We present the first fixed-parameter tractable (fpt) algorithms for precisely determining several central hypergraph decomposition parameters, including generalized hypertree width, fractional hypertree width, and adaptive width. Despite the recognized importance of these measures in complexity theory, databases, and constraint satisfaction, no exact fpt algorithms for any of them had previously been known. Our results are obtained for hypergraph classes of bounded rank and bounded degree. Our approach extends a recent algorithm for treewidth (Bojańcyk Pilipczuk, LMCS 2022) utilizing monadic second-order (MSO) transductions. Leveraging this framework, we overcome the significant technical hurdles presented by hypergraphs, whose structural decompositions are technically much more intricate than their graph counterparts.

In this paper we study syntactic branching programs of bounded repetition representing CNFs of bounded treewidth. For this purpose we introduce two new structural graph parameters d-pathwidth and clique preserving d-pathwidth denoted by pwd(G) and cpwd(G) where G is a graph. We show that cpw2(G)≤O(tw(G)Δ(G)) where tw(G) and Δ(G) are, respectively the treewidth and maximal degree of G. Using this upper bound, we demonstrate that each CNF ψ can be represented as a conjunction of two OBDDs (quite a restricted class of read-twice branching programs) of size 2O(Δ(ψ)⋅tw(ψ)2) where tw(ψ) is the treewidth of the primal graph of ψ and each variable occurs in ψ at most Δ(ψ) times.Next, we use d-pathwidth to obtain lower bounds for monotone branching programs. In particular, we consider the monotone version of syntactic nondeterministic read d times branching programs (just forbidding negative literals as edge labels) and introduce a further restriction that each computational path can be partitioned into at most d read-once subpaths. We call the resulting model separable monotone read d times branching programs and abbreviate them d-SMNBPs. For each graph G without isolated vertices, we introduce a CNF ψ(G) whose clauses are (u∨e∨v) for each edge e={u,v} of G. We prove that a d-SMNBP representing ψ(G) is of size at least Ω(cpwd(G)) where c=(8/7)1/12. We use this ’generic’ lower bound to obtain an exponential lower bound for a ’concrete’ class of CNFs ψ(Kn). In particular, we demonstrate that for each 0<a<1, the size of na-SMNBP representing ψ(Kn) is at least cnb where b is an arbitrary constant such that a+b<1. This lower bound is tight in the sense ψ(Kn) can be represented by a poly-sized n-SMNBP.

Decomposable Negation Normal Forms \textsc{dnnf} [Darwiche, 'Decomposable Negation Normal Form', JACM, 2001] is a landmark Knowledge Compilation (\textsc{kc}) model, highly important both in \textsc{ai} and Theoretical Computer Science. Numerous restrictions of the model have been studied. In this paper we consider the restriction where all the gates are $\alpha$-imbalanced that is, at most one input of each gate depends on more than $n^{\alpha}$ variables (where $n$ is the number if variables of the function being represented). The concept of imbalanced gates has been first considered in [Lai, Liu, Yin 'New canonical representations by augmenting OBDDs with conjunctive decomposition', JAIR, 2017]. We consider the idea in the context of representation of \textsc{cnf}s of bounded primal treewidth. We pose an open question as to whether \textsc{cnf}s of bounded primal treewidth can be represented as \textsc{fpt}-sized \textsc{dnnf} with $\alpha$-imbalanced gates. We answer the question negatively for Decision \textsc{dnnf} with $\alpha$-imbalanced conjunction gates. In particular, we establish a lower bound of $n^{\Omega((1-\alpha) \cdot k)}$ for the representation size (where $k$ is the primal treewidth of the input \textsc{cnf}). The main engine for the above lower bound is a combinatorial result that may be of an independent interest in the area of parameterized complexity as it introduces a novel concept of bidimensionality.

Decision dnnf (a.k.a. ∧_d-fbdd) is an important special case of Decomposable Negation Normal Form (dnnf), a landmark knowledge compilation model. Like other known dnnf restrictions, Decision dnnf admits fpt sized representation of cnfs of bounded primal treewidth. However, unlike other restrictions, the complexity of representation for cnfs of bounded incidence treewidth is wide open. In[arxiv:1708.07767], we resolved this question for two restricted classes of Decision dnnf that we name ∧_d-obdd and Structured Decision dnnf. In particular, we demonstrated that, while both these classes have fpt-sized representations for cnfs of bounded primal treewidth, they need xp-size for representation of cnfs of bounded incidence treewidth. In the main part of this paper we carry out an in-depth study of the ∧_d-obdd model. We formulate a generic methodology for proving lower bounds for the model. Using this methodology, we reestablish the xp lower bound provided in [arxiv:1708.07767]. We also provide exponential separations between fbdd and ∧_d-obdd and between ∧_d-obdd and an ordinary obdd. We study the complexity of Apply operation for ∧_d-obdd. While, in general, the Apply operation leads to exponential blow up of the resulting model, we identify a special restricted case where the Apply operation can be carried out efficiently. We introduce a relaxed version of Structured Decision dnnf that we name Structured ∧_d-fbdd and demonstrate that this model is quite powerful for cnfs of bounded incidence treewidth.

