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In the setting of DynFO, dynamic programs update the stored result of a query whenever the underlying data changes. This update is expressed in terms of first-order logic. We introduce a strategy for constructing dynamic programs that utilises periodic computation of auxiliary data from scratch and the ability to maintain a query for a limited number of change steps. We show that if some program can maintain a query for log n change steps after an AC 1 -computable initialisation, it can be maintained by a first-order dynamic program as well, i.e., in DynFO. As an application, it is shown that decision and optimisation problems defined by monadic second-order (MSO) formulas are in DynFO, if only change sequences that produce graphs of bounded treewidth are allowed. To establish this result, a Feferman-Vaught-type composition theorem for MSO is established that might be useful in its own right.

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For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph G with edge costs, a set R of terminal vertices, and an integer demand ds,t for every terminal pair s, t ∈ R. The task is to compute a subgraph H of G of minimum cost, such that there are at least ds,t disjoint paths between s and t in H. Depending on the type of disjointness we obtain several variants of SNDP that have been widely studied in the literature: if the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity variant (LC-SNDP), where the paths are disjoint on edges and non-terminals, i.e., they may only share terminals.In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size , the sum of demands D, the number of terminals k, and the maximum demand dmax. Using simple, elegant arguments, we prove the following results.• We give a complete picture of the parameterized tractability of the three variants w.r.t. parameter : both EC-SNDP and LC-SNDP are FPT, while VC-SNDP is W[1]-hard (even in the uniform single-source case with k = 3). • We identify some special cases of VC-SNDP that are FPT:when dmax ≤ 3 for parameter , on locally bounded treewidth graphs (e.g., planar graphs) for parameter , and on graphs of treewidth tw for parameter tw + D, which is in contrast to a result by Bateni et al. [JACM 2011] who show NP-hardness for tw = 3 (even if dmax = 1, i.e., the Steiner Forest problem). • The well-known Directed Steiner Tree (DST) problem can be seen as single-source EC-SNDP with dmax = 1 on directed graphs, and is FPT parameterized by k [Dreyfus & Wagner 1971]. We show that in contrast, the 2-DST problem, where dmax = 2, is W[1]-hard, even when parameterized by (which is always larger than k). * Supported by the Czech Science Foundation GA ČR (grant #19-27871X), and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).

Continuing the recent trend, in this article we design several space-efficient algorithms for two well-known graph search methods. Both these search methods share the same name breadth-depth search (henceforth BDS), although they work entirely in different fashion. The classical implementation for these graph search methods takes O(m + n) time and O(n lg n) bits of space in the standard word RAM model (with word size being Θ(lg n) bits), where m and n denotes the number of edges and vertices of the input graph respectively. Our goal here is to beat the space bound of the classical implementations, and design o(n lg n) space algorithms for these search methods by paying little to no penalty in the running time. Note that our space bounds (i.e., with o(n lg n) bits of space) do not even allow us to explicitly store the required information to implement the classical algorithms, yet our algorithms visits and reports all the vertices of the input graph in correct order.