In this paper we consider maintaining strongly connected components (SCCs) of a directed planar graph subject to edge insertions and deletions. We show a data structure maintaining an implicit representation of the SCCs within Õ(n^{6/7}) worst-case time per update. The data structure supports, in O(log²{n}) time, reporting vertices of any specified SCC (with constant overhead per reported vertex) and aggregating vertex information (e.g., computing the maximum label) over all the vertices of that SCC. Furthermore, it can maintain global information about the structure of SCCs, such as the number of SCCs, or the size of the largest SCC. To the best of our knowledge, no fully dynamic SCCs data structures with sublinear update time have been previously known for any major subclass of digraphs. Our result should be contrasted with the n^{1-o(1)} amortized update time lower bound conditional on SETH, which holds even for dynamically maintaining whether a general digraph has more than two SCCs.

The treedepth of a graph G is the least possible depth of an elimination forest of G: a rooted forest on the same vertex set where every pair of vertices adjacent in G is bound by the ancestor/descendant relation. We propose an algorithm that given a graph G and an integer d, either finds an elimination forest of G of depth at most d or concludes that no such forest exists; thus the algorithm decides whether the treedepth of G is at most d. The running time is 2^𝒪(d²)⋅n^𝒪(1) and the space usage is polynomial in n. Further, by allowing randomization, the time and space complexities can be improved to 2^𝒪(d²)⋅n and d^𝒪(1)⋅n, respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014], which also has time complexity 2^𝒪(d²)⋅n, but uses exponential space.

We study the Many-Visits Traveling Salesman Problem, where given a number k(v) for each of n cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city v exactly k(v) times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space O⁎(5n). They also show a polynomial-space algorithm running in time O(16n+o(n)). In this work, we show three main results: A randomized polynomial-space algorithm running in time O⁎(2nD), where D is the maximum distance between two cities. By using standard methods, this results in a (1+ϵ)-approximation running in time O⁎(2nϵ−1). A tight analysis of Berger et al.'s exponential-space algorithm, resulting in an O⁎(4n) running time bound. A new polynomial-space algorithm, running in time O(7.88n).

In this work, we present the first linear time deterministic algorithm computing the 4-edge-connected components of an undirected graph. First, we show an algorithm listing all 3-edge-cuts in a given 3-edge-connected graph, and then we use the output of this algorithm in order to determine the 4-edge-connected components of the graph.

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