Decomposable Negation Normal Forms dnnf is a landmark Knowledge Compilation (kc) model, highly important both in ai and Theoretical Computer Science. Numerous restrictions of the model have been studied. In this paper we consider the restriction where all the gates are α-imbalanced that is, at most one input of each gate depends on more than n^α variables (where n is the number if variables of the function being represented). The concept of imbalanced gates has been first considered in [Lai, Liu, Yin 'New canonical representations by augmenting OBDDs with conjunctive decomposition', JAIR, 2017]. We consider the idea in the context of representation of cnfs of bounded primal treewidth. We pose an open question as to whether cnfs of bounded primal treewidth can be represented as fpt-sized dnnf with α-imbalanced gates. We answer the question negatively for Decision dnnf with α-imbalanced conjunction gates. In particular, we establish a lower bound of n^Ω((1-α) · k) for the representation size (where k is the primal treewidth of the input cnf). The main engine for the above lower bound is a combinatorial result that may be of an independent interest in the area of parameterized complexity as it introduces a novel concept of bidimensionality.

Given two n -vertex graphs G1 and G2 of bounded treewidth, is there an n -vertex graph G of bounded treewidth having subgraphs isomorphic to G1 and G2? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if G1 is a binary tree and G2 is a ternary tree. We also provide an extensive study of cases where such ``gluing"" is possible. In particular, we prove that if G1 has treewidth k and G2 has pathwidth \ell , then there is an n -vertex graph of treewidth at most k + 3\ell + 1 containing both G1 and G2 as subgraphs.

Generalised hypertree width ($ghw$) is a hypergraph parameter that is central to the tractability of many prominent problems with natural hypergraph structure. Computing $ghw$ of a hypergraph is notoriously hard. The decision version of the problem, checking whether $ghw(H) \leq k$, is paraNP-hard when parameterised by $k$. Furthermore, approximation of $ghw$ is at least as hard as approximation of Set-Cover, which is known to not admit any fpt approximation algorithms. Research in the computation of ghw so far has focused on identifying structural restrictions to hypergraphs -- such as bounds on the size of edge intersections -- that permit XP algorithms for $ghw$. Yet, even under these restrictions that problem has so far evaded any kind of fpt algorithm. In this paper we make the first step towards fpt algorithms for $ghw$ by showing that the parameter can be approximated in fpt time for graphs of bounded edge intersection size. In concrete terms we show that there exists an fpt algorithm, parameterised by $k$ and $d$, that for input hypergraph $H$ with maximal cardinality of edge intersections $d$ and integer $k$ either outputs a tree decomposition with $ghw(H) \leq 4k(k+d+1+)(2k-1)$, or rejects, in which case it is guaranteed that $ghw(H) > k$. Thus, in the special case, of hypergraphs of bounded edge intersection, we obtain an fpt $O(k^3)$-approximation algorithm for $ghw$.

Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. We show how this combinatorial result can be used to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is ≤k for some constant k. Moreover, we show a dual version of our main result for fractional hitting sets.

We study the tractability of the maximum independent set problem from the viewpoint of graph width parameters, with the goal of defining a width parameter that is as general as possible and allows to solve independent set in polynomial-time on graphs where the parameter is bounded. We introduce two new graph width parameters: one-sided maximum induced matching-width (o-mim-width) and neighbor-depth. O-mim-width is a graph parameter that is more general than the known parameters mim-width and tree-independence number, and we show that independent set and feedback vertex set can be solved in polynomial-time given a decomposition with bounded o-mim-width. O-mim-width is the first width parameter that gives a common generalization of chordal graphs and graphs of bounded clique-width in terms of tractability of these problems. The parameter o-mim-width, as well as the related parameters mim-width and sim-width, have the limitation that no algorithms are known to compute bounded-width decompositions in polynomial-time. To partially resolve this limitation, we introduce the parameter neighbor-depth. We show that given a graph of neighbor-depth k, independent set can be solved in time n^O(k) even without knowing a corresponding decomposition. We also show that neighbor-depth is bounded by a polylogarithmic function on the number of vertices on large classes of graphs, including graphs of bounded o-mim-width, and more generally graphs of bounded sim-width, giving a quasipolynomial-time algorithm for independent set on these graph classes. This resolves an open problem asked by Kang, Kwon, Strømme, and Telle [TCS 2017].

We propose an algorithm whose input are parameters k and r and a hypergraph H of rank at most r . The algorithm either returns a tree decomposition of H of generalized hypertree width at most 4 k or ’NO’. In the latter case, it is guaranteed that the hypertree width of H is greater than k . Most importantly, the runtime of the algorithm is FPT in k and r . The approach extends to fractional hypertree width with a slightly worse approximation (4 k + 1 instead of 4 k ). We hope that the results of this paper will give rise to a new research direction whose aim is design of FPT algorithms for computation and approximation of hypertree width parameters for restricted classes of hypergraphs.

