arXiv subjects

Fionn Mc Inerney

Publications and source records attributed to Fionn Mc Inerney.

At least 19 recordsLinked to original sources

arXiv2026

Non-Clashing Teaching in Graphs: Algorithms, Complexity, and Bounds

Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and proved that it is the most efficient batch machine teaching model satisfying the collusion-avoidance benchmark established in the seminal work of Goldman and Mathias [COLT 1993]. Recently, (positive) non-clashing teaching was thoroughly studied for balls in graphs, yielding numerous algorithmic and combinatorial results. In particular, Chalopin et al. [COLT 2024] and Ganian et al. [ICLR 2025] gave an almost complete picture of the complexity landscape of the positive variant, showing that it is tractable only for restricted graph classes due to the non-trivial nature of the problem and concept class. In this work, we consider (positive) non-clashing teaching for closed neighborhoods in graphs. This concept class is not only extensively studied in various related contexts, but it also exhibits broad generality, as any finite binary concept class can be equivalently represented by a set of closed neighborhoods in a graph. In comparison to the works on balls in graphs, we provide improved algorithmic results, notably including FPT algorithms for more general classes of parameters, and we complement these results by deriving stronger lower bounds. Lastly, we obtain combinatorial upper bounds for wider classes of graphs.

cs.CC↗

arXiv2026

Makespan Minimization in Split Learning: From Theory to Practice

Split learning recently emerged as a solution for distributed machine learning with heterogeneous IoT devices, where clients can offload part of their training to computationally-powerful helpers. The core challenge in split learning is to minimize the training time by jointly devising the client-helper assignment and the schedule of tasks at the helpers. We first study the model where each helper has a memory cardinality constraint on how many clients it may be assigned, which represents the case of homogeneous tasks. Through complexity theory, we rule out exact polynomial-time algorithms and approximation schemes even for highly restricted instances of this problem. We complement these negative results with a non-trivial polynomial-time 5-approximation algorithm. Building on this, we then focus on the more general heterogeneous task setting considered by Tirana et al. [INFOCOM 2024], where helpers have memory capacity constraints and clients have variable memory costs. In this case, we prove that, unless P=NP, the problem cannot admit a polynomial-time approximation algorithm for any approximation factor. However, by adapting our aforementioned 5-approximation algorithm, we develop a novel heuristic for the heterogeneous task setting and show that it outperforms heuristics from prior works through extensive experiments.

cs.NI↗

arXiv2025

The Parameterized Complexity of Computing the VC-Dimension

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, we prove that the naive $2^{\mathcal{O}(|\mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a $1$-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $\mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We design a $2^{\mathcal{O}(\rm{tw}\cdot \log \rm{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\rm{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).

cs.CC↗

arXiv2025

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part I: Algorithmic Results

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $σ,ρ$ of non-negative integers, a $(σ,ρ)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in σ$ for every $u\in S$, and $|N(v)\cap S|\in ρ$ for every $v\not\in S$. The problem of finding a $(σ,ρ)$-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets $(σ,ρ)$, we determine (under standard complexity assumptions) the best possible value $c_{σ,ρ}$ such that there is an algorithm that counts $(σ,ρ)$-sets in time $c_{σ,ρ}^{\sf tw}\cdot n^{O(1)}$ (if a tree decomposition of width ${\sf tw}$ is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to $σ=\{0\}$ and $ρ=\{1\}$, we improve the $3^{\sf tw}\cdot n^{O(1)}$ algorithm of [van Rooij, 2020] to $2^{\sf tw}\cdot n^{O(1)}$. Despite the unusually delicate definition of $c_{σ,ρ}$, an accompanying paper shows that our algorithms are most likely optimal, that is, for any pair $(σ, ρ)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{σ,ρ}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-algorithm counting the number of $(σ,ρ)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $σ$ and $ρ$, these lower bounds also extend to the decision version, and hence, our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets.

cs.CC↗

arXiv2025

Problems in NP can Admit Double-Exponential Lower Bounds when Parameterized by Treewidth or Vertex Cover

Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $Σ_2^p$, $Π_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds.

