A signed tree model of a graph G is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of G, together with 2-colored edges xy, called transversal pairs, interpreted as bicliques or anti-bicliques whose sides are the leaves of the subtrees rooted at x and at y. We design an algorithm that, given such a representation of an unweighted n-vertex graph G with p transversal pairs, and given a source v ∈ V(G), computes a shortest-path tree rooted at v in G in time O(p log n). A wide variety of graph classes are such that for all n, their n-vertex graphs admit signed tree models with O(n) transversal pairs: for instance, those of bounded symmetric difference (hence, in particular, those of bounded flip-width, merge-width, twin-width, and degeneracy), more generally of bounded sd-degeneracy, as well as interval graphs. As applications of our Single-Source Shortest Path algorithm and new techniques, we - improve the runtime of the fixed-parameter algorithm for first-order model checking on graphs given with a witness of low merge-width from cubic [Dreier & Toruńczyk, STOC '25] to quadratic; - give an O(n² log n)-time algorithm for All-Pairs Shortest Path on graphs given with a witness of low merge-width, generalizing a result known for twin-width [Twin-Width III, SICOMP '24]; - significantly extend and simplify an O(n² log n)-time algorithm for multiplying two n × n matrices A, B of bounded twin-width in [Twin-Width V, STACS '23]: now A solely has to be an adjacency matrix of a graph of bounded twin-width and B can be arbitrary; - give an O(n² log² n)-time algorithm for All-Pairs Shortest Path on graphs of bounded twin-width, bypassing the need for contraction sequences in [Twin-Width III, SICOMP '24; Bannach et al. STACS '24]; - give an O(n^{7/3} log² n)-time algorithm for All-Pairs Shortest Path on graphs of symmetric difference O(n^{1/3}). The second and the last two items imply the same for Diameter, Radius, Eccentricity, Wiener Index, etc. The last three items do not assume any witness to be given as part of the input.

Let ${\mathbb{F}}$ be a finite field. We prove that there is an MSO-transduction which, given an ${\mathbb{F}}$-representable matroid of path-width k, produces a branch-decomposition of width at most f(k), for some function f. As a corollary, any recognizable property of ${\mathbb{F}}$-representable matroids with bounded path-width is definable in MSO logic, and therefore recognizability is equivalent to MSO-definability on classes of ${\mathbb{F}}$-representable matroids of bounded path-width. This generalizes the result of Bojańczyk, Grohe and Pilipczuk [Logical Methods in Computer Science 17(1), 2021] which asserts the equivalence of the two notions on graphs of bounded linear clique-width.

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.

We show that given a graph G we can CMSO-transduce its modular decomposition, its split decomposition and its bi-join decomposition. This improves results by Courcelle [Logical Methods in Computer Science, 2006] who gave such transductions using order-invariant MSO, a strictly more expressive logic than CMSO. Our methods more generally yield C_{2}MSO-transductions of the canonical decomposition of weakly-partitive set systems and weakly-bipartitive systems of bipartitions.

Abstract Given a graph G and an integer b, Bandwidth asks whether there exists a bijection $$\pi $$ π from V(G) to $$\{1, \ldots , |V(G)|\}$$ { 1 , … , | V ( G ) | } such that $$\max _{\{u, v \} \in E(G)} | \pi (u) - \pi (v) | \le b$$ max { u , v } ∈ E ( G ) | π ( u ) - π ( v ) | ≤ b . This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the tree-depth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number. In this paper we make progress in understanding the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number $$\omega $$ ω , thus significantly strengthening the previously mentioned result for vertex cover number. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results develop and generalize some of the methods of argumentation of the previous results and narrow some of the complexity gaps.

We show a flow-augmentation algorithm in directed graphs: There exists a randomized polynomial-time algorithm that, given a directed graph G , two vertices s, t ∈ V(G) , and an integer k , adds (randomly) to G a number of arcs such that for every minimal st -cut Z in G of size at most k , with probability 2 −poly( k ) the set Z becomes a minimum st -cut in the resulting graph. We also provide a deterministic counterpart of this procedure. The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set whose parameterized complexity status was repeatedly posed as open problems: Chain SAT , defined by Chitnis, Egri, and Marx [ESA’13, Algorithmica’17], a number of weighted variants of classic directed cut problems, such as Weighted st - Cut or Weighted Directed Feedback Vertex Set . By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph H , if the List H - Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable.

