arXiv subjects

Mateus de Oliveira Oliveira

Publications and source records attributed to Mateus de Oliveira Oliveira.

At least 19 recordsLinked to original sources

arXiv2026

State Canonization and Early Pruning in Width-Based Automated Theorem Proving

Width-based automated theorem proving is a framework where counterexamples to graph-theoretic conjectures are searched width-wise relative to some graph width measure, such as treewidth or pathwidth. In a recent work it has been shown that dynamic programming algorithms operating on tree decompositions can be combined together with the purpose of width-based theorem proving. This approach can be used to show that several long-standing conjectures in graph theory can be tested in time $2^{2^{k^{O(1)}}}$ on the class of graphs of treewidth at most $k$. In this work, we give the first steps towards evaluating the viability of this framework from a practical standpoint. At the same time, we advance the framework in two directions. First, we introduce a state-canonization technique that significantly reduces the number of states evaluated during the search for a counterexample of the conjecture. Second, we introduce an early-pruning technique that can be applied in the study of conjectures of the form $\mathcal{P}_1 \rightarrow \mathcal{P}_2$, for graph properties $\mathcal{P}_1$ and $\mathcal{P}_2$, where $\mathcal{P}_1$ is a property closed under subgraphs. As a concrete application, we use our framework in the study of graph-theoretic conjectures related to coloring triangle-free graphs. In particular, our algorithm is able to show that Reed's conjecture for triangle-free graphs is valid on the class of graphs of pathwidth at most 5, and on graphs of treewidth at most 3. Perhaps more interestingly, our algorithm is able to construct in a completely automated way counterexamples to invalid strengthenings of Reed's conjecture. These are the first results showing that width-based automated theorem proving is a promising avenue in the study of graph-theoretic conjectures.

cs.DS↗

arXiv2026

Optimal Extended Formulations from Optimal Dynamic Programming Algorithms

Vertex Subset Problems (VSPs) are a class of combinatorial optimization problems on graphs where the goal is to find a subset of vertices satisfying a predefined condition. Two prominent approaches for solving VSPs are dynamic programming over tree-like structures, such as tree decompositions or clique decompositions, and linear programming. In this work, we establish a sharp connection between both approaches by showing that if a vertex-subset problem $Π$ admits a solution-preserving dynamic programming algorithm that produces tables of size at most $α(k,n)$ when processing a tree decomposition of width at most $k$ of an $n$-vertex graph $G$, then the polytope $P_Π(G)$ defined as the convex-hull of solutions of $Π$ in $G$ has extension complexity at most $O(α(k,n)\cdot n)$. Additionally, this upper bound is optimal under the exponential time hypothesis (ETH). On the one hand, our results imply that ETH-optimal solution-preserving dynamic programming algorithms for combinatorial problems yield optimal-size parameterized extended formulations for the solution polytopes associated with instances of these problems. On the other hand, unconditional lower bounds obtained in the realm of the theory of extended formulations yield unconditional lower bounds on the table complexity of solution-preserving dynamic programming algorithms.

cs.DS↗

arXiv2026

Symbolic Functional Decomposition: A Reconfiguration Approach

Functional decomposition is the process of breaking down a function $f$ into a composition $f=g(f_1,\dots,f_k)$ of simpler functions $f_1,\dots,f_k$ belonging to some class $\mathcal{F}$. This fundamental notion can be used to model applications arising in a wide variety of contexts, ranging from machine learning to formal language theory. In this work, we study functional decomposition by leveraging on the notion of functional reconfiguration. In this setting, constraints are imposed not only on the factor functions $f_1,\dots,f_k$ but also on the intermediate functions arising during the composition process. We introduce a symbolic framework to address functional reconfiguration and decomposition problems. In our framework, functions arising during the reconfiguration process are represented symbolically, using ordered binary decision diagrams (OBDDs). The function $g$ used to specify the reconfiguration process is represented by a Boolean circuit $C$. Finally, the function class $\mathcal{F}$ is represented by a second-order finite automaton $\mathcal{A}$. Our main result states that functional reconfiguration, and hence functional decomposition, can be solved in fixed-parameter linear time when parameterized by the width of the input OBDD, by structural parameters associated with the reconfiguration circuit $C$, and by the size of the second-order finite automaton $\mathcal{A}$.

