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Barnaby Martin

Publications and source records attributed to Barnaby Martin.

At least 19 recordsLinked to original sources
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

On Detecting $H$-Induced Minors for Small $H$

We consider the $H$-Induced Minor problem: for a fixed graph~$H$, decide whether a given graph $G$ contains $H$ as an induced minor. While the problem is known to be NP-complete for some trees~$H$ on more than $2^{300}$ vertices, the complexity for small trees remains unresolved. In particular, the case where $H$ is the $7$-vertex tree consisting of a path on five vertices with a pendant vertex attached to the second and fourth vertex was a long-standing open problem. We show that this case is polynomial-time solvable by developing algorithms that detect a sequence of carefully chosen substructures. Complementing this, we prove that detecting some of these substructures individually is NP-hard. We also give polynomial-time algorithms for three cases where $H$ is a graph on five vertices (that is not a tree). In this way, we completed the classification of $H$-Induced Minor for graphs $H$ on five vertices and answered an open problem of Dallard, Dumas, Hilaire and Perez (2025).

math.CO↗
arXiv2026

Steiner Forest for $H$-Subgraph-Free Graphs

Our main result is a full classification, for every connected graph $H$, of the computational complexity of Steiner Forest on $H$-subgraph-free graphs. To obtain this dichotomy, we establish the following new algorithmic, hardness, and combinatorial results: Algorithms: We identify two new classes of graph-theoretical structures that make it possible to solve Steiner Forest in polynomial time. Roughly speaking, our algorithms handle the following cases: (1) a set $X$ of vertices of bounded size that are pairwise connected by subgraphs of treewidth $2$ or bounded size, possibly together with an independent set of arbitrary size that is connected to $X$ in an arbitrary way; (2) a set $X$ of vertices of arbitrary size that are pairwise connected in a cyclic manner by subgraphs of treewidth $2$ or bounded size. Hardness results: We show that Steiner Forest remains NP-complete for graphs with 2-deletion set number $3$. (The $c$-deletion set number is the size of a smallest cutset $S$ such that every component of $G-S$ has at most $c$ vertices.) Combinatorial results: To establish the dichotomy, we perform a delicate graph-theoretic analysis showing that if $H$ is a path or a subdivided claw, then excluding $H$ as a subgraph either yields one of the two algorithmically favourable structures described above, or yields a graph class for which NP-completeness of Steiner Forest follows from either our new hardness result or a previously known one. Along the way to classifying the hardness for excluded subgraphs, we establish a dichotomy for graphs with $c$-deletion set number at most $k$. Specifically, our results together with pre-existing ones show that Steiner Forest is polynomial-time solvable if (1) $c=1$ and $k\geq 0$, or (2) $c=2$ and $k\leq 2$, or (3) $c\geq 3$ and $k=1$, and is NP-complete otherwise.

math.CO↗
arXiv2025

Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification

We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains to consider when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided ``H''-graph, which is either a subdivided $\mathbb{H}_0$: a tree of maximum degree $4$ that is a star, or a subdivided $\mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases $\mathbb{H}_0$ and $\mathbb{H}_1$. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.

math.CO↗
arXiv2025

Restricted CSPs and F-free Digraph Algorithmics

In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this research line, by addressing the questions, is it true that digraph homomorphism problems CSP$({\mathbb H})$ have a P versus NP-complete dichotomy when the input is restricted to $\vec{\mathbb P}_k$-free (resp.\ $\vec{\mathbb P}_k$-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP$({\mathbb H})$ is NP-complete, then there is a positive integer $N$ such that CSP$({\mathbb H})$ remains NP-hard even for $\vec{\mathbb P}_N$-subgraph-free digraphs. Moreover, it remains NP-hard for acyclic $\vec{\mathbb P}_N$-subgraph-free digraphs, and becomes polynomial-time solvable for $\vec{\mathbb P}_{N-1}$-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between $\mathbb F$-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP$({\mathbb H})$ restricted to yes-instances of CSP$({\mathbb H}')$ -- these were called restricted homomorphism problems by Hell and Ne\v{s}et\v{r}il. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy.

