Abstract In traditional machine teaching, a teacher needs to teach a concept to a learner by means of a finite set of examples, the witness set. But concepts can have many equivalent representations. This redundancy strongly affects the search space, to the extent that teacher and learner may not be able to easily determine the equivalence class of each representation. In this common situation, instead of teaching concepts, we explore the idea of teaching representations. We work with several teaching schemas that exploit representation and witness size (Eager, Greedy and Optimal) and analyze the gains in teaching effectiveness, both theoretically, and also experimentally for languages where redundancy can vary (DNF expressions and Turing-complete P3 programs). Our theoretical and experimental results indicate that there are various types of redundancy, related, e.g,. to the spread of the redundant representations, handled better by the new Greedy schema introduced here than by the Eager schema. For P3 programs witness sets found by Greedy are usually smaller than the programs they identify, corroborating previous results that conveying information efficiently is a leitmotif of machine teaching.

Large language models have become multimodal, and many of them are said to integrate their modalities using common representations. If this were true, a drawing of a car as an image, for instance, should map to a similar area in the latent space as a textual description of the strokes that form the drawing. To explore this in a black-box access regime to these models, we propose the use of machine teaching, a theory that studies the minimal set of examples a teacher needs to choose so that the learner captures the concept. In this paper, we evaluate the complexity of teaching vision-language models a subset of objects in the Quick, Draw! dataset using two presentations: raw images as bitmaps and trace coordinates in TikZ format. The results indicate that image-based representations generally require fewer segments and achieve higher accuracy than coordinate-based representations. But, surprisingly, the teaching size usually ranks concepts similarly across both modalities, even when controlling for (a human proxy of) concept priors, suggesting that the simplicity of concepts may be an inherent property that transcends modality representations.

We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called teaching dimension. A recent paper (Ferri et al., 2024) conjectured that the worst case for this model, as a function of the size of the concept class, occurs when the consistency matrix contains the binary representations of numbers from zero and up. In this paper we prove their conjecture. The result can be seen as a generalization of a theorem resolving the edge isoperimetry problem for hypercubes (Hart, 1976), and our proof is based on a lemma of (Graham, 1970).

Imagine a learner $L$ who tries to infer a hidden concept from a collection of observations. Building on the work of Ferri et al we assume the learner to be parameterized by priors $P(c)$ and by $c$-conditional likelihoods $P(z|c)$ where $c$ ranges over all concepts in a given class $C$ and $z$ ranges over all observations in an observation set $Z$. $L$ is called a MAP-learner (resp.~an MLE-learner) if it thinks of a collection $S$ of observations as a random sample and returns the concept with the maximum a-posteriori probability (resp.~the concept which maximizes the $c$-conditional likelihood of $S$). Depending on whether $L$ assumes that $S$ is obtained from ordered or unordered sampling resp.~from sampling with or without replacement, we can distinguish four different sampling modes. Given a target concept $c^* \in C$, a teacher for a MAP-learner $L$ aims at finding a smallest collection of observations that causes $L$ to return $c^*$. This approach leads in a natural manner to various notions of a MAP- or MLE-teaching dimension of a concept class $C$. Our main results are as follows. First, we show that this teaching model has some desirable monotonicity properties. Second we clarify how the four sampling modes are related to each other. As for the (important!) special case, where concepts are subsets of a domain and observations are 0,1-labeled examples, we obtain some additional results. First of all, we characterize the MAP- and MLE-teaching dimension associated with an optimally parameterized MAP-learner graph-theoretically. From this central result, some other ones are easy to derive. It is shown, for instance, that the MLE-teaching dimension is either equal to the MAP-teaching dimension or exceeds the latter by $1$. It is shown furthermore that these dimensions can be bounded from above by the so-called antichain number, the VC-dimension and related combinatorial parameters. Moreover they can be computed in polynomial time.

Abstract In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in polynomial time.

