For a graph \(G\), the parameter treedepth measures the minimum depth among all forests \(F\), called elimination forests, such that \(G\) is a subgraph of the ancestor-descendant closure of \(F\). We introduce a logic, called neighborhood operator logic with acyclicity, connectivity and clique constraints \((\mathsf{NEO_2[FRec]\!+\!ACK}\) for short\()\), that captures all NP-hard problems—like Independent Set or Hamiltonian Cycle—that are known to be tractable in time \(2^{\mathcal{O}(\mathsf{td})} n^{\mathcal{O}(1)}\) and space \(n^{\mathcal{O}(1)}\) on \(n\)-vertex graphs provided with elimination forests of depth \(\mathsf{td}\). We provide a model checking algorithm for \(\mathsf{NEO_2[FRec]\!+\!ACK}\) with such complexity that unifies and extends these results. For \(\mathsf{NEO_2[FRec]\!+\!K}\), the fragment of the above logic that does not use acyclicity and connectivity constraints, we get a strengthening of this result, where the space complexity is reduced to \(\mathcal{O}(\mathsf{td}\log (n))\).

In this work we contribute to the study of the fine-grained complexity of problems parameterized by multi-clique-width, which was initiated by Fürer [ITCS 2017] and pursued further by Chekan and Kratsch [MFCS 2023]. Multi-clique-width is a parameter defined analogously to clique-width but every vertex is allowed to hold multiple labels simultaneously. This parameter is upper-bounded by both clique-width and treewidth (plus a constant), hence it generalizes both of them without an exponential blow-up. Conversely, graphs of multi-clique-width k have clique-width at most 2^k, and there exist graphs with clique-width at least 2^{Ω(k)}. Thus, while the two parameters are functionally equivalent, the fine-grained complexity of problems may differ relative to them. As our first and main result we show that under ETH the Max Cut problem cannot be solved in time n^{2^{o(k)}} ⋅ f(k) on graphs of multi-clique-width k for any computable function f. For clique-width k an n^{𝒪(k)} algorithm by Fomin et al. [SIAM J. Comput. 2014] is tight under ETH. This makes Max Cut the first known problem for which the tight running times differ for parameterization by clique-width and multi-clique-width and it contributes to the short list of known lower bounds of form n^{2^{o(k)}} ⋅ f(k). As our second contribution we show that Hamiltonian Cycle and Edge Dominating Set can be solved in time n^{𝒪(k)} on graphs of multi-clique-width k matching the tight running time for clique-width. These results answer three questions left open by Chekan and Kratsch [MFCS 2023].

We prove that Hamiltonian Path and Hamiltonian Cycle are NP -hard on graphs of linear mim-width 26, even when a linear order of the input graph with mim-width 26 is provided together with input. This fills a gap left by a broken proof of the para-NP-hardness of Hamiltonicity problems parameterized by mim-width.

F-branchwidth is a framework for width measures of graphs, recently introduced by Eiben et al. [ITCS 2022], that captures tree-width, co-tree-width, clique-width, and mim-width, and several of their generalizations and interpolations. In this work, we search for algorithmic applications of F-branchwidth measures that do not have an equivalent counterpart in the literature so far. Our first contribution is a minimal set of eleven F-branchwidth measures such that each of the infinitely many F-branchwidth measures is equivalent to one of the eleven. We observe that for the FO Model Checking problem, each F-branchwidth is either equivalent to clique-width (and therefore has an FPT-algorithm by formula length plus the width) or the problem remains as hard as on general graphs even on graphs of constant width. Next, we study the number of equivalence classes of the neighborhood equivalence in a decomposition, which upper bounds the run time of the model checking algorithm for ACDN logic recently introduced by Bergougnoux et al. [SODA 2023]. We give structural lower bounds that show that for each F-branchwidth, an efficient model checking algorithm was already known or cannot be obtained via this method. Lastly, we classify the complexity of Independent Set parameterized by any F-branchwidth except for one open case. Also here, our contributions are lower bounds. In this context, we also prove that Independent Set on graphs of mim-width w cannot be solved in time n^o(w) unless the Exponential Time Hypothesis fails, answering an open question in the literature.

We show that it is NP-hard to distinguish graphs of linear mim-width at most 1211 from graphs of sim-width at least 1216. This implies that Mim-Width, Sim-Width, One-Sided Mim-Width, and their linear counterparts are all paraNP-complete, i.e., NP-complete to compute even when upper bounded by a constant. A key intermediate problem that we introduce and show NP-complete, Linear Degree Balancing, inputs an edge-weighted graph G and an integer τ, and asks whether V(G) can be linearly ordered such that every vertex of G has weighted backward and forward degrees at most τ.

We prove that Hamiltonian Path and Hamiltonian Cycle are NP-hard on graphs of linear mim-width 26, even when a linear order of the input graph with mim-width 26 is provided together with input. This fills a gap left by a broken proof of the para-NP-hardness of Hamiltonicity problems parameterized by mim-width.