cs.CC↗

arXiv2024

From Width-Based Model Checking to Width-Based Automated Theorem Proving

In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including treewidth and cliquewidth. As a quantitative application of our framework, we prove analytically that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number $k$ as input and correctly determines in time double-exponential in $k^{O(1)}$ whether the conjecture is valid on all graphs of treewidth at most $k$. These upper bounds, which may be regarded as upper-bounds on the size of proofs/disproofs for these conjectures on the class of graphs of treewidth at most $k$, improve significantly on theoretical upper bounds obtained using previously available techniques.

cs.DS↗

arXiv2022

Learning from Positive and Negative Examples: New Proof for Binary Alphabets

One of the most fundamental problems in computational learning theory is the the problem of learning a finite automaton $A$ consistent with a finite set $P$ of positive examples and with a finite set $N$ of negative examples. By consistency, we mean that $A$ accepts all strings in $P$ and rejects all strings in $N$. It is well known that this problem is NP-complete. In the literature, it is stated that this NP-hardness holds even in the case of a binary alphabet. As a standard reference for this theorem, the work of Gold from 1978 is either cited or adapted. But as a crucial detail, the work of Gold actually considered Mealy machines and not deterministic finite state automata (DFAs) as they are considered nowadays. As Mealy automata are equipped with an output function, they can be more compact than DFAs which accept the same language. We show that the adaptions of Gold's construction for Mealy machines stated in the literature have some issues and give a new construction for DFAs with a binary alphabet ourselves.

cs.FL↗

arXiv2021

On Supergraphs Satisfying CMSO Properties

Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function $f$, there is an algorithm $\mathfrak{A}$ that takes as input a CMSO sentence $φ$, a positive integer $t$, and a connected graph $G$ of maximum degree at most $Δ$, and determines, in time $f(|φ|,t)\cdot 2^{O(Δ\cdot t)}\cdot |G|^{O(t)}$, whether $G$ has a supergraph $G'$ of treewidth at most $t$ such that $G'\models φ$. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time $f(d)\cdot 2^{O(Δ\cdot d)}\cdot |G|^{O(d)}$, whether a connected graph of maximum degree $Δ$ has a planar supergraph of diameter at most $d$. Additionally, we show that for each fixed $k$, the problem of determining whether $G$ has an $k$-outerplanar supergraph of diameter at most $d$ is strongly uniformly fixed parameter tractable with respect to the parameter $d$. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter $\mathbf{p}$. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.

cs.LO↗

arXiv2021

On the Complexity of Intersection Non-emptiness for Star-Free Language Classes

In the Intersection Non-Emptiness problem, we are given a list of finite automata $A_1,A_2,\dots,A_m$ over a common alphabet $Σ$ as input, and the goal is to determine whether some string $w\in Σ^*$ lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-Emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels zero or one half and already PSPACE-hard when all automata accept a language from the level one. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC$^0$ when restricted to level zero; complete for LOGSPACE or NLOGSPACE, depending on the input representation, when restricted to languages in the level one half; NP-complete when the input is given as DFAs accepting a language in from level one or three half; and finally, PSPACE-complete when the input automata accept languages in level two or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in the Straubing-Thérien hierarchy levels one ore three half does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation.

cs.FL↗

arXiv2021

Second-Order Finite Automata

Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite {\em sets of sets} of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for each $k$ and $w$ one can count the number of distinct functions computable by ODDs of width at most $w$ and length $k$ in time $h(|Σ|,w)\cdot k^{O(1)}$, for a suitable $h:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$. This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameter $w$ and in the length $k$ of the ODDs.

