Understand
Arising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms in various communities including combinatorics, integer-linear programming, and numerical analysis.
- Many NP-hard problems are known to be solvable in $\widetilde{O}(n \cdot 2^{O(\mathrm{tw})})$ time, where $\mathrm{tw}$ is the treewidth of the input graph.
- Analogously, many problems in P should be solvable in $\widetilde{O}(n \cdot \mathrm{tw}^{O(1)})$ time; however, due to the lack of appropriate tools, only a few such results are currently known.
- [Fom+18] conjectured this to hold as broadly as all linear programs; in our paper, we show this is true: Given a linear program of the form $\min_{Ax=b,\ell \leq x\leq u} c^{\top} x$, and a width-$\tau$ tree decomposition of a graph $G_A$ related to $A$, we show how to solve it in time $$\widetilde{O}(n \cdot \tau^2 \log (1/\varepsilon)),$$ where $n$ is the number of variables and $\varepsilon$ is the relative accuracy.