This paper considers four problems with possible applications in network design: Given a graph G with | G | = n and an integer k ≥ 0 , does G have a DFS tree with (i) ≤ k leaves, (ii) ≥ k leaves, (iii) ≤ n − k leaves, and (iv) ≥ n − k leaves? We show that all four problems are NP-hard. When parameterized by k , we prove that while (i) is para-NP-hard and (ii) is W[1]-hard, both (iii) and (iv) admit polynomial kernels with O ( k 3 ) vertices, implying FPT algorithms running in k O ( k ) ⋅ n O ( 1 ) time. Our polynomial kernels are based on a O ( k ) -sized vertex cover structure associated with the solution of these problems. As a byproduct, we obtain polynomial kernels for these problems parameterized by the vertex cover number of the input graph.

Abstract. We spotted an error in our publication More applications of the d-neighbor equivalence: Acyclicity and Connetivity constraints [ SIAM J. Discrete Math., 35 (2021), pp. 1881–1926]. We explain the problem and suggest a simple correction.

Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on $n$-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth $d$ and using $k$ labels, we can solve - Independent Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $O(dk^2\log n)$ space; - Max Cut in time $n^{O(dk)}$ using $O(dk\log n)$ space; and - Dominating Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $n^{O(1)}$ space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with $d$ if one wishes to keep the space complexity polynomial.

We introduce a logic called distance neighborhood logic with acyclicity and connectivity constraints (A&C DN for short) which extends existential MSO1 with predicates for querying neighborhoods of vertex sets in various powers of a graph and for verifying connectivity and acyclicity of vertex sets. Building upon [Bergougnoux and Kante, ESA 2019; SIDMA 2021], we show that the model checking problem for every fixed A&C DN formula is solvable in nO(w) time when the input graph is given together with a branch decomposition of mim-width W. Nearly all problems that are known to be solvable in polynomial time given a branch decomposition of constant mim-width can be expressed in this framework. We add several natural problems to this list, including problems asking for diverse sets of solutions. Our model checking algorithm is efficient whenever the given branch decomposition of the input graph has small index in terms of the d-neighborhood equivalence [Bui-Xuan, Telle, and Vatshelle, TCS 2013]. We therefore unify and extend known algorithms for tree-width, clique-width and rank-width. Our algorithm has a single-exponential dependence on these three width measures and asymptotically matches run times of the fastest known algorithms for several problems. This results in algorithms with tight run times under the Exponential Time Hypothesis (ETH) for tree-width, clique-width and rank-width; the above mentioned run time for mim-width is nearly tight under the ETH for several problems as well. Our results are also tight in terms of the expressive power of the logic: we show that already slight extensions of our logic make the model checking problem para-NP-hard when parameterized by mim-width plus formula length. * The full version of the paper can be accessed at https://arxiv.org/abs/2202.13335. This research is part of a project that has received funding from the Research Council of Norway Grant Agreement 274526 (LJ).

We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known $2^{O(k^2)} n^{O(1)}$ time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width $k$ are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for $n$-vertex graphs.

Abstract The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $$2^{O(rw^3)}\cdot n^{4}$$ 2 O ( r w 3 ) · n 4 , $$2^{O(q^2\log (q))}\cdot n^{4}$$ 2 O ( q 2 log ( q ) ) · n 4 , and $$n^{O(k^2)}$$ n O ( k 2 ) where rw is the rank-width, q the $${\mathbb {Q}}$$ Q -rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

Abstract In theDirected Feedback Vertex Set (DFVS)problem, the input is a directed graphDand an integerk. The objective is to determine whether there exists a set of at mostkvertices intersecting every directed cycle ofD. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity ofDFVSparameterized by a naturallargerparameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlyingundirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that solves all these problems and whose running time is upper bounded by $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$, and $n^{O(k)}$ where $k$ is respectively the clique-width, $\mathbb{Q}$-rank-width, rank-width, and maximum induced matching width of a given decomposition. Our approach simplifies and unifies the known algorithms for each of the parameters and its running time matches asymptotically also the running times of the best known algorithms for basic \sf NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the $d$-neighbor equivalence defined in [B. Bui-Xuan, J. A. Telle, and M. Vatshelle, Theoret. Comput. Sci., (2013), pp. 66--76] and the rank-based approach introduced in [H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof, Inform. and Comput., 243 (2015), pp. 86--111]. The results we obtain highlight the importance of the $d$-neighbor equivalence relation on the algorithmic applications of width measures. We also prove that our framework could be useful for ${\sf W}[1]$-hard problems parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For these latter problems, we obtain $n^{O(k)}$, $n^{O(k)}$, and $n^{2^{O(k)}}$ time algorithms where $k$ is respectively the clique-width, the $\mathbb{Q}$-rank-width, and the rank-width of the input graph.

The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connected versions of the classical graph problems (such as Vertex Cover and Dominating Set). We show that Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Restricted Edge-Subset Feedback Edge Set, Node Multiway Cut, and Multiway Cut are unlikely to have such running times. More precisely, we match algorithms running in time $2^{O(tw \log tw)}n^{O(1)}$ with tight lower bounds under the Exponential-Time Hypothesis (ETH), ruling out $2^{o(tw \log tw)}n^{O(1)}$, where $n$ is the number of vertices and $tw$ is the treewidth of the input graph. Our algorithms extend to the weighted case, while our lower bounds also hold for the larger parameter pathwidth and do not require weights. We also show that, in contrast to Odd Cycle Transversal, there is no $2^{o(tw \log tw)}n^{O(1)}$-time algorithm for Even Cycle Transversal under the ETH.

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width.

In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.