arXiv subjects

Benjamin Merlin Bumpus

Publications and source records attributed to Benjamin Merlin Bumpus.

18 recordsLinked to original sources

arXiv2026

Co-Decompositions and FO Model Checking

Tree decompositions describe large graphs as gluings of smaller component parts. Here we study their categorical duals, tree co-decompositions, which construct large graphs via fibered products of small constituent parts. Co-decompositions are compressed representations of graphs or relational structures: the decomposed object can be exponentially larger than the space required to specify the co-decomposition. Our main contribution is an algorithmic meta-theorem for first order logic: given only a co-decomposition of a $σ$-structure $G$ with $b$ bags, each of size at most $w$ and an FO sentence $ϕ$, there is an algorithm which, without ever constructing or inspecting $G$, operates on the co-decomposition and determines whether $G \models ϕ$ in time $f_σ(w,|φ|)\cdot b$. We pair this algorithm with structural results on tree co-width, including its relationship to tree-width and a separation from twin-width. Furthermore, we characterize bounded path co-width using a graph grammar. In contrast to our logical meta-theorem, we prove that $(s,t)$-Path becomes NP-hard on co-decomposition inputs. These results demonstrate how categorical constructions can enable efficient computation on exponentially compressed structures.

math.CO↗

arXiv2026

A Temporal Cellular Sheaf Framework for Multi-Agent Systems with Switching Communication Topologies

Cellular sheaves serve as a natural modeling framework for encoding heterogeneity in the state and interaction spaces of multi-agent systems. This work examines switching communication graphs with the use of temporal cellular sheaves, which we construct to represent heterogeneous agent state spaces, the maps that relate them, and their respective communication constraints. We then develop sufficient conditions under which a multi-agent system defined on a temporal cellular sheaf can solve the target tracking problem by generalizing the existing concept of joint connectivity. Using a Lyapunov-based stability analysis, we demonstrate that the target tracking error of the network converges to a neighborhood of the origin.

eess.SY↗

arXiv2026

Time-Varying Data as Sheaves: an Invitation to Narratives

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

math.CT↗

arXiv2026

Decomposing time-varying data into simple pieces: structured decompositions of narratives

Graphs that change over time arise throughout applications, but there is no single standard way to decompose them into smaller pieces. In this paper, we propose a systematic categorical method for doing so. The main idea is to combine structured decompositions, which generalize graph decompositions, such as tree-decompositions, with persistent narratives, which model time-varying data as diagrams. We prove that, under suitable categorical hypotheses, any static theory of decompositions can be lifted to a corresponding temporal theory. As case studies, we apply this construction to time-varying graphs and recover natural temporal analogues of ordinary tree-width, complemented tree-width, and the tree-independence number.

math.CT↗

arXiv2026

$O(n +f(k))$: Truly Linear FPT

Parameterized complexity has always been concerned with practical computing: by confining combinatorial explosion to a secondary parameter $k$, one can uncover why and how many NP-hard problems are effectively tackled in practice. Today, however, the scale of data has changed: scientists study Big Data, which is so large that even quadratic dependence in the total input size $n$ is unaffordable. Therefore, what constitutes a practical algorithm has also changed. Classically, parameterized complexity is blind to the difference between defining fixed parameter tractability multiplicatively (i.e. $f(k) \cdot n^c$) or additively (i.e. $f(k) + n^c$). But what if the constant $c$ is one and we require true linearity, is this distinction still inconsequential? Here, we define and explore Truly Linear FPT (TLFPT) -- that is $O(n)+f(k)$ -- and show that it is a strict subset of Linear FPT (LFPT) -- that is $O(n) \cdot f(k)$ -- via diagonalization. Populating TLFPT requires careful consideration of linear-time algorithmics and data structures. We meet many inhabitants of TLFPT: SAT, Vertex Cover, Min-Max Matching, $(n-k)$-Coloring, Diverse Pair of Matchings, $k$-Path, and $H$-Coloring. Our parameterizations are equally varied. Beyond classical parameters like solution size, we leverage two parameters, treedepth and BFS-width, which are particularly well-suited to the TLFPT regime. We do so by developing techniques based on depth- and breadth-first search. For parameterized complexity to be of service to the scientific community, we need to contend with Big Data. For sufficiently large inputs, FPT beyond linear may not suffice. Thus, there is a practical and theoretical need for more ambitious goals. TLFPT is a first step forward.

