Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space.

For many algorithmic problems on graphs of treewidth t, a standard dynamic programming approach gives algorithms with time and space complexity 2(O(t))center dot n(O(1)). It turns out that when one considers the more restrictive parameter treedepth, it is often the case that a variation of this technique can be used to reduce the space complexity to polynomial, while retaining time complexity of the form 2(O(d))center dot n(O(1)), where d is the treedepth. This transfer of methodology is, however, far from automatic. For instance, for problems with connectivity constraints, standard dynamic programming techniques give algorithms with time and space complexity 2(O(t) (log t))center dot n(O(1)) on graphs of treewidth t, but it is not clear how to convert them into time-efficient polynomial space algorithms for graphs of low treedepth. Cygan et al. [ACM Trans. Algorithms, 18 (2022), 17] introduced the Cut&Count technique and showed that a certain class of problems with connectivity constraints can be solved in time and space complexity 2(O(t))center dot n(O(1)). Recently, Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the Cut&Count technique can be also applied in the setting of treedepth, and it gives algorithms with running time 2(O(d))center dot n(O(1)) and polynomial space usage. However, several important problems eluded such a treatment, with the most prominent examples being Hamiltonian Cycle and Longest Path. In this paper, we clarify the situation by showing that Hamiltonian cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit 5d center dot n(O(1))-time and polynomial space algorithms on graphs of treedepth d. The algorithms are randomized Monte Carlo with only false negatives.