cs.CC↗
arXiv2025

Complexity Framework For Forbidden Subgraphs V: Beyond Simple Graphs

We continue the study of the recently-introduced C123-framework, for (simple) graph problems restricted to inputs specified by the forbidding of some finite set of subgraphs, to more general graph problems possibly involving multiedges and self-loops. We study specifically the problems Multigraph Matching Cut, Multigraph d-Cut and Partially Reflexive Stable Cut in this connection. The last may be seen as a Surjective Homomorphism problem to a path P_3 in which both leaves are looped while the interior vertex is loopless. We consider also another family of Surjective Homomorphism problems to a cycle in which only one vertex is loopless. When one forbids a single (simple) subgraph, our first three problems exhibit the same complexity behaviour as C123-problems, but on finite sets of forbidden subgraphs, the classification appears more complex. While Multigraph Matching Cut and Multigraph d-Cut have the same classification as C123-problems, already Partially Reflexive Stable Cut fails to have. This is witnessed by forbidding as subgraphs both C_3 and H_1. Indeed, the difference of behaviour occurs only around pendant subdivisions of nets and pendant subdivisions of H_1. We examine this area in close detail. Our other Surjective Homomorphism problem, ostensibly somewhat similar to Partially Reflexive Stable Cut, behaves very differently when the input is restricted to some class that is H-subgraph-free. For example, it is solvable in polynomial time on any class of bounded degree. Also, its hardness will never be preserved under any form of edge subdivision.

math.CO↗
arXiv2024

Model-checking positive equality free logic on a fixed structure (direttissima)

We give a new, direct proof of the tetrachotomy classification for the model-checking problem of positive equality-free logic parameterised by the model. The four complexity classes are Logspace, NP-complete, co-NP-complete and Pspace-complete. The previous proof of this result relied on notions from universal algebra and core-like structures called U-X-cores. This new proof uses only relations, and works for infinite structures also in the distinction between Logspace and NP-hard under Turing reductions. For finite domains, the membership in NP and co-NP follows from a simple argument, which breaks down already over an infinite set with a binary relation. We develop some interesting new algorithms to solve NP and co-NP membership for a variety of infinite structures. We begin with those first-order definable in (Q;=), the so-called equality languages, then move to those first-order definable in (Q;<), the so-called temporal languages. However, it is first-order expansions of the Random Graph (V,E) that provide the most interesting examples. In all of these cases, the derived classification is a tetrachotomy between Logspace, NP-complete, co-NP-complete and Pspace-complete.

cs.LO↗
arXiv2024

Graph Homomorphism, Monotone Classes and Bounded Pathwidth

In recent work by Johnson et al. (2022), a framework was described for the study of graph problems over classes specified by omitting each of a finite set of graphs as subgraphs. If a problem falls into the framework then its computational complexity can be described for all such graph classes, giving a dichotomy between those classes for which the problem is hard and those for which it is easy. In this article, we consider several variants of the homomorphism problem in relation to this framework. It is known that certain homomorphism problems, e.g. $C_5$-Colouring, do not sit in the framework. By contrast, we show that the more general problem of Graph Homomorphism does sit in the framework, with hard cases NP-complete and easy cases in P. We go on to consider several locally constrained variants of the homomorphism problem, namely the locally bijective, surjective and injective variants. Like $C_5$-Colouring, none of these is in the framework. However, where a bounded-degree restrictions are considered, we prove that each of these problems is in our framework, with hard cases NP-complete and easy cases in P Next, we give the first example of a problem in the framework such that hardness is in the polynomial hierarchy above NP. This comes from a list colouring game, realised through first-order logic as quantified constraints. We show that with the additional restriction of bounded alternation, the problem is contained in the framework. The hard cases are $\Pi_{2k}^\mathrm{P}$-complete and the easy cases are in P. Finally, we go on to consider an aforementioned problem from our framework, complete for the second level of the polynomial hierarchy, under the omission in the input of not just a graph, but rather a graph $H$ annotated with the types for each vertex: existential or universal.

cs.CC↗
arXiv2023

Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem

We study Steiner Forest on $H$-subgraph-free graphs, that is, graphs that do not contain some fixed graph $H$ as a (not necessarily induced) subgraph. We are motivated by a recent framework that completely characterizes the complexity of many problems on $H$-subgraph-free graphs. However, in contrast to e.g. the related Steiner Tree problem, Steiner Forest falls outside this framework. Hence, the complexity of Steiner Forest on $H$-subgraph-free graphs remained tantalizingly open. In this paper, we make significant progress towards determining the complexity of Steiner Forest on $H$-subgraph-free graphs. Our main results are four novel polynomial-time algorithms for different excluded graphs $H$ that are central to further understand its complexity. Along the way, we study the complexity of Steiner Forest for graphs with a small $c$-deletion set, that is, a small set $S$ of vertices such that each component of $G-S$ has size at most $c$. Using this parameter, we give two noteworthy algorithms that we later employ as subroutines. First, we prove Steiner Forest is FPT parameterized by $|S|$ when $c=1$ (i.e. the vertex cover number). Second, we prove Steiner Forest is polynomial-time solvable for graphs with a 2-deletion set of size at most 2. The latter result is tight, as the problem is NP-complete for graphs with a 3-deletion set of size 2.