Over the past decades in the field of machine teaching, several restrictions have been introduced to avoid ‘cheating’, such as collusion-free or non-clashing teaching. However, these restrictions forbid several teaching situations that we intuitively consider natural and fair, especially those ‘changes of mind’ of the learner as more evidence is given, affecting the likelihood of concepts and ultimately their posteriors. Under a new generalised probabilistic teaching, not only do these non-cheating constraints look too narrow but we also show that the most relevant machine teaching models are particular cases of this framework: the consistency graph between concepts and elements simply becomes a joint probability distribution. We show a simple procedure that builds the witness joint distribution from the ground joint distribution. We prove a chain of relations, also with a theoretical lower bound, on the teaching dimension of the old and new models. Overall, this new setting is more general than the traditional machine teaching models, yet at the same time more intuitively capturing a less abrupt notion of non-cheating teaching.

An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known undirected intersection graphs like interval graphs and permutation graphs, not many algorithmic applications on intersection digraphs have been developed. Motivated by the successful story on algorithmic applications of intersection graphs using a graph width parameter called mim-width, we introduce its directed analogue called `bi-mim-width' and prove that various classes of reflexive intersection digraphs have bounded bi-mim-width. In particular, we show that as a natural extension of $H$-graphs, reflexive $H$-digraphs have linear bi-mim-width at most $12|E(H)|$, which extends a bound on the linear mim-width of $H$-graphs [On the Tractability of Optimization Problems on $H$-Graphs. Algorithmica 2020]. For applications, we introduce a novel framework of directed versions of locally checkable problems, that streamlines the definitions and the study of many problems in the literature and facilitates their common algorithmic treatment. We obtain unified polynomial-time algorithms for these problems on digraphs of bounded bi-mim-width, when a branch decomposition is given. Locally checkable problems include Kernel, Dominating Set, and Directed $H$-Homomorphism.

Abstract The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $$2^{O(rw^3)}\cdot n^{4}$$ 2 O ( r w 3 ) · n 4 , $$2^{O(q^2\log (q))}\cdot n^{4}$$ 2 O ( q 2 log ( q ) ) · n 4 , and $$n^{O(k^2)}$$ n O ( k 2 ) where rw is the rank-width, q the $${\mathbb {Q}}$$ Q -rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

We investigate the teaching of infinite concept classes through the effect of the learning prior (which is used by the learner to derive posteriors giving preference of some concepts over others and by the teacher to devise the teaching examples) and the sampling prior (which determines how the concepts are sampled from the class). We analyse two important classes: Turing machines and finite-state machines. We derive bounds for the teaching dimension when the learning prior is derived from a complexity measure (Kolmogorov complexity and minimal number of states respectively) and analyse the sampling distributions that lead to finite expected teaching dimensions. The learning prior goes beyond a complexity or preference choice when we use it to increase the confidence of identification, expressed as a posterior, which increases as more examples are given. We highlight the existing trade-off between three elements: the bound on teaching dimension, the representativeness of the sample and the certainty of the identification. This has implications for the understanding of what teaching from rich concept classes to machines (and humans) entails.

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph $G$ and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs $H$ having the property that for any $k$ the graph $H(k)$ being the join of $k$ copies of $H$ has an optimal hierarchical clustering that splits each copy of $H$ in the same optimal way. To optimally cluster such a graph $H(k)$ we thus only need to optimally cluster the smaller graph $H$. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

We initialize a series of papers deepening the understanding of algorithmic properties of the width parameter maximum induced matching width (mim-width) of graphs. In this first volume we provide the first polynomial-time algorithms on graphs of bounded mim-width for problems that are not locally checkable. In particular, we givenO(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H -Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, k -Trapezoid, Circular k -Trapezoid, k -Polygon, Dilworth-k and Co- k -Degenerate graphs for fixed k. We contrast these positive results to the fact that problems about finding long non-induced paths remain hard on graphs of bounded mim-width: We show that Hamiltonian Cycle (and hence Hamiltonian Path) is NP-hard on graphs of linear mim-width 1; this further hints at the expressive power of the mim-width parameter.