The Weighted F - Vertex Deletion for a class F of graphs asks, weighted graph G , for a minimum weight vertex set S such that G − S ∈ F . The case when F is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted F - Vertex Deletion . Only three cases of minor-closed F are known to admit constant-factor approximations, namely Vertex Cover , Feedback Vertex Set and Diamond Hitting Set . We study the problem for the class F of θ c -minor-free graphs, under the equivalent setting of the Weighted c -Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted F - Vertex Deletion , our result may be useful as a template for algorithms for other minor-closed families.

Abstract. We study the parameterized problem of satisfying “almost all” constraints of a given formula [Formula: see text] over a fixed, finite Boolean constraint language [Formula: see text], with or without weights. More precisely, for each finite Boolean constraint language [Formula: see text], we consider the following two problems. In Min SAT([Formula: see text]), the input is a formula [Formula: see text] over [Formula: see text] and an integer [Formula: see text], and the task is to find an assignment [Formula: see text] that satisfies all but at most [Formula: see text] constraints of [Formula: see text], or determine that no such assignment exists. In Weighted Min SAT([Formula: see text]), the input additionally contains a weight function [Formula: see text] and an integer [Formula: see text], and the task is to find an assignment [Formula: see text] such that (1) [Formula: see text] satisfies all but at most [Formula: see text] constraints of [Formula: see text], and (2) the total weight of the violated constraints is at most [Formula: see text]. We give a complete dichotomy for the fixed-parameter tractability of these problems: We show that for every Boolean constraint language [Formula: see text], either Weighted Min SAT([Formula: see text]) is [Formula: see text]; or Weighted Min SAT([Formula: see text]) is [Formula: see text]-hard but Min SAT([Formula: see text]) is [Formula: see text]; or Min SAT([Formula: see text]) is [Formula: see text]-hard. This generalizes recent work of Kim et al. [in SODA 2021, SIAM, Philadelphia, 2021, pp. 149–168], which did not consider weighted problems and only considered languages [Formula: see text] that cannot express implications [Formula: see text] (as is used to, e.g., model digraph cut problems). Our result generalizes and subsumes multiple previous results, including the FPT algorithms for Weighted Almost 2-SAT, weighted and unweighted [Formula: see text]-Chain SAT, and Coupled Min-Cut, as well as weighted and directed versions of the latter. The main tool used in our algorithms is the recently developed method of directed flow-augmentation [E. J. Kim et al., in STOC 2022, ACM, 2022, pp. 938–947].

No abstract available.

Abstract. We recently introduced the notion of twin-width, a novel graph invariant, and showed that first-order model checking can be solved in time [Formula: see text] for [Formula: see text]-vertex graphs given with a witness that the twin-width is at most [Formula: see text], called [Formula: see text]-contraction sequence or [Formula: see text]-sequence, and formulas of size [Formula: see text] [Bonnet et al., JACM ’22]. The inevitable price to pay for such a general result is that [Formula: see text] is a tower of exponentials of height roughly [Formula: see text]. In this paper, we show that algorithms based on twin-width need not be impractical. We present [Formula: see text]-time algorithms for [Formula: see text]-independent set, [Formula: see text]-scattered set, [Formula: see text]-clique, and [Formula: see text]-dominating set when an [Formula: see text]-sequence of the graph is given in input. We further show how to solve the weighted version of [Formula: see text]-independent set, subgraph isomorphism, and induced subgraph isomorphism in the slightly worse running time [Formula: see text]. Up to logarithmic factors in the exponent, all these running times are optimal unless the exponential time hypothesis fails. Like our first-order model checking algorithm, these new algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence by starting at its end and rewinding it. As an example of such a reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, thereby establishing that bounded twin-width classes are [Formula: see text]-bounded. This significantly extends the [Formula: see text]-boundedness of bounded rank-width classes and does so with a very concise proof. It readily yields a constant approximation for max independent set on [Formula: see text]-free graphs of bounded twin-width and a [Formula: see text]-approximation for min coloring on bounded twin-width graphs. We further observe that a constant approximation for max independent set on bounded twin-width graphs (but arbitrarily large clique number) would actually imply a polynomial-time approximation scheme. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques such that both sides of the bicliques are on consecutive vertices in a fixed vertex ordering. This property is trivially shared with graphs of bounded average degree. Given that biclique edge-partition, we show how to solve the unweighted single-source shortest paths, and hence all-pairs shortest paths, in time [Formula: see text] and time [Formula: see text], respectively. In sharp contrast, even diameter does not admit a truly subquadratic algorithm on bounded twin-width graphs unless the strong exponential time hypothesis fails. The fourth algorithmic use of twin-width builds on the so-called versatile tree of contractions [Bonnet et al., Comb. Theory ’22], a branching and more robust witness of low twin-width. We present constant-approximation algorithms for min dominating set and related problems on bounded twin-width graphs by showing that the integrality gap is constant. This is done by going down the versatile tree and stopping according to a problem-dependent criterion. At the reached node, a greedy approach yields the desired approximation.

