Finding a few solutions for a given problem that are diverse, as opposed to finding a single best solution to solve the problem, has recently become a notable topic in theoretical computer science. Recently, Baste, Fellows, Jaffke, Masařík, Oliveira, Philip, and Rosamond showed that under a standard structural parameterization by treewidth, one can find a set of diverse solutions for many problems with only a very small additional cost [Artificial Intelligence 2022]. In this paper, we investigate a much stronger graph parameter, the cliquewidth, which can additionally describe some dense graph classes. Broadly speaking, it describes graphs that can be recursively constructed by a few operations defined on graphs whose vertices are divided into a bounded number of groups, while each such group behaves uniformly with respect to any operation. We show that for any vertex problem, if we are given a dynamic program solving that problem on cliquewidth decomposition, we can modify it to produce a few solutions that are as diverse as possible with as little overhead as in the above-mentioned treewidth paper. As a consequence, we prove that a diverse version of any MSO1 expressible problem can be solved in linear FPT time parameterized by the cliquewidth, the number of sought solutions, and the number of quantifiers in the formula, which was a natural missing piece in the complexity landscape of structural graph parameters and logic for the diverse problems. We prove our results, allowing for a more general natural collection of diversity functions compared to only two mostly studied diversity functions previously. That might be of independent interest as a larger pool of different diversity functions can highlight various aspects of different solutions to a problem.

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique K_{𝓁,𝓁} forces tree-independence number at least 𝓁. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers t and 𝓁, {P_t,K_{𝓁,𝓁}}-free graphs have bounded tree-independence number. We prove this conjecture for t = 5 by showing that every {P₅,K_{𝓁,𝓁}}-free graph has tree-independence number at most 4𝓁. We also obtain related bounds for the weaker parameter of α-degeneracy.

The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan's theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC '90, Journal of the AMS '90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.

We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroché [Discrete Appl. Math. 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice. We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for "almost-subcubic" graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates (FO+DP) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].

We study a generalization of the classic Global Min-Cut problem, called Global Label Min-Cut (or sometimes Global Hedge Min-Cut ): the edges of the input (multi)graph are labeled (or partitioned into color classes or hedges), and removing all edges of the same label (color or from the same hedge) costs one. The problem asks to disconnect the graph at minimum cost. While the \(st\) -cut version of the problem is known to be \(\mathsf{NP}\) -hard, the above global cut version is known to admit a quasi-polynomial randomized \(n^{\mathcal{O}(\log\mathrm{OPT})}\) -time algorithm due to Ghaffari, Karger, and Panigrahi [SODA 2017]. They consider this as “strong evidence that this problem is in P ” . We show that this is actually not the case. We complete the study of the complexity of the Global Label Min-Cut problem by showing that the quasi-polynomial running time is probably optimal: We show that the existence of an algorithm with running time \((np)^{o(\log n/(\log\log n)^{2})}\) would contradict the Exponential Time Hypothesis, where \(n\) is the number of vertices, and \(p\) is the number of labels in the input. The key step for the lower bound is a proof that Global Label Min-Cut is \(\mathsf{W}\) [1]-hard when parameterized by the number of uncut labels . In other words, the problem is difficult in the regime where almost all labels need to be cut to disconnect the graph.

Distributed networks are prone to errors so verifying their output is critical. We develop local certification protocols for graph properties in which nodes are given certificates that allow them to check whether the network as a whole satisfies some fixed property while only communicating with their local network. Instead of considering a specific problem and developing a local certification protocol tailor-made for the problem, we aim for generic protocols that can certify any property expressible in a certain logical framework.

Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masařík and Hliněný [GD 2023] recently asked for each edge of $G$ to be drawn without crossings while allowing multiple different drawings of $G$. More formally, a collection $\mathcal{D}$ of drawings of $G$ is uncrossed if, for each edge $e$ of $G$, there is a drawing in $\mathcal{D}$ such that $e$ is uncrossed. The uncrossed number $\mathrm{unc}(G)$ of $G$ is then the minimum number of drawings in some uncrossed collection of $G$. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hliněný and Masařík. We also present a strong general lower bound on $\mathrm{unc}(G)$ in terms of the number of vertices and edges of $G$. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph $G$, which is the smallest number of edges of $G$ taken over all drawings of $G$ that participate in a crossing. This problem was posed as open by Schaefer in his book [Crossing Numbers of Graphs 2018].

We consider a voting model, where a number of candidates need to be selected subject to certain feasibility constraints. The model generalizes committee elections (where there is a single constraint on the number of candidates that need to be selected), various elections with diversity constraints, the model of public decisions (where decisions need to be taken on a number of independent issues), and the model of collective scheduling. A critical property of voting is that it should be fair---not only to individuals but also to groups of voters with similar opinions on the subject of the vote; in other words, the outcome of an election should proportionally reflect the voters' preferences.

A random 2-cell embedding of a connected graph G in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes - those of a bouquet of n loops and those of n parallel edges connecting two vertices - have been extensively studied and are well-understood. However, little is known about more general graphs despite their important connections with central problems in mainstream mathematics and in theoretical physics (see [Lando & Zvonkin, Graphs on surfaces and their applications, Springer 2004]). There are also tight connections with problems in computing (random generation, approximation algorithms). The results of this paper, in particular, explain why Monte Carlo methods (see, e.g., [Gross & Tucker, Local maxima in graded graphs of imbeddings, Ann. NY Acad. Sci 1979] and [Gross & Rieper, Local extrema in genus stratified graphs, JGT 1991]) cannot work for approximating the minimum genus of graphs.

We show that the Maximum Weight Independent Set problem (MWIS) can be solved in quasi-polynomial time on H-free graphs (graphs excluding a fixed graph H as an induced subgraph) for every H whose every connected component is a path or a subdivided claw (i.e., a tree with at most three leaves). This completes the dichotomy of the complexity of MWIS in F-free graphs for any finite set F of graphs into NP-hard cases and cases solvable in quasi-polynomial time, and corroborates the conjecture that the cases not known to be NP-hard are actually polynomial-time solvable. The key graph-theoretic ingredient in our result is as follows. Fix an integer t ≥ 1. Let St,t,t be the graph created from three paths on t edges by identifying one endpoint of each path into a single vertex. We show that, given a graph G, one can in polynomial time find either an induced St,t,t in G, or a balanced separator consisting of O(log|V(G)|) vertex neighborhoods in G, or an extended strip decomposition of G (a decomposition almost as useful for recursion for MWIS as a partition into connected components) with each particle of weight multiplicatively smaller than the weight of G. This is a strengthening of a result of Majewski, Masařík, Novotná, Okrasa, Pilipczuk, Rzążewski, and Sokołowski [Transactions on Computation Theory ‍2024] which provided such an extended strip decomposition only after the deletion of O(log|V(G)|) vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above.

Abstract. One of the first applications of the recently introduced technique of flow augmentation [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of Directed Feedback Vertex Set, a landmark problem in parameterized complexity. In this article, we explore the applicability of flow augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following: In weighted undirected graphs, Multicut is fixed-parameter tractable (FPT) in both the edge- and the vertex-deletion version. The weighted version of Group Feedback Vertex Set is FPT, even with oracle access to group operations. The weighted version of Directed Subset Feedback Vertex Set is FPT. Our study reveals Directed Symmetric Multicut as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting.

We show fixed-parameter tractability of the DIRECTED MULTICUT problem with three terminal pairs (with a randomized algorithm). In this problem we are given a directed graph G, three pairs of vertices (called terminals) (s1, t1), (s2, t2), (s3, t3), and an integer k and we want to find a set of at most k non-terminal vertices in G that intersect all s1t1-paths, all s2t2-paths, and all s3t3-paths. The parameterized complexity of this problem has been open since Chitnis, Hajiaghayi, and Marx proved fixed-parameter tractability of the two-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the four-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the “middle” box in the grid minor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.07425. The research leading to the results presented in this paper was partially carried out during the Parameterized Algorithms Retreat of the University of Warsaw, PARUW 2022, held in Bedlewo in April 2022. This research is a part of projects that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant Agreement 714704 (TM, MP) and 648527 (MH), from the Alexander von Humboldt Foundation (MS), from the Research Council of Norway (LJ), and by the Federal Ministry of Education and Research (BMBF) and by a fellowship within the IFI programme of the German Academic Exchange Service (DAAD). (MH).

