arXiv subjects

Igor Razgon

Publications and source records attributed to Igor Razgon.

At least 19 recordsLinked to original sources

arXiv2025

FPT Parameterisations of Fractional and Generalised Hypertree Width

We present the first fixed-parameter tractable (FPT) algorithms for exact computation of generalized hypertree width (ghw) and fractional hypertree width (fhw). Our algorithms are parameterized by the target width, the rank, and the maximum degree of the input hypergraph. More generally, we show that testing f-width is in FPT for a broad class of width functions that we call manageable. This class contains the edge cover number $\rho$ and its fractional relaxation $\rho^*$, and thus covers both generalized and fractional hypertree width. We additionally extend our framework to also obtain an fpt algorithm for computing a discretized version of adaptive width. Our approach extends a recent algorithm for treewidth (Boja\'ncyk & Pilipczuk, LMCS 2022) that utilises monadic second-order transductions. To extend this idea beyond treewidth we develop new combinatorial machinery around elimination forests in hypergraphs, culminating in a structural normal form for optimal witnesses that makes transduction-based optimisation applicable in the much more general context of manageable width functions. This yields the first exact FPT algorithms for these measures under any nontrivial parameterisation and provides structural tools that may enable more direct optimisation algorithms

cs.DS↗

arXiv2025

Decision DNNFs with imbalanced conjunction cannot efficiently represent CNFs of bounded width

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.

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arXiv2025

On complexity of restricted fragments of Decision DNNF

Decision \textsc{dnnf} (a.k.a. $\wedge_d$-\textsc{fbdd}) is an important special case of Decomposable Negation Normal Form (\textsc{dnnf}), a landmark knowledge compilation model. Like other known \textsc{dnnf} restrictions, Decision \textsc{dnnf} admits \textsc{fpt} sized representation of \textsc{cnf}s of bounded \emph{primal} treewidth. However, unlike other restrictions, the complexity of representation for \textsc{cnf}s of bounded \emph{incidence} treewidth is wide open. In[arxiv:1708.07767], we resolved this question for two restricted classes of Decision \textsc{dnnf} that we name $\wedge_d$-\textsc{obdd} and Structured Decision \textsc{dnnf}. In particular, we demonstrated that, while both these classes have \textsc{fpt}-sized representations for \textsc{cnf}s of bounded primal treewidth, they need \textsc{xp}-size for representation of \textsc{cnf}s of bounded incidence treewidth. In the main part of this paper we carry out an in-depth study of the $\wedge_d$-\textsc{obdd} model. We formulate a generic methodology for proving lower bounds for the model. Using this methodology, we reestablish the \textsc{xp} lower bound provided in [arxiv:1708.07767]. We also provide exponential separations between \textsc{fbdd} and $\wedge_d$-\textsc{obdd} and between $\wedge_d$-\textsc{obdd} and an ordinary \textsc{obdd}. We study the complexity of Apply operation for $\wedge_d$-\textsc{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 \textsc{dnnf} that we name Structured $\wedge_d$-\textsc{fbdd} and demonstrate that this model is quite powerful for \textsc{cnf}s of bounded incidence treewidth.

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arXiv2023

FPT Approximation of Generalised Hypertree Width for Bounded Intersection Hypergraphs

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 $ghw$.

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arXiv2023

New Width Parameters for Independent Set: One-sided-mim-width and Neighbor-depth

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, Str{\o}mme, and Telle [TCS 2017].

cs.DS↗

arXiv2022

Constant ratio FPT approximation of hypertree width parameters for hypergraphs of bounded rank

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.

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arXiv2022

The treewidth and pathwidth of graph unions

Given two $n$-vertex graphs $G_1$ and $G_2$ of bounded treewidth, is there an $n$-vertex graph $G$ of bounded treewidth having subgraphs isomorphic to $G_1$ and $G_2$? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if $G_1$ is a binary tree and $G_2$ is a ternary tree. We also provide an extensive study of cases where such `gluing' is possible. In particular, we prove that if $G_1$ has treewidth $k$ and $G_2$ has pathwidth $\ell$, then there is an $n$-vertex graph of treewidth at most $k + 3 \ell + 1$ containing both $G_1$ and $G_2$ as subgraphs.

math.CO↗

arXiv2022

The splitting power of branching programs of bounded repetition and CNFs of bounded width

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 $d-pw(G)$ and $d-cpw(G)$ where $G$ is a graph. We show that $2-cpw(G) \leq O(tw(G) Δ(G))$ 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 of size $2^{O(Δ(ψ)*tw(ψ)^2)}$ 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 $d$-pathwdith 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 $ψ(G)$ whsose clauses are $(u \vee e \vee v)$ for each edge $e=\{u,v\}$ of $G$. We prove that a $d$-SMNBP representing $ψ(G)$ is of size at least $Ω(c^{d-pw(G)})$ where $c=(8/7)^{1/12}$. We use this 'generic' lower bound to obtain an exponential lower bound for a 'concrete' class of CNFs $ψ(K_n)$. In particular, we demonstrate that for each $0<a<1$, the size of $n^{a}$-SMNBP representing $ψ(K_n)$ is at least $c^{n^b}$ where $b$ is an arbitrary constant such that $a+b<1$. This lower bound is tight in the sense $ψ(K_n)$ can be represented by a poly-sized $n$-SMNBP.

math.CO↗

arXiv2021

Classification of OBDD size for monotone 2-CNFs

We introduce a new graph parameter called linear upper maximum induced matching width \textsc{lu-mim width}, denoted for a graph $G$ by $lu(G)$. We prove that the smallest size of the \textsc{obdd} for $φ$, 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.