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 present an undirected version of the recently introduced flow-augmentation technique: Given an undirected multigraph G with distinguished vertices s,t ∈ V(G) and an integer k , one can in randomized k 𝒪(1) ⋅ (|V(G)| + |E(G)|) time sample a set A ⊆ \(\binom{V(G)}{2}\) such that the following holds: for every inclusion-wise minimal st -cut Z in G of cardinality at most k , Z becomes a minimum-cardinality cut between s and t in G+A (i.e., in the multigraph G with all edges of A added) with probability 2 -𝒪( k log k ). Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability (2 -𝒪( k log k ) instead of 2 -𝒪( k 4 log k ) ), linear dependency on the graph size in the running time bound, and an arguably simpler proof. An immediate corollary is that the Bi-objective st -Cut problem can be solved in randomized FPT time 2 𝒪( k log k ) (|V(G)|+|E(G)|) on undirected graphs.

Given a graph G and an integer b, Bandwidth asks whether there exists a bijection π from V(G) to {1, …, |V(G)|} such that max_{{u, v} ∈ E(G)} | π(u) - π(v) | ≤ b. This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the treedepth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number of the input graph. In this paper, we make progress towards the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number ω of the input graph, thus generalizing the previously mentioned result for vertex cover. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results generalize some of the previous results and narrow some of the complexity gaps.

We study the parameterized problem of satisfying “almost all” constraints of a given formula F over a fixed, finite Boolean constraint language Γ, with or without weights. More precisely, for each finite Boolean constraint language Γ, we consider the following two problems. In MIN SAT(T), the input is a formula F over Γ and an integer k, and the task is to find an assignment α : V(F) → {0,1} that satisfies all but at most k constraints of F, or determine that no such assignment exists. In WEIGHTED MIN SAT(Γ), the input additionally contains a weight function ω : F → ℤ+ and an integer W, and the task is to find an assignment α such that (1) α satisfies all but at most k constraints of F, and (2) the total weight of the violated constraints is at most W. We give a complete dichotomy for the fixed-parameter tractability of these problems: We show that for every Boolean constraint language Γ, either WEIGHTED MIN SAT(Γ) is FPT; or WEIGHTED MIN SAT(Γ) is W[1]-hard but MIN SAT(Γ) is FPT; or MIN SAT (Γ) is W[1]-hard. This generalizes recent work of Kim et al. (SODA 2021) which did not consider weighted problems, and only considered languages Γ that cannot express implications (u → v) (as is used to, e.g., model digraph cut problems). Our result generalizes and subsumes multiple previous results, including the FPT algorithms for WEIGHTED Almost 2-SAT, weighted and unweighted ℓ-CHAIN SAT, and COUPLED MIN-CUT, as well as weighted and directed versions of the latter. The main tool used in our algorithms is the recently developed method of directed flow-augmentation (Kim et al., STOC 2022). * This research is a part of a project that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant Agreement 714704 (M. Pilipczuk). Eun Jung Kim is supported by the grant from French National Research Agency under JCJC program (ASSK: ANR-18-CE40-0025-01). The full version of the paper can be accessed at https://arxiv.org/abs/2207.07422.