We study a generalization of the classic GLOBAL MIN-CUT problem, called GLOBAL LABEL MIN-CUT (or sometimes GLOBAL HEDGE MIN-CUT): the edges of the input (multi)graph are labeled (or partitioned into color classes or hedges), and removing all edges of the same label (color or from the same hedge) costs one. The problem asks to disconnect the graph at minimum cost. While the st-cut version of the problem is known to be NP-hard, the above global cut version is known to admit a quasi-polynomial randomized nO(log OPT)-time algorithm due to Ghaffari, Karger, and Panigrahi [SODA 2017]. They consider this as “strong evidence that this problem is in P”. We show that this is actually not the case. We complete the study of the complexity of the Global Label Min-Cut problem by showing that the quasi-polynomial running time is probably optimal: We show that the existence of an algorithm with running time (np)o(log n/(log log n)2) would contradict the Randomized Exponential Time Hypothesis, where n is the number of vertices, and p is the number of labels in the input. The key step for the lower bound is a proof that Global Label Min-Cut is W[1]-hard when parameterized by the number of uncut labels. In other words, the problem is difficult in the regime where almost all labels need to be cut to disconnect the graph. To turn this lower bound into a quasi-polynomial-time lower bound, we also needed to revisit the framework due to Marx [Theory Comput. 2010] of proving lower bounds assuming Exponential Time Hypothesis through the SUBGRAPH ISOMORPHISM problem parameterized by the number of edges of the pattern. Here, we provide an alternative simplified proof of the hardness of this problem that is more versatile with respect to the choice of the regimes of the parameters. * This research is a part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant Agreement 714704 (LJ, TM, MP, US) and from the Research Council of Norway (LJ).

Abstract. Motivated by the Beck–Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on [Formula: see text] vertices and [Formula: see text] edges. In the first model, each of the [Formula: see text] edges is constructed by placing each vertex into the edge independently with probability [Formula: see text], where [Formula: see text] is a parameter satisfying [Formula: see text] and [Formula: see text]. In the second model, each vertex independently chooses a subset of [Formula: see text] edge labels from [Formula: see text] uniformly at random. Edge [Formula: see text] is then defined to be exactly those vertices whose [Formula: see text]-subsets include label [Formula: see text]. In the sparse regime, i.e., when [Formula: see text], we show that with high probability a random hypergraph from either model has discrepancy at least [Formula: see text]. In the dense regime, i.e., when [Formula: see text], we show that with high probability a random hypergraph from either model has discrepancy at least [Formula: see text], where [Formula: see text]. Furthermore, we obtain nearly matching asymptotic upper bounds on the discrepancy. Specifically, we apply the partial coloring lemma of Lovett and Meka to show that, in the dense regime, with high probability the two random hypergraph models each have discrepancy [Formula: see text]. In fact, in a significant parameter range we can tighten our analysis to get an upper bound which matches our lower bound up to a constant factor. This result is algorithmic, and together with the work of Bansal and Meka [ On the discrepancy of random low degree set systems, in Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms, 2019, pp. 2557–2564] characterizes how the discrepancy of each random hypergraph transitions from [Formula: see text] to [Formula: see text] as [Formula: see text] increases from [Formula: see text] to [Formula: see text].