math.CO↗

arXiv2021

Tree-width dichotomy

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.

math.CO↗

arXiv2020

Fractional Covers of Hypergraphs with Bounded Multi-Intersection

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. This allows us to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is $\leq k$ for some constant $k$. We also show how our results translate to fractional vertex covers.

cs.DM↗

arXiv2020

Complexity Analysis of Generalized and Fractional Hypertree Decompositions

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 $fhw$.

cs.CC↗

arXiv2019

Regular resolution for CNFs with almost bounded one-sided treewidth

We introduce a one-sided incidence tree decomposition of a CNF $\varphi$. This is a tree decomposition of the incidence graph of $\varphi$ where the underlying tree is rooted and the set of bags containing each clause induces a directed path in the tree. The one-sided treewidth is the smallest width of a one-sided incidence tree decomposition. We consider a class of unsatisfiable CNF $\varphi$ that can be turned into one of one sided treewidth at most $k$ by removal of at most $p$ clauses. We show that the size of regular resolution for this class of CNFs is FPT parameterized by $k$ and $p$. The results contributes to understanding the complexity of resolution for CNFs of bounded incidence treewidth, an open problem well known in the areas of proof complexity and knowledge compilation. In particular, the result significantly generalizes all the restricted classes of CNFs of bounded incidence treewidth that are known to admit an FPT sized resolution. The proof includes an auxiliary result and several new notions that may be of an independent interest.

cs.CC↗

arXiv2018

Linear read-once and related Boolean functions

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 func- tions 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. conjec- tured 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.

math.CO↗

arXiv2017

Partial matching width and its application to lower bounds for branching programs

We introduce a new structural graph parameter called \emph{partial matching width}. For each (sufficiently large) integer $k \geq 1$, we introduce a class $\mathcal{G}_k$ of graphs of treewidth at most $k$ and max-degree $7$ such that for each $G \in \mathcal{G}_k$ and each (sufficiently large) $V \subseteq V(G)$, the partial matching width of $V$ is $Ω(k \log |V|)$. We use the above lower bound to establish a lower bound on the size of non-deterministic read-once branching programs (NROBPs). In particular, for each sufficiently large ineteger $k$, we introduce a class ${\bf Φ}_k$ of CNFs of (primal graph) treewidth at most $k$ such that for any $φ\in {\bf Φ}_k$ and any Boolean function $F \subseteq φ$ and such that $|φ|/|F| \leq 2^{\sqrt{n}}$ (here the functions are regarded as sets of assignments on which they are true), a NROBP implementing $F$ is of size $n^{Ω(k)}$. This result significantly generalises an earlier result of the author showing a non-FPT lower bound for NROBPs representing CNFs of bounded treewidth. Intuitively, we show that not only those CNFs but also their arbitrary one side approximations with an exponential ratio still attain that lower bound. The non-trivial aspect of this approximation is that due to a small number of satisfying assignments for $F$, it seems difficult to establish a large bottleneck: the whole function can `sneak' through a single rectangle corresponding to just \emph{one} vertex of the purported bottleneck. We overcome this problem by simultaneously exploring $\sqrt{n}$ bottlenecks and showing that at least one of them must be large. This approach might be useful for establishing other lower bounds for branching programs.

cs.CC↗

arXiv2017

Non-FPT lower bounds for structural restrictions of decision DNNF

We give a non-FPT lower bound on the size of structured decision DNNF and OBDD with decomposable AND-nodes representing CNF-formulas of bounded incidence treewidth. Both models are known to be of FPT size for CNFs of bounded primal treewidth. To the best of our knowledge this is the first parameterized separation of primal treewidth and incidence treewidth for knowledge compilation models.

cs.AI↗

arXiv2017

Specifying a positive threshold function via extremal points

An extremal point of a positive threshold Boolean function $f$ is either a maximal zero or a minimal one. It is known that if $f$ depends on all its variables, then the set of its extremal points completely specifies $f$ within the universe of threshold functions. However, in some cases, $f$ can be specified by a smaller set. The minimum number of points in such a set is the specification number of $f$. It was shown in [S.-T. Hu. Threshold Logic, 1965] that the specification number of a threshold function of $n$ variables is at least $n+1$. In [M. Anthony, G. Brightwell, and J. Shawe-Taylor. On specifying Boolean functions by labelled examples. Discrete Applied Mathematics, 1995] it was proved that this bound is attained for nested functions and conjectured that for all other threshold functions the specification number is strictly greater than $n+1$. In the present paper, we resolve this conjecture negatively by exhibiting threshold Boolean functions of $n$ variables, which are non-nested and for which the specification number is $n+1$. On the other hand, we show that the set of extremal points satisfies the statement of the conjecture, i.e., a positive threshold Boolean function depending on all its $n$ variables has $n+1$ extremal points if and only if it is nested. To prove this, we reveal an underlying structure of the set of extremal points.

math.CO↗

arXiv2016

Two types of branching programs with bounded repetition that cannot efficiently compute monotone 3-CNFs

It is known that there are classes of 2-CNFs requiring exponential size non-deterministic read-once branching programs to compute them. However, to the best of our knowledge, there are no superpolynomial lower bounds for branching programs of a higher repetition computing a class of 2-CNFs. This work is an attempt to make a progress in this direction. We consider a class of monotone 3-CNFs that are almost 2-CNFs in the sense that in each clause there is a literal occurring in this clause only. We prove exponential lower bounds for two classes of non-deterministic branching programs. The first class significantly generalizes monotone read-$k$-times {\sc nbp}s and the second class generalizes oblivious read $k$ times branching programs. The lower bounds remain exponential for $k \leq \log n/a$ where $a$ is a sufficiently large constant.

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