cs.FL↗

arXiv2021

Diversity in Kemeny Rank Aggregation: A Parameterized Approach

In its most traditional setting, the main concern of optimization theory is the search for optimal solutions for instances of a given computational problem. A recent trend of research in artificial intelligence, called solution diversity, has focused on the development of notions of optimality that may be more appropriate in settings where subjectivity is essential. The idea is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. When combined with techniques from parameterized complexity theory, the paradigm of diversity of solutions offers a powerful algorithmic framework to address problems of practical relevance. In this work, we investigate the impact of this combination in the field of Kemeny Rank Aggregation, a well-studied class of problems lying in the intersection of order theory and social choice theory and also in the field of order theory itself. In particular, we show that the Kemeny Rank Aggregation problem is fixed-parameter tractable with respect to natural parameters providing natural formalizations of the notions of diversity and of the notion of a sufficiently good solution. Our main results work both when considering the traditional setting of aggregation over linearly ordered votes, and in the more general setting where votes are partially ordered.

cs.AI↗

arXiv2021

On the Width of Regular Classes of Finite Structures

In this work, we introduce the notion of decisional width of a finite relational structure and the notion of decisional width of a regular class of finite structures. Our main result states that given a first-order formula ψ over a vocabulary τ, and a finite automaton F over a suitable alphabet B(Σ,w,τ) representing a width-w regular-decisional class of τ-structures C, one can decide in time f(τ,Σ,ψ,w)|F| whether some τ-structure in C satisfies ψ. Here, f is a function that depends on the parameters τ,Σ,ψ,w, but not on the size of the automaton F representing the class. Therefore, besides implying that the first-order theory of any given regular-decisional class of finite structures is decidable, it also implies that when the parameters τ, ψ, Σ and w are fixed, decidability can be achieved in linear time on the size of the input automaton F. Building on the proof of our main result, we show that the problem of counting satisfying assignments for a first-order logic formula in a given structure A of width w is fixed-parameter tractable with respect to w, and can be solved in quadratic time on the length of the input representation of A.

cs.LO↗

arXiv2020

Diversity of Solutions: An Exploration Through the Lens of Fixed-Parameter Tractability Theory

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

cs.DS↗

arXiv2020

Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach

It can be shown that each permutation group $G \sqsubseteq S_n$ can be embedded, in a well defined sense, in a connected graph with $O(n+|G|)$ vertices. Some groups, however, require much fewer vertices. For instance, $S_n$ itself can be embedded in the $n$-clique $K_n$, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group $G \sqsubseteq S_n$ can be upper bounded by three structural parameters of connected graphs embedding $G$: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group $G \sqsubseteq S_n$ that can be embedded into a connected graph with $m$ vertices, treewidth k, and maximum degree $Δ$, can also be generated by a context-free grammar of size $2^{O(kΔ\logΔ)}\cdot m^{O(k)}$. By combining our upper bound with a connection between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity $2^{O(k Δ\log Δ)}\cdot m^{O(k)}$. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated $2^{Ω(n)}$ lower bound on the grammar complexity of the symmetric group $S_n$ we have that connected graphs of treewidth $o(n/\log n)$ and maximum degree $o(n/\log n)$ embedding subgroups of $S_n$ of index $2^{cn}$ for some small constant $c$ must have $n^{ω(1)}$ vertices. This lower bound can be improved to exponential on graphs of treewidth $n^{\varepsilon}$ for $\varepsilon<1$ and maximum degree $o(n/\log n)$.

cs.FL↗

arXiv2016

A Near-Quadratic Lower Bound for the Size of Quantum Circuits of Constant Treewidth

We show that any quantum circuit of treewidth $t$, built from $r$-qubit gates, requires at least $Ω(\frac{n^{2}}{2^{O(r\cdot t)}\cdot \log^4 n})$ gates to compute the element distinctness function. Our result generalizes a near-quadratic lower bound for quantum formula size obtained by Roychowdhury and Vatan [SIAM J. on Computing, 2001]. The proof of our lower bound follows by an extension of Nečiporuk's method to the context of quantum circuits of constant treewidth. This extension is made via a combination of techniques from structural graph theory, tensor-network theory, and the connected-component counting method, which is a classic tool in algebraic geometry.