cs.CC↗

arXiv2026

A Parameterized Algorithm for Testing whether the Limit of a Diagram is Empty

A limit of a (small) diagram $d : J \to E$ in a complete category $E$ can be thought of as specifying a set of equations involving the objects of $E$. To motivate this intuitively, one can think of each object $d(j)$ as a "variable" and each morphism in $J$ as a "constraint" connecting these variables. If $E$ has an initial object, a natural question arises: does our set of equations have any solution at all? Equivalently, we can ask: is the limit of $d$ initial? In this paper we consider the computational problem that, given finite diagram $d$ in a finitely complete category $E$, asks whether its limit is empty. We construct a fast algorithm (in the sense of parameterized complexity theory) that solves this problem when $E$ is of the form $\mathbf{FinSet}^{J}$ for a finite category $J$ and $d$ is a structured co-decomposition, i.e. a diagram arising from the opposite of the Grothendieck construction of a simple graph.

math.CT↗

arXiv2025

Algorithmic and Extremal Obstructions Through the Language of Cohomology

We model problems as presheaves that assign sets of certificates to input instances, and we show how to use presheaf Čech cohomology to capture the precise ways in which local solutions fail to patch into global ones. Applied to problems like Vertex Cover, Cycle Cover, and Odd Cycle Transversal, our framework exposes emergent phenomena such as hidden cycles or the inflation of small, local solutions. This approach not only rephrases classical results like König's Theorem in cohomological terms, but also reveals how to systematically account for failures of compositionality. Although our main focus is on presheaves of sets, the methods generalize naturally to Abelian presheaves, suggesting a rich interplay between graph theory, cohomology, and complexity. This work represents a first step toward a systematic, sheaf-theoretic theory of algorithmic structure and related obstructions.

math.AC↗

arXiv2025

Additive Invariants of Open Petri Nets

We classify all additive invariants of open Petri nets: these are $\mathbb{N}$-valued invariants which are additive with respect to sequential and parallel composition of open Petri nets. In particular, we prove two classification theorems: one for open Petri nets and one for monically open Petri nets (i.e. open Petri nets whose interfaces are specified by monic maps). Our results can be summarized as follows. The additive invariants of open Petri nets are completely determined by their values on a particular class of single-transition Petri nets. However, for monically open Petri nets, the additive invariants are determined by their values on transitionless Petri nets and all single-transition Petri nets. Our results confirm a conjecture of John Baez (stated during the AMS' 2022 Mathematical Research Communities workshop).

math.CT↗

arXiv2025

Lassos: Pushing Tree Decompositions Forward Along Homomorphisms

It is folklore that tree-width is monotone under taking subgraphs (i.e. injective graph homomorphisms) and contractions (certain kinds of surjective graph homomorphisms). However, although tree-width is obviously not monotone under any surjective graph homomorphism, it is not clear whether contractions are canonically the only class of surjections with respect to which it is monotone. Under the requirement that the decomposition shape must be preserved, we prove that this is indeed the case. Our results provide a framework for answering questions of this sort for many other kinds of combinatorial data structures (such as directed multigraphs, hypergraphs, Petri nets, circular port graphs, half-edge graphs, databases, simplicial sets etc.) for which natural analogues of tree decompositions can be defined. Furthermore and of independent interest, we prove these results by introducing the notion of a lasso, a generalization of contractions of graphs to arbitrary categories with pushouts of monomorphisms.

math.CO↗

arXiv2025

Structured Decompositions: Structural and Algorithmic Compositionality

We introduce structured decompositions, category-theoretic structures which simultaneously generalize notions from graph theory (including treewidth, layered treewidth, co-treewidth, graph decomposition width, tree independence number, hypergraph treewidth and H-treewidth), geometric group theory (specifically Bass-Serre theory), and dynamical systems (e.g. hybrid dynamical systems). We define width functors, which provide a compositional way to analyze and relate different structural complexity measures, and establish a general duality between decompositions and completions of objects.

math.CT↗

arXiv2025

Towards a Unified Theory of Time-Varying Data

What is a time-varying graph, a time-varying topological space, or, more generally, a mathematical structure that evolves over time? In this work, we lay the foundations for a general theory of temporal data by introducing categories of narratives. These are sheaves on posets of time intervals that encode snapshots of a temporal object along with the relationships between them. This theory satisfies five desiderata distilled from the burgeoning field of time-varying graphs: (D1) it defines both time-varying objects and their morphisms; (D2) it distinguishes between cumulative and persistent interpretations and provides principled methods for transitioning between them; (D3) it systematically lifts static notions to their temporal analogues; (D4) it is object agnostic; (D5) it integrates with theories of dynamical systems. To achieve this, we build upon existing categorical and sheaf-theoretic approaches to temporal graph theory, generalizing them to any category with limits and colimits. We also formalize tacit intuitions that, while present, often remain implicit in temporal graph theory. Beyond synthesizing and reformulating existing ideas in categorical language, we introduce sheaf-theoretic constructions and prove results that, to our knowledge, have not appeared in the temporal data literature - such as the adjunction between persistent and cumulative narratives. More importantly, we integrate these existing and novel elements into a consistent and coherent framework, setting the stage for a unified theory of time-varying data.