We introduce the notion of delineation. A graph class C is said delineated by twin-width (or simply, delineated) if for every hereditary closure D of a subclass of C, it holds that D has bounded twin-width if and only if D is monadically dependent. An effective strengthening of delineation for a class C implies that tractable FO model checking on C is perfectly understood: On hereditary closures of subclasses D of C, FO model checking on D is fixed-parameter tractable (FPT) exactly when D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that K_{t,t}-free segment graphs, and axis-parallel H_t-free unit segment graphs have bounded twin-width, where H_t is the half-graph or ladder of height t. In contrast, axis-parallel H₄-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated. More broadly, we explore which structures (large bicliques, half-graphs, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results, combined with the FPT algorithm for first-order model checking on graphs given with O(1)-sequences [BKTW, JACM '22], give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If p is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C, then p(G) ⩾ k can be decided in FPT time f(k) ⋅ |V(G)|^O(1). For instance, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width.

A contraction sequence of a graph consists of iteratively merging two of its vertices until only one vertex remains. The recently introduced twin-width graph invariant is based on contraction sequences. More precisely, if one puts error edges, henceforth red edges, between two vertices representing non-homogeneous subsets, the twin-width is the minimum integer d such that a contraction sequence exists that keeps red degree at most d. By changing the condition imposed on the trigraphs (i.e., graphs with some edges being red) and possibly slightly tweaking the notion of contractions, we show how to characterize the well-established bounded rank-width, tree-width, linear rank-width, path-width –usually defined in the framework of branch-decompositions–, and proper minor-closed classes by means of contraction sequences. Contraction sequences hold a crucial advantage over branch-decompositions: While one can scale down contraction sequences to capture classical width notions, the more general bounded twin-width goes beyond their scope, as it contains planar graphs in particular, a class with unbounded rank-width. As an application we give a transparent alternative proof of the celebrated Courcelle's theorem (actually of its generalization by Courcelle, Makowsky, and Rotics), that MSO2 (resp. MSO1) model checking on graphs with bounded tree-width (resp. bounded rank-width) is fixed-parameter tractable in the size of the input sentence. We are hopeful that our characterizations can help in other contexts. We then explore new avenues along the general theme of contraction sequences both in order to refine the landscape between bounded tree-width and bounded twin-width (via spanning twin-width) and to capture more general classes than bounded twin-width. To this end, we define an oriented version of twin-width, where appearing red edges are oriented away from the newly contracted vertex, and the mere red out-degree should remain bounded. Surprisingly, classes of bounded oriented twin-width coincide with those of bounded twin-width. This greatly simplifies the task of showing that a class has bounded twin-width. As an example, using a lemma by Norine, Seymour, Thomas, and Wollan, we give a 5-line proof that Kt-minor free graphs have bounded twin-width. Without oriented twin-width, this fact was shown by a somewhat intricate 4-page proof in the first paper of the series. Finally we explore the concept of partial contraction sequences, instead of terminating on a single-vertex graph, the sequence ends when reaching a particular target class. We show that FO model checking (resp. ∃FO model checking) is fixed-parameter tractable on classes with partial contraction sequences to a class of bounded degree (resp. bounded expansion), provided such a sequence is given. Efficiently finding such partial sequences could turn out simpler than finding a (complete) sequence.

Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field F, the list contains only finitely many F-representable matroids, due to the well-quasi-ordering of F-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these F-representable excluded minors in general. We consider the class of matroids of path-width at most k for fixed k. We prove that for a finite field F, every F-representable excluded minor for the class of matroids of path-width at most k has at most 2^{|𝔽|^{O(k²)}} elements. We can therefore compute, for any integer k and a fixed finite field F, the set of F-representable excluded minors for the class of matroids of path-width k, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an F-represented matroid is at most k. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most k has at most 2^{2^{O(k²)}} vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.

We show a flow-augmentation algorithm in directed graphs: There exists a randomized polynomial-time algorithm that, given a directed graph G, two integers s,t ∈ V(G), and an integer k, adds (randomly) to G a number of arcs such that for every minimal st-cut Z in G of size at most k, with probability 2−poly(k) the set Z becomes a minimum st-cut in the resulting graph.

For a family of graphs $$\mathcal {F}$$ , Weighted $$\mathcal {F}$$ -Deletion is the problem for which the input is a vertex weighted graph $$G = (V, E)$$ and the goal is to delete $$S \subseteq V$$ with minimum weight such that $$G \setminus S \in \mathcal {F}$$ . Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when $$\mathcal {F}$$ is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.

