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- W4366282323 abstract "We prove that there is a randomized polynomial-time algorithm that given an edge-weighted graph $G$ excluding a fixed-minor $Q$ on $n$ vertices and an accuracy parameter $varepsilon>0$, constructs an edge-weighted graph~$H$ and an embedding $etacolon V(G)to V(H)$ with the following properties: * For any constant size $Q$, the treewidth of $H$ is polynomial in $varepsilon^{-1}$, $log n$, and the logarithm of the stretch of the distance metric in $G$. * The expected multiplicative distortion is $(1+varepsilon)$: for every pair of vertices $u,v$ of $G$, we have $mathrm{dist}_H(eta(u),eta(v))geq mathrm{dist}_G(u,v)$ always and $mathrm{Exp}[mathrm{dist}_H(eta(u),eta(v))]leq (1+varepsilon)mathrm{dist}_G(u,v)$. Our embedding is the first to achieve polylogarithmic treewidth of the host graph and comes close to the lower bound by Carroll and Goel, who showed that any embedding of a planar graph with $mathcal{O}(1)$ expected distortion requires the host graph to have treewidth $Omega(log n)$. It also provides a unified framework for obtaining randomized quasi-polynomial-time approximation schemes for a variety of problems including network design, clustering or routing problems, in minor-free metrics where the optimization goal is the sum of selected distances. Applications include the capacitated vehicle routing problem, and capacitated clustering problems." @default.
- W4366282323 created "2023-04-20" @default.
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- W4366282323 date "2023-04-14" @default.
- W4366282323 modified "2023-09-28" @default.
- W4366282323 title "Planar and Minor-Free Metrics Embed into Metrics of Polylogarithmic Treewidth with Expected Multiplicative Distortion Arbitrarily Close to 1" @default.
- W4366282323 doi "https://doi.org/10.48550/arxiv.2304.07268" @default.
- W4366282323 hasPublicationYear "2023" @default.
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