cs.CC↗

arXiv2016

On the Satisfiability of Quantum Circuits of Small Treewidth

It has been known for almost three decades that many $\mathrm{NP}$-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit $C$ with $n$ uninitialized inputs, $\mathit{poly}(n)$ gates, and treewidth $t$, one can compute in time $(\frac{n}δ)^{\exp(O(t))}$ a classical assignment $y\in \{0,1\}^n$ that maximizes the acceptance probability of $C$ up to a $δ$ additive factor. In particular, our algorithm runs in polynomial time if $t$ is constant and $1/poly(n) < δ< 1$. For unrestricted values of $t$, this problem is known to be complete for the complexity class $\mathrm{QCMA}$, a quantum generalization of MA. In contrast, we show that the same problem is $\mathrm{NP}$-complete if $t=O(\log n)$ even when $δ$ is constant. On the other hand, we show that given a $n$-input quantum circuit $C$ of treewidth $t=O(\log n)$, and a constant $δ<1/2$, it is $\mathrm{QMA}$-complete to determine whether there exists a quantum state $\mid\!φ\rangle \in (\mathbb{C}^d)^{\otimes n}$ such that the acceptance probability of $C\mid\!φ\rangle$ is greater than $1-δ$, or whether for every such state $\mid\!φ\rangle$, the acceptance probability of $C\mid\!φ\rangle$ is less than $δ$. As a consequence, under the widely believed assumption that $\mathrm{QMA} \neq \mathrm{NP}$, we have that quantum witnesses are strictly more powerful than classical witnesses with respect to Merlin-Arthur protocols in which the verifier is a quantum circuit of logarithmic treewidth.

cs.CC↗

arXiv2015

An Algorithmic Metatheorem for Directed Treewidth

The notion of directed treewidth was introduced by Johnson, Robertson, Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as a first step towards an algorithmic metatheory for digraphs. They showed that some NP-complete properties such as Hamiltonicity can be decided in polynomial time on digraphs of constant directed treewidth. Nevertheless, despite more than one decade of intensive research, the list of hard combinatorial problems that are known to be solvable in polynomial time when restricted to digraphs of constant directed treewidth has remained scarce. In this work we enrich this list by providing for the first time an algorithmic metatheorem connecting the monadic second order logic of graphs to directed treewidth. We show that most of the known positive algorithmic results for digraphs of constant directed treewidth can be reformulated in terms of our metatheorem. Additionally, we show how to use our metatheorem to provide polynomial time algorithms for two classes of combinatorial problems that have not yet been studied in the context of directed width measures. More precisely, for each fixed $k,w \in \mathbb{N}$, we show how to count in polynomial time on digraphs of directed treewidth $w$, the number of minimum spanning strong subgraphs that are the union of $k$ directed paths, and the number of maximal subgraphs that are the union of $k$ directed paths and satisfy a given minor closed property. To prove our metatheorem we devise two technical tools which we believe to be of independent interest. First, we introduce the notion of tree-zig-zag number of a digraph, a new directed width measure that is at most a constant times directed treewidth. Second, we introduce the notion of $z$-saturated tree slice language, a new formalism for the specification and manipulation of infinite sets of digraphs.

cs.DS↗

arXiv2014

Automated Verification, Synthesis and Correction of Concurrent Systems via MSO Logic

