Improved Approximation Algorithms for $k$-Submodular Maximization via Multilinear Extension
Keywords: $k$-submodular maximization, approximation algorithm, $k$-multilinear extension
TL;DR: We provide improved or optimal approximation algorithms for $k$-submodular maximization with various constraints, via a novel framework of multilinear extension.
Abstract: We investigate a generalized form of submodular maximization, referred to as $k$-submodular maximization, with applications across the domains of social networks and machine learning. In this work, we propose the multilinear extension of $k$-submodular functions and unified Frank-Wolfe-type frameworks based on that. This continuous framework accommodates 1) monotone or non-monotone functions, and 2) various constraint types including matroid constraints, knapsack constraints, and their combinations. Notably, we attain an asymptotically optimal $1/2$-approximation for monotone $k$-submodular maximization problems with knapsack constraints, surpassing previous $1/3$-approximation results, and a factor-$1/3$ approximation for non-monotone $k$-submodular maximization problems with knapsack constraints and matroid constraints which outperforms previous $0.245$-approximation results. The foundation for our analysis stems from new insights into specific linear and monotone properties pertaining to the multilinear extension.
Primary Area: optimization
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Submission Number: 3295
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