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw $S_{t,t,t}$ as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time $2^{\mathcal{O}(\sqrt{n}\log n)}$ and a quasipolynomial-time approximation scheme with improved running time $2^{\mathcal{O}(\varepsilon^{-1} \log^{5} n)}$. The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in $P_t$-free graphs, ensures that given an $n$-vertex $P_t$-free graph, in polynomial time we can find a set $P$ of at most $t-1$ vertices, such that every connected component of $G-N[P]$ has at most $n/2$ vertices. Our main technical contribution is an analog of this result for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, in polynomial time we can find a set $P$ of $\mathcal{O}(t \log n)$ vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of $G-N[P]$ such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most $n/2$ vertices.

A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time. We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family 𝒮 consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P₃,C₄} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ∈ 𝒮 (unless the ETH fails).

A graph where each vertex v has a list L(v) of available colors is L-colorable if there is a proper coloring such that the color of v is in L(v) for each v. A graph is k-choosable if every assignment L of at least k colors to each vertex guarantees an L-coloring. Given a list assignment L, an L-request for a vertex v is a color c∈L(v). In this paper, we look at a variant of the widely studied class of precoloring extension problems from Dvořák, Norin, and Postle (J. Graph Theory, 2019), wherein one must satisfy “enough”, as opposed to all, of the requested set of precolors. A graph G is ɛ-flexible for list size k if for any k-list assignment L, and any set S of L-requests, there is an L-coloring of G satisfying ɛ-fraction of the requests in S. It is conjectured that planar graphs are ɛ-flexible for list size 5, yet it is proved only for list size 6 and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masařík and show that planar graphs without K4− are ɛ-flexible for list size 5. We also prove that planar graphs without 4-cycles and 3-cycle distance at least 2 are ɛ-flexible for list size 4. Finally, we introduce a new (slightly weaker) form of ɛ-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are ɛ-flexible for list size 5.

Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Equivalently, the same problem can be formulated as the minimum connected dominating set problem, where the task is to find a smallest subset of vertices $D\subseteq V(G)$ such that every vertex of $G$ is in the closed neighborhood of $D$. Throughout recent decades, these two equivalent problems have received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks. Recently, a similar but stronger notion was defined by Bradshaw, Masařík, and Stacho [ Flexible list colorings in graphs with special degeneracy conditions, in Proceedings of the 31st International Symposium on Algorithms and Computation (ISAAC 2020), LIPIcs. Leibniz Int. Proc. Inform. 181, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2020, article 31]. They introduced a new invariant for a graph $G$, called the robust connectivity and written as $\kappa_\rho(G)$, defined as the minimum value $\frac{|R \cap \Lambda (T)|}{|R|}$ taken over all nonempty subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to maximize $|R \cap \Lambda(T)|$. Large robust connectivity was originally used to show flexible choosability in nonregular graphs. In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of $\Omega(\gamma^{-\frac{1}{r}})$ for the robust connectivity of $r$-connected graphs of Euler genus $\gamma$. Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman [ A conjecture on planar graphs, in Graph Theory and Related Topics, Academic Press, San Diego, CA, 1979, p. 57], which states that every planar graph on $n$ vertices contains an induced forest of size at least $n/2$.

The Directed Grid Theorem, stating that there is a function $f$ such that a directed graph of directed treewidth at least $f(k)$ contains a directed grid of size at least $k$ as a butterfly minor, after being a conjecture for nearly 20 years, was proved in 2015 by Kawarabayashi and Kreutzer. However, the function $f$ obtained in the proof is very fast growing. In this work, we show that if one relaxes directed grid to bramble of constant congestion, one can obtain a polynomial bound. More precisely, we show that for every $k \geq 1$ there exists $t = \mathcal{O}(k^{48} \log^{13} k)$ such that every directed graph of directed treewidth at least $t$ contains a bramble of congestion at most 8 and size at least $k$.