Weighted Counting for Constraint Satisfaction with Default Values (#CSPD) is a powerful special case of the sum-of-products problem that admits succinct encodings of #CSP, #SAT, and inference in probabilistic graphical models. We investigate #CSPD under the fundamental parameter of incidence treewidth (i.e., the treewidth of the incidence graph of the constraint hypergraph). We show that if the incidence treewidth is bounded, #CSPD can be solved in polynomial time. More specifically, we show that the problem is fixed-parameter tractable for the combined parameter incidence treewidth, domain size, and support size (the maximum number of non-default tuples in a constraint). This generalizes known results on the fixed-parameter tractability of #CSPD under the combined parameter primal treewidth and domain size. We further prove that the problem is not fixed-parameter tractable if any of the three components is dropped from the parameterization.

Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the “price of generality” of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable (FPT) to intractable? This type of question is by now very well studied, but somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that, for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes “for free.” Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy first-fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP--hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.

The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper, we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems that are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions and provide FPT algorithms for the showcase problems Capacitated Vertex Cover, Capacitated Dominating Set, and Imbalance parameterized by the tree-cut width of an input graph. On the other hand, we show that List Coloring, Precoloring Extension, and Boolean CSP (the last parameterized by the tree-cut width of the incidence graph) are W[1]-hard and hence unlikely to be FPT when parameterized by tree-cut width.

We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-sequence, and formulas of size $k$ [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Clique, and $k$-Dominating Set when an $O(1)$-sequence is provided. We further show how to solve weighted $k$-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in time $2^{O(k \log k)}n$. These algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example of this reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, establishing that bounded twin-width classes are $\chi$-bounded. This significantly extends the $\chi$-boundedness of bounded rank-width classes, and does so with a very concise proof. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in sublinear time $O(n \log n)$ and time $O(n^2 \log n)$, respectively.

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for $k$-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected $k$-Dominating Set and Total $k$-Dominating Set (albeit with a worse upper bound on the twin-width). The $k$-Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP '21], which extends to $k$-Independent Dominating Set, $k$-Path, $k$-Induced Path, $k$-Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected $k$-Vertex Cover and Capacitated $k$-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik-Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate $O(k^{1.5})$ vertex kernel for Connected $k$-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.

The recently introduced twin-width of a graph G is the minimum integer d such that G has a d-contraction sequence, that is, a sequence of |V(G)| – 1 iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most d, where a red edge appears between two sets of identified vertices if they are not homogeneous in G (not fully adjacent nor fully non-adjacent). We show that if a graph admits a d-contraction sequence, then it also has a linear-arity tree of f(d)-contractions, for some function f. Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of Θ(|V(G)|) pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i.e., has at most n!cn graphs labeled by [n], for some constant c. This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an O(log n)-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the small conjecture that, conversely, every small hereditary class has bounded twin-width. The conjecture passes many tests. Inspired by sorting networks of logarithmic depth, we show that logΘ(log log d) n-subdivisions of Kn (a small class when d is constant) have twin-width at most d. We obtain a rather sharp converse with a surprisingly direct proof: the logd+1 n-subdivision of Kn has twin-width at least d. Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. These sparse classes are surprisingly rich since they contain certain (small) classes of expanders. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from K4 [Bilu and Linial, Combinatorica '06] also have bounded twin-width. These graphs are related to so-called separable permutations and also form a small class. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width. We show that for a hereditary class of bounded twin-width the five following conditions are equivalent: every graph in (1) is Kt,t-free for some fixed t, (2) has an adjacency matrix without a d-by-d division with a 1 entry in each d2 cells for some fixed d, (3) has at most linearly many edges, (4) the subgraph closure of has bounded twin-width, and (5) has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.