math.CT↗

arXiv2025

Directed branch-width: A directed analogue of tree-width

Gurski and Wanke showed that a graph class C has bounded tree-width if and only if its associated class of directed line graphs has bounded clique-width. Inevitably -- asking whether this relationship lifts to directed graphs -- we introduce a new digraph width measure: we obtain it by investigating digraphs whose directed line graphs have bounded cliquewidth. Thus, to generalize Gurski and Wanke's aforementioned result, we introduce a natural generalization of branch-width to digraphs and we name it accordingly. Directed branch-width is a genuinely directed width-measure insofar as it cannot be used to bound the value of the underlying undirected tree-width. Despite this, the two measures are still closely related: the directed branch-width of a digraph D can differ from the branch-width of its underlying undirected graph only at sources and sinks. This relationship allows us to extend a range of algorithmic results from directed graphs with bounded underlying treewidth to the strictly larger class of digraphs having bounded directed branch-width.

math.CO↗

arXiv2024

Degree of Satisfiability in Heyting Algebras

Given a finite structure $M$ and property $p$, it is a natural to study the degree of satisfiability of $p$ in $M$; i.e. to ask: what is the probability that uniformly randomly chosen elements in $M$ satisfy $p$? In group theory, a well-known result of Gustafson states that the equation $xy=yx$ has a finite satisfiability gap: its degree of satisfiability is either $1$ (in Abelian groups) or no larger than $\frac{5}{8}$. Degree of satisfiability has proven useful in the study of (finite and infinite) group-like and ring-like algebraic structures, but finite satisfiability gap questions have not been considered in lattice-like, order-theoretic settings yet. Here we investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies $x \vee \neg x = \top$ is no larger than $\frac{2}{3}$. Finally, we generalize our results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic.

math.LO↗

arXiv2023

Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves

Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to mathematical structures writ large. We achieve this horizontal generality by adopting a categorial perspective which allows us to show that: (1) structured decompositions (a recent, abstract generalization of many graph decompositions) define Grothendieck topologies on categories of data (adhesive categories) and that (2) any computational problem which can be represented as a sheaf with respect to these topologies can be decided in linear time on classes of inputs which admit decompositions of bounded width and whose decomposition shapes have bounded feedback vertex number. This immediately leads to algorithms on objects of any C-set category; these include -- to name but a few examples -- structures such as: symmetric graphs, directed graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases, simplicial complexes, circular port graphs and half-edge graphs. Thus we initiate the bridging of tools from sheaf theory, structural graph theory and parameterized complexity theory; we believe this to be a very fruitful approach for a general, algebraic theory of dynamic programming algorithms. Finally we pair our theoretical results with concrete implementations of our main algorithmic contribution in the AlgebraicJulia ecosystem.

cs.CC↗

arXiv2022

Search-Space Reduction via Essential Vertices

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.

cs.DS↗

arXiv2022

Spined categories: generalizing tree-width beyond graphs

Tree-width is an invaluable tool for computational problems on graphs. But often one would like to compute on other kinds of objects (e.g. decorated graphs or even algebraic structures) where there is no known tree-width analogue. Here we define an abstract analogue of tree-width which provides a uniform definition of various tree-width-like invariants including graph tree-width, hypergraph tree-width, complemented tree-width and even new constructions such as the tree-width of modular quotients. We obtain this generalization by developing a general theory of categories that admit abstract analogues of both tree decompositions and tree-width; we call these pseudo-chordal completions and the triangulation functor respectively.

math.CO↗

arXiv2021

Edge exploration of temporal graphs

We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open question of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars. Furthermore, we introduce a vertex-variant of interval-membership-width (which can be arbitrarily larger than its edge-counterpart) and use it to obtain an FPT-time algorithm for a natural vertex-exploration problem that remains hard even when interval-membership-width is bounded.

cs.CC↗

arXiv2021

Treewidth via Spined Categories (extended abstract)

Treewidth is a well-known graph invariant with multiple interesting applications in combinatorics. On the practical side, many NP-complete problems are polynomial-time (sometimes even linear-time) solvable on graphs of bounded treewidth. On the theoretical side, treewidth played an essential role in the proof of the celebrated Robertson-Seymour graph minor theorem. While defining treewidth-like invariants on graphs and treewidth analogues on other sorts of combinatorial objects (incl. hypergraphs, digraphs) has been a fruitful avenue of research, a direct, categorial description capturing multiple treewidth-like invariants is yet to emerge. Here we report on our recent work on spined categories (arXiv:2104.01841): categories equipped with extra structure that permits the definition of a functorial analogue of treewidth, the triangulation functor. The usual notion of treewidth is recovered as a special case, the triangulation functor of a spined category with graphs as objects and graph monomorphisms as arrows. The usual notion of treewidth for hypergraphs arises as the triangulation functor of a similar category of hypergraphs.

math.CT↗