Adapting the Directed Grid Theorem into an FPT Algorithm

The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the Grid Theorem in digraphs was conjectured by Johnson et al. [JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely, they showed that there is a function $f(k)$ such that every digraph of directed tree-width at least $f(k)$ contains a cylindrical grid of size $k$ as a butterfly minor and stated that their proof can be turned into an XP algorithm, with parameter $k$, that either constructs a decomposition of the appropriate width, or finds the claimed large cylindrical grid as a butterfly minor. In this paper, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our main technical contributions are two FPT algorithms with parameter $k$. The first one either produces an arboreal decomposition of width $3k-2$ or finds a haven of order $k$ in a digraph $D$, improving on the original result for arboreal decompositions by Johnson et al. The second algorithm finds a well-linked set of order $k$ in a digraph $D$ of large directed tree-width. As tools to prove these results, we show how to solve a generalized version of the problem of finding balanced separators for a given set of vertices $T$ in FPT time with parameter $|T|$, a result that we consider to be of its own interest.Comment: 31 pages, 9 figure

Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large

We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC\u2708]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT\u2721] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP

A Trichotomy for Regular Simple Path Queries on Graphs

Regular path queries (RPQs) select nodes connected by some path in a graph. The edge labels of such a path have to form a word that matches a given regular expression. We investigate the evaluation of RPQs with an additional constraint that prevents multiple traversals of the same nodes. Those regular simple path queries (RSPQs) find several applications in practice, yet they quickly become intractable, even for basic languages such as (aa)* or a*ba*. In this paper, we establish a comprehensive classification of regular languages with respect to the complexity of the corresponding regular simple path query problem. More precisely, we identify the fragment that is maximal in the following sense: regular simple path queries can be evaluated in polynomial time for every regular language L that belongs to this fragment and evaluation is NP-complete for languages outside this fragment. We thus fully characterize the frontier between tractability and intractability for RSPQs, and we refine our results to show the following trichotomy: Evaluations of RSPQs is either AC0, NL-complete or NP-complete in data complexity, depending on the regular language L. The fragment identified also admits a simple characterization in terms of regular expressions. Finally, we also discuss the complexity of the following decision problem: decide, given a language L, whether finding a regular simple path for L is tractable. We consider several alternative representations of L: DFAs, NFAs or regular expressions, and prove that this problem is NL-complete for the first representation and PSPACE-complete for the other two. As a conclusion we extend our results from edge-labeled graphs to vertex-labeled graphs and vertex-edge labeled graphs.Comment: 15 pages, conference submissio

Finding detours is fixed-parameter tractable

We consider the following natural "above guarantee" parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study the related problem Exact Detour that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k, and a deterministic algorithm with running time about 6.745^k, showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs.Comment: Extended abstract appears at ICALP 201

New Menger-like dualities in digraphs and applications to half-integral linkages

We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs

Evaluation and Enumeration Problems for Regular Path Queries

Regular path queries (RPQs) are a central component of graph databases. We investigate decision- and enumeration problems concerning the evaluation of RPQs under several semantics that have recently been considered: arbitrary paths, shortest paths, and simple paths. Whereas arbitrary and shortest paths can be enumerated in polynomial delay, the situation is much more intricate for simple paths. For instance, already the question if a given graph contains a simple path of a certain length has cases with highly non-trivial solutions and cases that are long-standing open problems. We study RPQ evaluation for simple paths from a parameterized complexity perspective and define a class of simple transitive expressions that is prominent in practice and for which we can prove a dichotomy for the evaluation problem. We observe that, even though simple path semantics is intractable for RPQs in general, it is feasible for the vast majority of RPQs that are used in practice. At the heart of our study on simple paths is a result of independent interest: the two disjoint paths problem in directed graphs is W[1]-hard if parameterized by the length of one of the two paths

The complexity of general-valued CSPs seen from the other side

The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the $k$-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory.Comment: v2: Full version of a FOCS'18 paper; improved presentation and small correction