cs.CC↗

arXiv2025

Metric Dimension and Geodetic Set Parameterized by Vertex Cover

For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on $n$ vertices and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. In another metric-based graph problem, Geodetic Set, the input is a graph $G$ and an integer $k$, and the objective is to determine whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that, for any vertex $u \in V(G)$, there are two vertices $s_1, s_2 \in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{\mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}\cdot n^{\mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

cs.DS↗

arXiv2025

The Computational Complexity of Positive Non-Clashing Teaching in Graphs

We study the classical and parameterized complexity of computing the positive non-clashing teaching dimension of a set of concepts, that is, the smallest number of examples per concept required to successfully teach an intelligent learner under the considered, previously established model. For any class of concepts, it is known that this problem can be effortlessly transferred to the setting of balls in a graph G. We establish (1) the NP-hardness of the problem even when restricted to instances with positive non-clashing teaching dimension k=2 and where all balls in the graph are present, (2) near-tight running time upper and lower bounds for the problem on general graphs, (3) fixed-parameter tractability when parameterized by the vertex integrity of G, and (4) a lower bound excluding fixed-parameter tractability when parameterized by the feedback vertex number and pathwidth of G, even when combined with k. Our results provide a nearly complete understanding of the complexity landscape of computing the positive non-clashing teaching dimension and answer open questions from the literature.

cs.CC↗

arXiv2025

A Structural Complexity Analysis of Hierarchical Task Network Planning

We perform a refined complexity-theoretic analysis of three classical problems in the context of Hierarchical Task Network Planning: the verification of a provided plan, whether an executable plan exists, and whether a given state can be reached. Our focus lies on identifying structural properties which yield tractability. We obtain new polynomial algorithms for all three problems on a natural class of primitive networks, along with corresponding lower bounds. We also obtain an algorithmic meta-theorem for lifting polynomial-time solvability from primitive to general task networks, and prove that its preconditions are tight. Finally, we analyze the parameterized complexity of the three problems.

cs.CC↗

arXiv2024

Parameterized Complexity of Caching in Networks

The fundamental caching problem in networks asks to find an allocation of contents to a network of caches with the aim of maximizing the cache hit rate. Despite the problem's importance to a variety of research areas -- including not only content delivery, but also edge intelligence and inference -- and the extensive body of work on empirical aspects of caching, very little is known about the exact boundaries of tractability for the problem beyond its general NP-hardness. We close this gap by performing a comprehensive complexity-theoretic analysis of the problem through the lens of the parameterized complexity paradigm, which is designed to provide more precise statements regarding algorithmic tractability than classical complexity. Our results include algorithmic lower and upper bounds which together establish the conditions under which the caching problem becomes tractable.

cs.NI↗

arXiv2024

Complexity of Maker-Breaker Games on Edge Sets of Graphs

We study the algorithmic complexity of Maker-Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the $H$-game. Maker wins if she claims the edges of a perfect matching in the first, and a copy of a fixed graph $H$ in the second. We prove that deciding who wins the perfect matching game and the $H$-game is PSPACE-complete, even for the latter in small-diameter graphs if $H$ is a tree. Toward finding the smallest graph $H$ for which the $H$-game is PSPACE-complete, we also prove that such an $H$ of order 51 and size 57 exists. We then give several positive results for the $H$-game. As the $H$-game is already PSPACE-complete when $H$ is a tree, we mainly consider the case where $H$ belongs to a subclass of trees. In particular, we design two linear-time algorithms, both based on structural characterizations, to decide the winners of the $P_4$-game in general graphs and the $K_{1,\ell}$-game in trees. Then, we prove that the $K_{1,\ell}$-game in any graph, and the $H$-game in trees are both FPT parameterized by the length of the game, notably adding to the short list of games with this property, which is of independent interest. Another natural direction to take is to consider the $H$-game when $H$ is a cycle. While we were unable to resolve this case, we prove that the related arboricity-$k$ game is polynomial-time solvable. In particular, when $k=2$, Maker wins this game if she claims the edges of any cycle.