We present a new technique for designing fixed-parameter algorithms for graph cut problems in undirected graphs, which we call flow augmentation. Our technique is applicable to problems that can be phrased as a search for an (edge) (s, t)-cut of cardinality at most k in an undirected graph G with designated terminals s and t. More precisely, we consider problems where an (unknown) solution is a set Z ⊆ E(G) of size at most k such that •in G – Z, s and t are in distinct connected components,•every edge of Z connects two distinct connected components of G – Z, and•if we define the set Zs, t ⊆ Z as those edges e ∊ Z for which there exists an (s, t)-path Pe with E(Pe) ∩ Z = {e}, then Zs, t separates s from t. We prove that in the above scenario one can in randomized time k(1) (|V(G)| + |E(G)|) add a number of edges to the graph so that with probably at least 2–(k log k) no added edge connects two components of G – Z, and Zs, t becomes a minimum cut between s and t. This additional property becomes a handy lever in applications. For example, consider the question of an (s, t)-cut of cardinality at most k and of minimum possible weight (assuming edge weights in G). While the problem is NP-hard in general, it easily reduces to the maximum flow / minimum cut problem if we additionally assume that k is the minimum possible cardinality of an (s, t)-cut in G. Hence, we immediately obtain that the aforementioned problem admits an 2(k log k) n(1)-time randomized fixed-parameter algorithm. We apply our method to obtain a randomized fixed-parameter algorithm for a notorious “hard nut” graph cut problem we call Coupled Min-Cut. This problem emerges out of the study of FPT algorithms for Min CSP problems (see below), and was unamenable to other techniques for parameterized algorithms in graph cut problems, such as Randomized Contractions, Treewidth Reduction or Shadow Removal. In fact, we go one step further. To demonstrate the power of the approach, we consider more generally the Boolean Min CSP(Γ)-problems, a.k.a. Min SAT(Γ), parameterized by the solution cost. This is a framework of optimization problems that includes problems such as Almost 2-SAT and the notorious i-Chain SAT problem. We are able to show that every problem Min SAT(Γ) is either (1) FPT, (2) W[1]-hard, or (3) able to express the soft constraint (u → v), and thereby also the min-cut problem in directed graphs. All the W[1]-hard cases were known or immediate, and the main new result is an FPT algorithm for a generalization of Coupled Min-Cut. In other words, flow-augmentation is powerful enough to let us solve every fixed-parameter tractable problem in the class, except those that explicitly encompass directed graph cuts.

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field ${\mathbb F}$, we wish to find a “branch-decomposition” of these subspaces of width at most $k$ that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated to the leaves in one component of $T-e$ and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of ${\mathbb F}$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over ${\mathbb F}$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks [ J. Algorithms, 21 (1996), pp. 358--402] on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of ${\mathbb F}$-represented matroids was due to Hliněný and Oum [ SIAM J. Comput., 38 (2008), pp. 1012--1032] that runs in time $O(n^3)$ where $n$ is the number of elements of the input ${\mathbb F}$-represented matroid. But their method is highly indirect. Their algorithm uses the nontrivial fact by Geelen et al. [ J. Combin. Theory Ser. B, 88 (2003), pp. 261--265] that the number of forbidden minors is finite and uses the algorithm of Hliněný [ J. Combin. Theory Ser. B, 96 (2006), pp. 325--351] on checking monadic second-order formulas on ${\mathbb F}$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.

Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, Kt-free unit d-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d-contractions, witness that the twin-width is at most d. We show that FO model checking, that is deciding if a given first-order formula φ evaluates to true for a given binary structure G on a domain D, is FPT in |φ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d-contraction sequence for G, our algorithm runs in time f(d,|φ|)·|D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS '15].

The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k)n^{2^{k-1}}-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K_{t,t}-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.

We consider the complexity of the Independent Set Reconfiguration problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area. We then go on to consider the c-Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set c-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed c >= 1 on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time (n^{O(c)}) algorithm for all fixed values of c, except c=1, for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that c-Colorable Reconfiguration is W[2]-hard on split graphs parameterized by c and the length of the solution, as well as a tight ETH-based lower bound for both parameters.

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} on disk graphs. In stark contrast, \textsc{Maximum Clique} on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant approximation which is not attainable even in time $2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails.

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathbb F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$ that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated to the leaves in one component of $T-e$ and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathbb F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathbb F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathbb F$-represented matroids was due to Hliněný and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathbb F$-represented matroid. But their method is highly indirect. Their algorithm uses the nontrivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hliněný (2006) on checking monadic second-order formulas on $\mathbb F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.

Weighted Counting for Constraint Satisfaction with Default Values (#CSPD) is a powerful special case of the sum-of-products problem that admits succinct encodings of #CSP, #SAT, and inference in probabilistic graphical models. We investigate #CSPD under the fundamental parameter of incidence treewidth (i.e., the treewidth of the incidence graph of the constraint hypergraph). We show that if the incidence treewidth is bounded, then #CSPD can be solved in polynomial time. More specifically, we show that the problem is fixed-parameter tractable for the combined parameter incidence treewidth, domain size, and support size (the maximum number of non-default tuples in a constraint), generalizing a known result on the fixed-parameter tractability of #CSPD under the combined parameter primal treewidth and domain size. We further prove that the problem is not fixed-parameter tractable if any of the three components is dropped from the parameterization.

