Greedy Region Growing: Improved Approximations for Cutwidth, Pathwidth, Flow-Expander Decomposition, and More

We give a new rounding technique for spreading-metric linear programs, called Greedy Region Growing, which adapts the classical technique of region growing to min-max graph layout problems and flow-expander decomposition. Using this technique, we obtain state-of-the-art approximation algorithms for several fundamental problems. (1) An $O(\log n)$-approximation for Cutwidth and Pathwidth, improving upon the previous bound of $O(\log n\exp (O(\sqrt{\log \log n})))$ due to Bansal, Katzelnick, and Schwartz (FOCS 2024). This matches the $\mathrm{\Omega }(\log n)$ integrality gap of the standard LP relaxations. (2) An $O(\log n)$-approximation for their directed analogues, Minimum Storage and Minimum Register Sufficiency on directed acyclic graphs, improving the previous bound of $O({\log }^{3\mathrm{/}2}n)$. (3) A $\phi$-flow-expander decomposition that cuts at most an $O(\phi \log n\log \log n)$ fraction of the edges. This improves the $O(\phi \log n\exp (O(\sqrt{\log \log n})))$ bound of Bansal, Jambulapati, and Saranurak (ICALP 2026), and is within an $O(\log \log n)$ factor of the graph-theoretic lower bound. The vertex version of our algorithm also gives an $O(\log n)$-approximation for Treedepth. In particular, our results answer several open problems of Bansal, Katzelnick, and Schwartz (FOCS 2024). Technically, we also solve the conceptual question raised by their work: our technique shows how to adapt Seymour recursion to min-max objectives, and constructs $O(1)$-weak local partitions in spreading metrics.