We prove that the tree-width of graphs in a hereditary class defined by a finite set $F$ of forbidden induced subgraphs is bounded if and only if $F$ includes a complete graph, a complete bipartite graph, a tripod (a forest in which every connected component has at most 3 leaves) and the line graph of a tripod.

We propose an algorithm whose input are parameters $k$ and $r$ and a hypergraph $H$ of rank at most $r$. The algorithm either returns a tree decomposition of $H$ of generalized hypertree width at most $4k$ or 'NO'. In the latter case, it is guaranteed that the hypertree width of $H$ is greater than $k$. Most importantly, the runtime of the algorithm is \emph{FPT} in $k$ and $r$. The approach extends to fractional hypertree width with a slightly worse approximation ($4k+1$ instead of $4k$). We hope that the results of this paper will give rise to a new research direction whose aim is design of FPT algorithms for computation and approximation of hypertree width parameters for restricted classes of hypergraphs.

Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph $H$ has a width relative to each of these methods: its hypertree width $hw(H)$, its generalized hypertree width $ghw(H)$, and its fractional hypertree width $fhw(H)$, respectively. It is known that $hw(H)\leq k$ can be checked in polynomial time for fixed $k$, while checking $ghw(H)\leq k$ is NP-complete for $k \geq 3$. The complexity of checking $fhw(H)\leq k$ for a fixed $k$ has been open for over a decade. We settle this open problem by showing that checking $fhw(H)\leq k$ is NP-complete, even for $k=2$. The same construction allows us to prove also the NP-completeness of checking $ghw(H)\leq k$ for $k=2$. After that, we identify meaningful restrictions which make checking for bounded $ghw$ or $fhw$ tractable or allow for an efficient approximation of the $fhw$.

We introduce a new graph parameter called linear upper maximum induced matching width, denoted for a graph $G$ by $lu(G)$. We prove that the smallest size of the \textsc{obdd} for $\varphi$, the monotone 2-\textsc{cnf} corresponding to $G$, is sandwiched between $2^{lu(G)}$ and $n^{O(lu(G))}$. The upper bound is based on a combinatorial statement that might be of an independent interest. We show that the bounds in terms of this parameter are best possible.

We prove that the tree-width of graphs in a hereditary class defined by a finite set F of forbidden induced subgraphs is bounded if and only if F includes a complete graph, a complete bipartite graph, a tripod (a forest in which every connected component has at most 3 leaves) and the line graph of a tripod.

We demonstrate that Regular Resolution is FPT for two restricted families of CNFs of bounded incidence treewidth. The first includes CNFs having at most $p$ clauses whose removal results in a CNF of primal treewidth at most $k$. The parameters we use in this case are $p$ and $k$. The second class includes CNFs of bounded one-sided (incidence) treewdth, a new parameter generalizing both primal treewidth and incidence pathwidth. The parameter we use in this case is the one-sided treewidth.

Recently, Daligault, Rao and Thomass\'e asked in [3] if every hereditary class which is well-quasi-ordered by the induced subgraph relation is of bounded clique-width. There are two reasons why this questions is interesting. First, it connects two seemingly unrelated notions. Second, if the question is answered affirmatively, this will have a strong algorithmic consequence. In particular, this will mean (through the use of Courcelle theorem [2]), that any problem definable in Monadic Second Order Logic can be solved in a polynomial time on any class well-quasi-ordered by the induced subgraph relation. In the present paper, we answer this question affirmatively for graphs without large bicliques. Thus the above algorithmic consequence is true, for example, for classes of graphs of bounded degree.

It is known that a positive Boolean function f depending on n variables has at least n+1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n+1 extremal points if and only if it is linear read-once. The class of linear read-once functions is known to be the intersection of the classes of read-once and threshold functions. Generalizing this result we show that the class of linear read-once functions is the intersection of read-once and Chow functions. We also find the set of minimal read-once functions which are not linear read-once and the set of minimal threshold functions which are not linear read-once. In other words, we characterize the class of linear read-once functions by means of minimal forbidden subfunctions within the universe of read-once and the universe of threshold functions. Within the universe of threshold functions the importance of linear read-once functions is due to the fact that they attain the minimum value of the specification number, which is n+1 for functions depending on n variables. In 1995 Anthony et al. conjectured that for all other threshold functions the specification number is strictly greater than n+1. We disprove this conjecture by exhibiting a threshold non-linear read-once function depending on n variables whose specification number is n+1.

Does well-quasi-ordering by induced subgraphs imply bounded clique-width for hereditary classes? This question was asked by Daligault, Rao, and Thomassé [7]. We answer this question negatively by presenting a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered by the induced subgraph relation. We also show that graphs in our class have at most logarithmic clique-width and that the number of minimal forbidden induced subgraphs for our class is infinite. These results lead to a conjecture relaxing the above question and to a number of related open questions connecting well-quasi-ordering and clique-width.