Optimization ] [ time complexity ]

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Wed 30 Nov 2 p.m. PST — 4 p.m. PST

Abstract: Augmenting algorithms with learned predictions is a promising approach for going beyond worst-case bounds. Dinitz, Im, Lavastida, Moseley, and Vassilvitskii~(2021) have demonstrated that warm-starts with learned dual solutions can improve the time complexity of the Hungarian method for weighted perfect bipartite matching. We extend and improve their framework in a principled manner via \textit{discrete convex analysis} (DCA), a discrete analog of convex analysis. We show the usefulness of our DCA-based framework by applying it to weighted perfect bipartite matching, weighted matroid intersection, and discrete energy minimization for computer vision. Our DCA-based framework yields time complexity bounds that depend on the $\ell_\infty$-distance from a predicted solution to an optimal solution, which has two advantages relative to the previous $\ell_1$-distance-dependent bounds: time complexity bounds are smaller, and learning of predictions is more sample efficient. We also discuss whether to learn primal or dual solutions from the DCA perspective.

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