We study the concept of compactor, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function F:Sigma^* -> N and a parameterization kappa: Sigma^* -> N, a compactor (P,M) consists of a polynomial-time computable function P, called condenser, and a computable function M, called extractor, such that F=M o P, and the condensing P(x) of x has length at most s(kappa(x)), for any input x in Sigma^*. If s is a polynomial function, then the compactor is said to be of polynomial-size. Although the study on counting-analogue of kernelization is not unprecedented, it has received little attention so far. We study a family of vertex-certified counting problems on graphs that are MSOL-expressible; that is, for an MSOL-formula phi with one free set variable to be interpreted as a vertex subset, we want to count all A subseteq V(G) where |A|=k and (G,A) models phi. In this paper, we prove that every vertex-certified counting problems on graphs that is MSOL-expressible and treewidth modulable, when parameterized by k, admits a polynomial-size compactor on H-topological-minor-free graphs with condensing time O(k^2n^2) and decoding time 2^{O(k)}. This implies the existence of an FPT-algorithm of running time O(n^2 k^2)+2^{O(k)}. All aforementioned complexities are under the Uniform Cost Measure (UCM) model where numbers can be stored in constant space and arithmetic operations can be done in constant time.

We study a family of generalizations of Edge Dominating Set on directed graphs called Directed (p,q)-Edge Dominating Set. In this problem an arc (u,v) is said to dominate itself, as well as all arcs which are at distance at most q from v, or at distance at most p to u. First, we give significantly improved FPT algorithms for the two most important cases of the problem, (0,1)-dEDS and (1,1)-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that (p,q)-dEDS is FPT parameterized by p+q+tw, but W-hard parameterized just by tw, where tw is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of p,q, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case (p=q=1) which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

35 pages, 11 figures, accepted to JCTB

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour (J Comb Theory Ser B 96(4):514–528, 2006). Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most 1 and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time $$f(k)\cdot n^3$$ for some function f, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time $$8^k\cdot n^{{\mathcal {O}}(1)}$$ . The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time $$2^{{\mathcal {O}}(k)}\cdot n^4$$ . We also prove that the running time cannot be improved to $$2^{o(k)}\cdot n^{{\mathcal {O}}(1)}$$ under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.

In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with $${\mathcal {O}}(k^{6})$$ vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of ‘complete degree’ of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time $$10^{k}\cdot n^{{\mathcal {O}}(1)}$$ .

Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a linear layout V1, V2,..., Vnof the subspaces such that dim((V1+ V2+ ··· + Vi) ∩ (Vi+1+ ··· + Vn)) ≤ k for all i; such a linear layout is said to have width at most k. When restricted to 1-dimensional subspaces, this problem is equivalent to computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory and computing the path-width of an F-represented matroid in matroid theory. We present a fixed-parameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over F. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input F-represented matroid of path-width at most k, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most k for an input graph of linear rank-width at most k. In both corollaries, no such algorithms were known previously. Our approach is based on dynamic programming combined with the idea developed by Bodlaender and Kloks (1996) for their work on path-width and tree-width of graphs. It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width; a theorem by Geelen, Gerards, and Whittle (2002) implies that for each fixed finite field F, there are finitely many forbidden F-representable minors for the class of matroids of path-width at most k. An algorithm by Hlinený (2006) can detect a minor in an input F-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.

Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a linear layout V1, V2, …, Vn of the subspaces such that dim((V1 + V2 + ⃛ + Vi)∩(Vi+1 + ⃛ + Vn)) ≤ k for all i; such a linear layout is said to have width at most k. When restricted to 1-dimensional subspaces, this problem is equivalent to computing the path-width of an F-represented matroid in matroid theory and computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory. We present a fixed-parameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over F. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input F-represented matroid of path-width at most k, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most k for an input graph of linear rank-width at most k. In both corollaries, no such algorithms were known previously. Our approach is based on dynamic programming combined with the idea developed by Bodlaender and Kloks (1996) for their work on path-width and tree-width of graphs. It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width; a theorem by Geelen, Gerards, and Whittle (2002) implies that for each fixed finite field F, there are finitely many forbidden F-representable minors for the class of matroids of path-width at most k. An algorithm by Hliněný (2006) can detect a minor in an input F-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition even if the complete list of forbidden minors were known. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.