Given two disjoint sets W1 and W2 of points in the plane, the Optimal Discretization problem asks for the minimum size of a family of horizontal and vertical lines that separate W1 from W2, that is, in every region into which the lines partition the plane there are either only points of W1, or only points of W2, or the region is empty. Equivalently, Optimal Discretization can be phrased as a task of discretizing continuous variables: We would like to discretize the range of x-coordinates and the range of y-coordinates into as few segments as possible, maintaining that no pair of points from W1 × W2 are projected onto the same pair of segments under this discretization. We provide a fixed-parameter algorithm for the problem, parameterized by the number of lines in the solution. Our algorithm works in time , where k is the bound on the number of lines to find and n is the number of points in the input. Our result answers in positive a question of Bonnet, Giannopolous, and Lampis [IPEC 2017] and of Froese (PhD thesis, 2018) and is in contrast with the known intractability of two closely related generalizations: the Rectangle Stabbing problem and the generalization in which the selected lines are not required to be axis-parallel.

Abstract The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $$H_1,H_2,\ldots $$ H 1 , H 2 , … ; the graphs in the class are called $$(H_1,H_2,\ldots )$$ ( H 1 , H 2 , … ) -free. The complexity of 3-coloring is far from being understood, even for classes defined by a few small forbidden induced subgraphs. For H-free graphs, the complexity is settled for any H on up to seven vertices. There are only two unsolved cases on eight vertices, namely $$2P_4$$ 2 P 4 and $$P_8$$ P 8 . For $$P_8$$ P 8 -free graphs, some partial results are known, but to the best of our knowledge, $$2P_4$$ 2 P 4 -free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on $$(2P_4,C_5)$$ ( 2 P 4 , C 5 ) -free graphs.

Abstract Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007 ). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parameterization. In particular, on one hand Steiner Tree is known to be ${APX}$-hard, and ${W[2]}$-hard on the other, if parameterized by the number of nonterminals ( Steiner vertices) in the optimum solution. In contrast to this, we give an efficient parameterized approximation scheme (${EPAS}$), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (${PSAKS}$) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For none of these is an ${EPAS}$ likely to exist for the studied parameter. For Steiner Forest an easy observation shows that the problem is ${APX}$-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is ${W[1]}$-hard. Nevertheless, we show that an ${EPAS}$ exists for Unweighted Directed Steiner Tree, but a ${PSAKS}$ does not. We also prove that there is an ${EPAS}$ and a ${PSAKS}$ for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

For a given ε > 0, we say that a graph G is ε-flexibly k-choosable if the following holds: for any assignment L of lists of size k on V(G), if a preferred color is requested at any set R of vertices, then at least ε |R| of these requests are satisfied by some L-coloring. We consider flexible list colorings in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree Δ that are ε-flexibly Δ-choosable for some ε = ε(Δ) > 0, which answers a question of Dvořák, Norin, and Postle [List coloring with requests, JGT 2019]. We also show that graphs of treewidth 2 are 1/3-flexibly 3-choosable, answering a question of Choi et al. [arXiv 2020], and we give conditions for list assignments by which graphs of treewidth k are 1/(k+1)-flexibly (k+1)-choosable. We show furthermore that graphs of treedepth k are 1/k-flexibly k-choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well-understood class of exceptions, 3-connected non-regular graphs of maximum degree Δ are flexibly (Δ - 1)-degenerate.

In the Steiner Tree problem, we are given an edge-weighted undirected graph G = (V,E) and a set of terminals R ⊆ V. The task is to find a connected subgraph of G containing R and minimizing the sum of weights of its edges. Steiner Tree is well known to be NP-complete and is undoubtedly one of the most studied problems in (applied) computer science. We observe that many approximation algorithms for Steiner Tree follow a similar scheme (meta-algorithm) and perform (exhaustively) a similar routine which we call star contraction. Here, by a star contraction, we mean finding a star-like subgraph in (the metric closure of) the input graph minimizing the ratio of its weight to the number of contained terminals minus one; and contract. It is not hard to see that the well-known MST-approximation seeks the best star to contract among those containing two terminals only. Zelikovsky’s approximation algorithm follows a similar workflow, finding the best star among those containing three terminals. We perform an empirical study of star contractions with the relaxed condition on the number of terminals in each star contraction motivated by a recent result of Dvořák et al. [Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices, STACS 2018]. Furthermore, we propose two improvements of Zelikovsky’s 11/6-approximation algorithm and we empirically confirm that the quality of the solution returned by any of these is better than the one returned by the former algorithm. However, such an improvement is exchanged for a slower running time (up to a multiplicative factor of the number of terminals).

