Genus and the geometry of the cut graph

We study the quantitative geometry of graphs in terms of their genus, using the structure of certain "cut graphs," i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow/min-cut theorems, approximations for uniform and nonuniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus <i>g</i>, unless otherwise noted. • We show that such graphs admit an <i>O</i>(log <i>g</i>)-approximate multi-commodity max-flow/min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of <i>O</i>(<i>g</i>) [KPR93, FT03]. For general demands, we show that the worst possible gap is <i>O</i>(log <i>g</i> + <i>C</i><sub><i>P</i></sub>), where <i>C</i><sub><i>P</i></sub> is the gap for planar graphs. This dependence is optimal, and already yields a bound of <i>O</i>(log <i>g</i> + √log <i>n</i>), improving over the previous bound of <i>O</i>(√<i>g</i> log <i>n</i>) [KLMN04]. • We give an <i>O</i>(√log <i>g</i>)-approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of <i>O</i>(<i>g</i>) [FHL05]. • If a graph <i>G</i> has genus <i>g</i> and maximum degree <i>D</i>, we show that the <i>k</i>th Laplacian eigenvalue of <i>G</i> is (log <i>g</i>)<sup>2</sup> · <i>O</i>(<i>kgD/n</i>), improving over the previous bound of <i>g</i><sup>2</sup>·<i>O</i>(<i>kgD/n</i>) [KLPT09]. There is a lower bound of Ω(<i>kgD/n</i>), making this result almost tight. • We show that if (<i>X, d</i>) is the shortest-path metric on a graph of genus <i>g</i> and <i>S</i> ⊆ <i>X</i>, then every <i>L</i>-Lipschitz map <i>f</i>: <i>S</i> → <i>Z</i> into a Banach space <i>Z</i> admits an <i>O</i>(<i>L</i> log <i>g</i>)-Lipschitz extension <i>f</i>: <i>X</i> → <i>Z</i>. This improves over the previous bound of <i>O</i>(<i>Lg</i>) [LN05], and compares to a lower bound of Ω(<i>L</i>√log <i>g</i>). In a related way, we show that there is an <i>O</i>(log <i>g</i>)-approximation for the 0-extension problem on such graphs, improving over the previous <i>O</i>(<i>g</i>) bound. • We show that every <i>n</i>-vertex shortest-path metric on a graph of genus <i>g</i> embeds into <i>L</i><sub>2</sub> with distortion <i>O</i>(log <i>g</i> + √log <i>n</i>), improving over the previous bound of <i>O</i>(√<i>g</i> log <i>n</i>). Our result is asymptotically optimal for every dependence <i>g</i> = <i>g</i>(<i>n</i>).

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