Isolation schemes for problems on decomposable graphs

Abstract

The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much e ort has been dedicated towards derandomization of the Isolation Lemma for speci c classes of problems. So far, the focus was mainly on problems solvable in polynomial time.In this paper, we study a setting that is more typical for NP-complete problems, and obtain partial derandomizations in the form of signi cantly 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( √ 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 xed H we obtain an algorithm for detecting a Hamiltonian cycle in an Hminor-free graph that runs in deterministic time 2 O( √ n) and uses polynomial space; this is the rst algorithm to achieve such complexity guarantees. For problems of more local nature, such as nding 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 ndings with several unconditional and conditional lower bounds, which show that many of the results cannot be signi cantly improved.