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arXiv2024

The Complexity of Optimizing Atomic Congestion

Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory, and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies -- that is, a centrally planned routing that minimizes the average cost of agents -- is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem.

cs.GT↗

arXiv2024

Non-Clashing Teaching Maps for Balls in Graphs

Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and showed it is the most efficient machine teaching model satisfying the Goldman-Mathias collusion-avoidance criterion. A teaching map $T$ for a concept class $\mathcal{C}$ assigns a (teaching) set $T(C)$ of examples to each concept $C \in \mathcal{C}$. A teaching map is non-clashing if no pair of concepts are consistent with the union of their teaching sets. The size of a non-clashing teaching map (NCTM) $T$ is the maximum size of a teaching set $T(C)$, $C \in \mathcal{C}$. The non-clashing teaching dimension NCTD$(\mathcal{C})$ of $\mathcal{C}$ is the minimum size of an NCTM for $\mathcal{C}$. NCTM$^+$ and NCTD$^+(\mathcal{C})$ are defined analogously, except the teacher may only use positive examples. We study NCTMs and NCTM$^+$s for the concept class $\mathcal{B}(G)$ consisting of all balls of a graph $G$. We show that the associated decision problem B-NCTD$^+$ for NCTD$^+$ is NP-complete in split, co-bipartite, and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails, B-NCTD$^+$ does not admit an algorithm running in time $2^{2^{o(\text{vc})}}\cdot n^{O(1)}$, nor a kernelization algorithm outputting a kernel with $2^{o(\text{vc})}$ vertices, where vc is the vertex cover number of $G$. We complement these lower bounds with matching upper bounds. These are extremely rare results: it is only the second problem in NP to admit such a tight double-exponential lower bound parameterized by vc, and only one of very few problems to admit such an ETH-based conditional lower bound on the number of vertices in a kernel. For trees, interval graphs, cycles, and trees of cycles, we derive NCTM$^+$s or NCTMs for $\mathcal{B}(G)$ of size proportional to its VC-dimension, and for Gromov-hyperbolic graphs, we design an approximate NCTM$^+$ of size 2.

cs.CC↗

arXiv2024

Enumerating minimal solution sets for metric graph problems

Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to enumerating minimal transversals in hypergraphs (denoted Trans-Enum), whose solvability in total-polynomial time is one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in recent works: for which vertex (or edge) set graph property $Π$ is the enumeration of minimal (or maximal) subsets satisfying $Π$ equivalent to Trans-Enum? As very few properties are known to fit within this context -- namely, those related to minimal domination -- our results make significant progress in characterizing such properties, and provide new angles to approach Trans-Enum. In contrast, we observe that minimal strong resolving sets can be enumerated with polynomial delay. Additionally, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show both positive and negative results related to the enumeration and extension of partial solutions.

cs.DM↗

arXiv2024

Hypergraph dualization with FPT-delay parameterized by the degeneracy and dimension

At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an $n^{O(d)}$-delay algorithm listing all minimal transversals of an $n$-vertex hypergraph of degeneracy $d$. Recently at IWOCA 2019, Conte, Kanté, Marino, and Uno asked whether this XP-delay algorithm parameterized by $d$ could be made FPT-delay for a weaker notion of degeneracy, or even parameterized by the maximum degree $Δ$, i.e., whether it can be turned into an algorithm with delay $f(Δ)\cdot n^{O(1)}$ for some computable function $f$. Moreover, and as a first step toward answering that question, they note that they could not achieve these time bounds even for the particular case of minimal dominating sets enumeration. In this paper, using ordered generation, we show that an FPT-delay algorithm can be devised for minimal transversals enumeration parameterized by the degeneracy and dimension, giving a positive and more general answer to the latter question.

cs.DS↗

arXiv2024

Sample compression schemes for balls in graphs

One of the open problems in machine learning is whether any set-family of VC-dimension $d$ admits a sample compression scheme of size $O(d)$. In this paper, we study this problem for balls in graphs. For a ball $B=B_r(x)$ of a graph $G=(V,E)$, a realizable sample for $B$ is a signed subset $X=(X^+,X^-)$ of $V$ such that $B$ contains $X^+$ and is disjoint from $X^-$. A proper sample compression scheme of size $k$ consists of a compressor and a reconstructor. The compressor maps any realizable sample $X$ to a subsample $X'$ of size at most $k$. The reconstructor maps each such subsample $X'$ to a ball $B'$ of $G$ such that $B'$ includes $X^+$ and is disjoint from $X^-$. For balls of arbitrary radius $r$, we design proper labeled sample compression schemes of size $2$ for trees, of size $3$ for cycles, of size $4$ for interval graphs, of size $6$ for trees of cycles, and of size $22$ for cube-free median graphs. For balls of a given radius, we design proper labeled sample compression schemes of size $2$ for trees and of size $4$ for interval graphs. We also design approximate sample compression schemes of size 2 for balls of $δ$-hyperbolic graphs.