math.CO↗
arXiv2023

Complexity Framework for Forbidden Subgraphs III: When Problems are Tractable on Subcubic Graphs

For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity is determined on classes of $\mathcal{H}$-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most~$3$ and examine their complexity on $H$-subgraph-free graph classes where $H$ is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree $3$. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

cs.DS↗
arXiv2022

Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-graphs

For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal H}$-subgraph-free graphs (for finite sets ${\cal H}$) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity in ${\cal H}$-subgraph-free graphs is unknown. We study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: Hamilton Cycle, $k$-Induced Disjoint Paths, $C_5$-Colouring and Star $3$-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and also from problems that do satisfy all three conditions of the framework, in particular when we forbid certain subdivisions of the ``H''-graph (the graph that looks like the letter ``H''). Hence, we exhibit a rich complexity landscape among problems for ${\cal H}$-subgraph-free graph classes.

cs.DM↗
arXiv2022

Complexity Framework For Forbidden Subgraphs I: The Framework

For any particular class of graphs, algorithms for computational problems restricted to the class often rely on structural properties that depend on the specific problem at hand. This begs the question if a large set of such results can be explained by some common problem conditions. We propose such conditions for $HH$-subgraph-free graphs. For a set of graphs $HH$, a graph $G$ is $HH$-subgraph-free if $G$ does not contain any of graph from $H$ as a subgraph. Our conditions are easy to state. A graph problem must be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness must be preserved under edge subdivision of subcubic graphs. Our meta-classification says that if a graph problem satisfies all three conditions, then for every finite set $HH$, it is ``efficiently solvable'' on $HH$-subgraph-free graphs if $HH$ contains a disjoint union of one or more paths and subdivided claws, and is ``computationally hard'' otherwise. We illustrate the broad applicability of our meta-classification by obtaining a dichotomy between polynomial-time solvability and NP-completeness for many well-known partitioning, covering and packing problems, network design problems and width parameter problems. For other problems, we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). The proposed framework thus gives a simple pathway to determine the complexity of graph problems on $HH$-subgraph-free graphs. This is confirmed even more by the fact that along the way, we uncover and resolve several open questions from the literature.

math.CO↗
arXiv2022

Complexity Classification Transfer for CSPs via Algebraic Products

We study the complexity of infinite-domain constraint satisfaction problems: our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure $\mathfrak A$ can be transferred to a classification of the CSPs of first-order expansions of another structure $\mathfrak B$. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of the $n$-fold algebraic power of $(\mathbb{Q};<)$. This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of $(\mathbb{Q};<)$ and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen's Interval Algebra, the $n$-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyse with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the AI literature.

math.LO↗
arXiv2022

Few Induced Disjoint Paths for $H$-Free Graphs

Paths $P^1,\ldots,P^k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P^i$ and $P^j$ have neither common vertices nor adjacent vertices. For a fixed integer $k$, the $k$-Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P^i$ such that each $P^i$ starts from $s_i$ and ends at $t_i$. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer $k$, a classical result from the literature states that even $2$-Induced Disjoint Paths is NP-complete. We prove new complexity results for $k$-Induced Disjoint Paths if the input is restricted to $H$-free graphs, that is, graphs without a fixed graph $H$ as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where $k$ is part of the input.

math.CO↗
arXiv2022

Induced Disjoint Paths and Connected Subgraphs for $H$-Free Graphs

Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.

math.CO↗
arXiv2021

Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter

For a fixed integer, the $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for an integer $k$, such that no two adjacent vertices are coloured alike. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. It is known that for all $k\geq 3$, the $k$-Colouring problem is NP-complete for $H$-free graphs if $H$ contains an induced claw or cycle. The case where $H$ contains a cycle follows from the known result that the problem is NP-complete even for graphs of arbitrarily large fixed girth. We examine to what extent the situation may change if in addition the input graph has bounded diameter.

math.CO↗
arXiv2021

The complexity of quantified constraints: collapsibility, switchability and the algebraic formulation

Let A be an idempotent algebra on a finite domain. By mediating between results of Chen and Zhuk, we argue that if A satisfies the polynomially generated powers property (PGP) and B is a constraint language invariant under A (that is, in Inv(A)), then QCSP(B) is in NP. In doing this we study the special forms of PGP, switchability and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now the original Chen Conjecture is known to be false. Switchability was introduced by Chen as a generalisation of the already-known collapsibility. For three-element domain algebras A that are switchable and omit a G-set, we prove that, for every finite subset D of Inv(A), Pol(D) is collapsible. The significance of this is that, for QCSP on finite structures (over a three-element domain), all QCSP tractability (in P) explained by switchability is already explained by collapsibility.

cs.CC↗