Isolation schemes for problems on decomposable graphs

In this paper, we study a setting that is more typical for NP-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use

- $O(t\log n + \log^2{n})$ random bits on graphs of treewidth at most $t$;
- $O(\sqrt{n})$ random bits on planar or $H$-minor free graphs; and
- $O(n)$-random bits on general graphs.

In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed $H$ we obtain an algorithm for detecting a Hamiltonian cycle in an $H$-minor-free graph that runs in deterministic time $2^{O(\sqrt{n})}$ and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most $d$ that use $O(d)$ random bits and assign polynomially-bounded weights.

We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.

This is joint work with Jesper Nederlof, Michał Pilipczuk and Céline Swennenhuis.