cs.DM↗

arXiv2024

The Maker-Breaker Largest Connected Subgraph Game

Given a graph $G$ and $k \in \mathbb{N}$, we introduce the following game played in $G$. Each round, Alice colours an uncoloured vertex of $G$ red, and then Bob colours one blue (if any remain). Once every vertex is coloured, Alice wins if there is a connected red component of order at least $k$, and otherwise, Bob wins. This is a Maker-Breaker version of the Largest Connected Subgraph game introduced in [Bensmail et al. The Largest Connected Subgraph Game. {\it Algorithmica}, 84(9):2533--2555, 2022]. We want to compute $c_g(G)$, which is the maximum $k$ such that Alice wins in $G$, regardless of Bob's strategy. Given a graph $G$ and $k\in \mathbb{N}$, we prove that deciding whether $c_g(G)\geq k$ is PSPACE-complete, even if $G$ is a bipartite, split, or planar graph. To better understand the Largest Connected Subgraph game, we then focus on {\it A-perfect} graphs, which are the graphs $G$ for which $c_g(G)=\lceil|V(G)|/2\rceil$, {\it i.e.}, those in which Alice can ensure that the red subgraph is connected. We give sufficient conditions, in terms of the minimum and maximum degrees or the number of edges, for a graph to be A-perfect. Also, we show that, for any $d \geq 4$, there are arbitrarily large A-perfect $d$-regular graphs, but no cubic graph with order at least $18$ is A-perfect. Lastly, we show that $c_g(G)$ is computable in linear time when $G$ is a $P_4$-sparse graph (a superclass of cographs).

math.CO↗

arXiv2023

Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters

For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set} if for any two vertices $u,v \in V(G)$, there exists a vertex $w \in S$ such that $d(w,u) \neq d(w,v)$. The {\sc Metric Dimension} problem takes as input a graph $G$ and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. This problem was introduced in the 1970s and is known to be \NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the problem is \W[2]-hard when parameterized by the natural parameter $k$. They also observed that it is \FPT\ when parameterized by the vertex cover number and asked about its complexity under \emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {\sc Metric Dimension} is \W[1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit~[IPEC 2019] which states that the problem is \W[1]-hard parameterized by the pathwidth. On the positive side, we show that {\sc Metric Dimension} is \FPT\ when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

cs.DM↗

arXiv2023

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part II: Hardness Results

For a well-studied family of domination-type problems, in bounded-treewidth graphs, we investigate whether it is possible to find faster algorithms. For sets $σ,ρ$ of non-negative integers, a $(σ,ρ)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in σ$ for every $u\in S$, and $|N(v)\cap S|\in ρ$ for every $v\not\in S$. The problem of finding a $(σ,ρ)$-set (of a certain size) unifies common problems like $\text{Independent Set}$, $\text{Dominating Set}$, $\text{Independent Dominating Set}$, and many others. In an accompanying paper, it is proven that, for all pairs of finite or cofinite sets $(σ,ρ)$, there is an algorithm that counts $(σ,ρ)$-sets in time $(c_{σ,ρ})^{\text{tw}}\cdot n^{O(1)}$ (if a tree decomposition of width $\text{tw}$ is given in the input). Here, $c_{σ,ρ}$ is a constant with an intricate dependency on $σ$ and $ρ$. Despite this intricacy, we show that the algorithms in the accompanying paper are most likely optimal, i.e., for any pair $(σ, ρ)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{σ,ρ}-\varepsilon)^{\text{tw}}\cdot n^{O(1)}$-algorithm counting the number of $(σ,ρ)$-sets would violate the Counting Strong Exponential-Time Hypothesis ($\#$SETH). For finite sets $σ$ and $ρ$, our lower bounds also extend to the decision version, showing that those algorithms are optimal in this setting as well.

cs.CC↗