In this work we provide algorithmic solutions to five fundamental problems concerning the verification, synthesis and correction of concurrent systems that can be modeled by bounded p/t-nets. We express concurrency via partial orders and assume that behavioral specifications are given via monadic second order logic. A c-partial-order is a partial order whose Hasse diagram can be covered by c paths. For a finite set T of transitions, we let P(c,T,ϕ) denote the set of all T-labelled c-partial-orders satisfying ϕ. If N=(P,T) is a p/t-net we let P(N,c) denote the set of all c-partially-ordered runs of N. A (b, r)-bounded p/t-net is a b-bounded p/t-net in which each place appears repeated at most r times. We solve the following problems: 1. Verification: given an MSO formula ϕand a bounded p/t-net N determine whether P(N,c)\subseteq P(c,T,ϕ), whether P(c,T,ϕ)\subseteq P(N,c), or whether P(N,c)\cap P(c,T,ϕ)=\emptyset. 2. Synthesis from MSO Specifications: given an MSO formula ϕ, synthesize a semantically minimal (b,r)-bounded p/t-net N satisfying P(c,T,ϕ)\subseteq P(N, c). 3. Semantically Safest Subsystem: given an MSO formula ϕdefining a set of safe partial orders, and a b-bounded p/t-net N, possibly containing unsafe behaviors, synthesize the safest (b,r)-bounded p/t-net N' whose behavior lies in between P(N,c)\cap P(c,T,ϕ) and P(N,c). 4. Behavioral Repair: given two MSO formulas ϕand ψ, and a b-bounded p/t-net N, synthesize a semantically minimal (b,r)-bounded p/t net N' whose behavior lies in between P(N,c) \cap P(c,T,ϕ) and P(c,T,ψ). 5. Synthesis from Contracts: given an MSO formula ϕ^yes specifying a set of good behaviors and an MSO formula ϕ^no specifying a set of bad behaviors, synthesize a semantically minimal (b,r)-bounded p/t-net N such that P(c,T,ϕ^yes) \subseteq P(N,c) but P(c,T,ϕ^no ) \cap P(N,c)=\emptyset.

cs.LO↗

arXiv2013

Subgraphs Satisfying MSO Properties on z-Topologically Orderable Digraphs

We introduce the notion of z-topological orderings for digraphs. We prove that given a digraph G on n vertices admitting a z-topological order- ing, together with such an ordering, one may count the number of subgraphs of G that at the same time satisfy a monadic second order formula ϕ and are the union of k directed paths, in time f (ϕ, k, z) * n^O(k*z) . Our result implies the polynomial time solvability of many natural counting problems on digraphs admitting z-topological orderings for constant values of z and k. Concerning the relationship between z-topological orderability and other digraph width measures, we observe that any digraph of directed path-width d has a z- topological ordering for z <= 2d + 1. On the other hand, there are digraphs on n vertices admitting a z-topological order for z = 2, but whose directed path-width is Θ(log n). Since graphs of bounded directed path-width can have both arbitrarily large undirected tree-width and arbitrarily large clique width, our result provides for the first time a suitable way of partially trans- posing metatheorems developed in the context of the monadic second order logic of graphs of constant undirected tree-width and constant clique width to the realm of digraph width measures that are closed under taking subgraphs and whose constant levels incorporate families of graphs of arbitrarily large undirected tree-width and arbitrarily large clique width.

cs.CC↗

arXiv2012

Canonizable Partial Order Generators and Regular Slice Languages

In a previous work we introduced slice graphs as a way to specify both infinite languages of directed acyclic graphs (DAGs) and infinite languages of partial orders. Therein we focused on the study of Hasse diagram generators, i.e., slice graphs that generate only transitive reduced DAGs, and showed that they could be used to solve several problems related to the partial order behavior of p/t-nets. In the present work we show that both slice graphs and Hasse diagram generators are worth studying on their own. First, we prove that any slice graph SG can be effectively transformed into a Hasse diagram generator HG representing the same set of partial orders. Thus from an algorithmic standpoint we introduce a method of transitive reducing infinite families of DAGs specified by slice graphs. Second, we identify the class of saturated slice graphs. By using our transitive reduction algorithm, we prove that the class of partial order languages representable by saturated slice graphs is closed under union, intersection and even under a suitable notion of complementation (cut-width complementation). Furthermore partial order languages belonging to this class can be tested for inclusion and admit canonical representatives in terms of Hasse diagram generators. As an application of our results, we give stronger forms of some results in our previous work, and establish some unknown connections between the partial order behavior of $p/t$-nets and other well known formalisms for the specification of infinite families of partial orders, such as Mazurkiewicz trace languages and message sequence chart (MSC) languages.

cs.FL↗