We present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X ⊆ V ( G ), called a treewidth-modulator , such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition , is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k ) that has a finite integer index and such that Y es -instances have a treewidth-modulator of size O ( k ) admits a linear kernel on the class of H -topological-minor-free graphs, for any fixed graph H . This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H -minor-free graphs. Let F be a fixed finite family of graphs containing at least one planar graph. Given an n -vertex graph G and a non-negative integer k , P lanar - F -D eletion asks whether G has a set X ⊆ V ( G ) such that | X | ⩽ k and G − X is H -minor-free for every H ϵ F . As our second application, we present the first single-exponential algorithm to solve P lanar - F -D eletion . Namely, our algorithm runs in time 2 O ( k ) · n 2 , which is asymptotically optimal with respect to k . So far, single-exponential algorithms were only known for special cases of the family F .

In this paper we design FPT-algorithms for two parameterized problems. The first is List Digraph Homomorphism: given two digraphs G and H and a list of allowed vertices of H for every vertex of G, the question is whether there exists a homomorphism from G to H respecting the list constraints. The second problem is a variant of Multiway Cut, namely Min-Max Multiway Cut: given a graph G, a non-negative integer l, and a set T of r terminals, the question is whether we can partition the vertices of G into r parts such that (a) each part contains one terminal and (b) there are at most l edges with only one endpoint in this part. We parameterize List Digraph Homomorphism by the number w of edges of G that are mapped to non-loop edges of H and we give a time 2^{O(l * log(h) + l^{2 * log(l)}} * n^{4} * log(n) algorithm, where h is the order of the host graph H.We also prove that Min-Max Multiway Cut can be solved in time 2^{O((l * r)^2 * log(l *r))} * n^{4} * log(n). Our approach introduces a general problem, called List Allocation, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an FPT-algorithm for the List Allocation problem that is designed using a suitable adaptation of the randomized contractions technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).

We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k. We attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT-time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPT-AS), which computes a (1-epsilon)-approximation in time f(k,epsilon) * poly(n). 3. There is no FPT-AS unless FPT=W[1]. For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approxi-mating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514-528, 2006.], and it is similar to pathwidth, which is the linearized variant of treewidth. Motivated from the results on graph modification problems into graphs of bounded treewidth or pathwidth, we investigate a graph modification problem into the class of graphs having linear rankwidth at most one, called the Linear Rankwidth-1 Vertex Deletion (shortly, LRW1-Vertex Deletion). In this problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most one and if one exists, find such a vertex set. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time f (k) · n 3 for some function f , it is not clear whether this problem allows a runtime with a modest exponential function. We establish that LRW1-Vertex Deletion can be solved in time 8 k · n O(1). The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define the necklace graphs and investigate their structural properties. We also show that the LRW1-Vertex Deletion has a polynomial kernel.

Local Search is one of the fundamental approaches to combinatorial optimization and it is used throughout AI. Several local search algorithms are based on searching the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naive brute-force search of the k-exchange neighborhood requires n(O(k)) time, which is not practical even for very small values of k. Fellows et al. (IJCAI 2009) studied whether this brute-force search is avoidable and gave positive and negative answers for several combinatorial problems. They used the notion of local search in a strict sense. That is, an improved solution needs to be found in the k-exchange neighborhood even if a global optimum can be found efficiently. In this paper we consider a natural relaxation of local search, called permissive local search (Marx and Schlotter, IWPEC 2009) and investigate whether it enhances the domain of tractable inputs. We exemplify this approach on a fundamental combinatorial problem, Vertex Cover. More precisely, we show that for a class of inputs, finding an optimum is hard, strict local search is hard, but permissive local search is tractable. We carry out this investigation in the framework of parameterized complexity.

International audience

Value-based argumentation frameworks, as introduced by Bench-Capon, allow the abstract representation of persuasive argumentation. This formalism takes into account the relative strength of arguments with respect to some ordering which represents an audience. Deciding subjective or objective acceptance (i.e., acceptance with respect to at least one or with respect to all orderings) are intractable computational problems.

We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2r – 1)m + k)/2r clauses. Thus Max-r-Sat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1 − 2−r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k2), then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k)/4 clauses. We also outline how the fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems.