Minimizing Monotone Norms in Fractional Load Balancing
Daniel Blankenburg, Antonia Ellerbrock, Thomas Kesselheim, Jens Vygen
原文摘要
We devise a fully polynomial time approximation scheme for minimizing a monotone norm $\left\lVert\cdot\right\rVert$ over a convex set $X\subseteq \mathbb{R}^d_{\geq 0}$. We study the multi-customer setting, where $X=\sum_{i=1}^n X_i$ and the solution set $X_i\subseteq \mathbb{R}^d_{\geq 0}$ is given by a linear minimization oracle for each customer $i$. Our algorithm computes, for any given $\varepsilon>0$, a $(1+\varepsilon)$-approximate solution. With high probability, it uses $\tilde{\mathcal{O}}\left( \varepsilon^{-2}(n+d) \right)$ single-customer linear minimization oracle calls and the same number of generalized KL projections to the dual space $Y=\{y\in \mathbb{R}^d_{\geq 0}: \left\lVert y \right\rVert^* \le 1\}$. We assume no prior knowledge on $X$, and our running time depends only on $n$, $d$, and $\varepsilon$. Our result is robust in the sense that it allows for approximate linear minimization oracles and approximate KL projections. We show that the (approximate) KL projection can be performed efficiently for large classes of monotone norms, demonstrating the applicability of our approach. While such a result (with exact projection) is well-known for the $\ell_{\infty}$ norm via multiplicative weight updates and was recently obtained for ordered norms via a smooth norm approximation and a novel randomization, existing techniques do not extend to general monotone norms. In particular, a key feature used in previous proofs was dual price stability, which was derived from a global contraction property of the KL projection to the dual space $Y$. We show that this property does not extend to general monotone norms. Therefore, we use a different price mechanism, based on the well-known mirror descent paradigm, for which a local contraction property holds and guarantees stability.
math.OC · cs.DS