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs - a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

Abstract The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $$L(u)\subseteq \{1,\ldots ,k\},$$ L ( u ) ⊆ { 1 , … , k } , then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph $$P_r+P_s$$ P r + P s is the disjoint union of the r-vertex path $$P_r$$ P r and the s-vertex path $$P_s.$$ P s . We prove that List 3-Colouring is polynomial-time solvable for $$(P_2+P_5)$$ ( P 2 + P 5 ) -free graphs and for $$(P_3+P_4)$$ ( P 3 + P 4 ) -free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices.

The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size O(k log k). After being known for long as Younger’s conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary. We show that if we compare the size of a minimum feedback vertex set in a directed graph with quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most four of the cycles, then there exists a feedback vertex set in G of size O(k^4). On the way there we prove a more general result about quarter-integral packing of subgraphs of high directed treewidth: for every pair of positive integers a and b, if a directed graph G has directed treewidth Omega(a^6 b^8 log^2(ab)), then one can find in G a family of a subgraphs, each of directed treewidth at least b, such that every vertex of G is in at most four subgraphs.

A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure [Lin&Sahni: Fair edge deletion problems. IEEE Trans. Comput. 89]. The measure is k if the number of solution neighbors does not exceed k for any vertex in the graph. One possible way to study graph problems is by defining the property in a certain logic. For a given objective an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. In the presented paper we show that there is an FPT algorithm for the MSO Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph. Here, the free variable corresponds to the solution sought. One may define an extended variant of MSO Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is W[1]-hard for parameter l even on graphs with twin cover one. Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is W[1]-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width.

This paper settles the computational complexity of model checking of several extensions of the monadic second order (MSO) logic on two classes of graphs: graphs of bounded treewidth and graphs of bounded neighborhood diversity. A classical theorem of Courcelle states that any graph property definable in MSO is decidable in linear time on graphs of bounded treewidth. Algorithmic metatheorems like Courcelle's serve to generalize known positive results on various graph classes. We explore and extend three previously studied MSO extensions: global and local cardinality constraints (CardMSO and MSO-LCC) and optimizing the fair objective function (fairMSO). First, we show how these extensions of MSO relate to each other in their expressive power. Furthermore, we highlight a certain "linearity" of some of the newly introduced extensions which turns out to play an important role. Second, we provide parameterized algorithm for the aforementioned structural parameters. On the side of neighborhood diversity, we show that combining the linear variants of local and global cardinality constraints is possible while keeping the linear (FPT) runtime but removing linearity of either makes this impossible. Moreover, we provide a polynomial time (XP) algorithm for the most powerful of studied extensions, i.e. the combination of global and local constraints. Furthermore, we show a polynomial time (XP) algorithm on graphs of bounded treewidth for the same extension. In addition, we propose a general procedure of deriving XP algorithms on graphs on bounded treewidth via formulation as Constraint Satisfaction Problems (CSP). This shows an alternate approach as compared to standard dynamic programming formulations. Comment: An extended abstract appeared in Proceedings of WG 2017

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $L(u) \subseteq \{1,\cdots, k\}$, then we obtain the List $k$-Colouring problem. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. We continue an extensive study into the complexity of these two problems for $H$-free graphs. The graph $P_r+P_s$ is the disjoint union of the $r$-vertex path $P_r$ and the $s$-vertex path $P_s$. We prove that List $3$-Colouring is polynomial-time solvable for $(P_2+P_5)$-free graphs and for $(P_3+P_4)$-free graphs. Combining our results with known results yields complete complexity classifications of $3$-Colouring and List $3$-Colouring on $H$-free graphs for all graphs $H$ up to seven vertices.