We generalize the family of (σ,ρ) problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as Distance-r Dominating Set and Distance-r Independent Set. We show that these distance problems are in XP parameterized by the structural parameter mim-width, and hence polynomial-time solvable on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. We obtain these results by showing that taking any power of a graph never increases its mim-width by more than a factor of two. To supplement these findings, we show that many classes of (σ,ρ) problems are W[1]-hard parameterized by mim-width + solution size. We show that powers of graphs of tree-width w−1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1.

We generalize the family of $(σ, ρ)$-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-$r$ dominating set and distance-$r$ independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as $k$-trapezoid graphs, Dilworth $k$-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, $k$-polygon graphs, circular arc graphs, complements of $d$-degenerate graphs, and $H$-graphs if given an $H$-representation. To supplement these findings, we show that many classes of (distance) $(σ, ρ)$-problems are W[1]-hard parameterized by mim-width + solution size.

We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an n^{O(w)}-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs for fixed k. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013]. We show that powers of graphs of tree-width w-1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results extensively provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1. Given a tree decomposition of width w-1, a path decomposition of width w, or a clique-width w-expression of a graph G, one can for any value of k find a mim-width decomposition of its k-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the k-power in time n^{O(w)}. In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width 1, which further hints at the expressive power of the mim-width parameter.

We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give $n^{\mathcal{O}(w)}$-time algorithms on graphs of mim-width at most $w$, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and $H$-Induced Topological Minor for fixed $H$. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, $k$-Trapezoid, Circular $k$-Trapezoid, $k$-Polygon, Dilworth-$k$ and Co-$k$-Degenerate graphs for fixed $k$.

It was recently shown [Sæther, Telle, and Vatshelle, JAIR 54, 2015] that satisfiability is polynomially solvable when the incidence graph is an interval bipartite graph (an interval graph turned into a bipartite graph by omitting all edges within each partite set). Here we relax this condition in several directions: First, we show an FPT algorithm parameterized by k for k-interval bigraphs, bipartite graphs which can be converted to interval bipartite graphs by adding to each node of one side at most k edges; the same result holds for the counting and the weighted maximization version of satisfiability. Second, given two linear orders, one for the variables and one for the clauses, we show how to find, in polynomial time, the smallest k such that there is a k-interval bigraph compatible with these two orders. On the negative side we prove that, barring complexity collapses, no such extensions are possible for CSPs more general than satisfiability. We also show NP-hardness of recognizing 1-interval bigraphs.

We give alternative definitions for maximum matching width, e.g., a graph G has mmw(G) <= k if and only if it is a subgraph of a chordal graph H and for every maximal clique X of H there exists A,B,C \subseteq X with A \cup B \cup C=X and |A|,|B|,|C| <= k such that any subset of X that is a minimal separator of H is a subset of either A, B or C. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing on nodes and on edges. Based on this we prove that given a graph G and a branch decomposition of mm-width k we can solve Dominating Set in time O^*(8^k), thereby beating O^*(3^{tw(G)}) whenever tw(G) > log_3(8) * k ~ 1.893 k. Note that mmw(G) <= tw(G)+1 <= 3 mmw(G) and these inequalities are tight. Given only the graph G and using the best known algorithms to find decompositions, maximum matching width will be better for solving Dominating Set whenever tw(G) > 1.549 * mmw(G).

We look at dynamic programming algorithms for propositional model counting, also called #SAT, and MaxSAT. Tools from graph structure theory, in particular treewidth, have been used to successfully identify tractable cases in many subfields of AI, including SAT, Constraint Satisfaction Problems (CSP), Bayesian reasoning, and planning. In this paper we attack #SAT and MaxSAT using similar, but more modern, graph structure tools. The tractable cases will include formulas whose class of incidence graphs have not only unbounded treewidth but also unbounded clique-width. We show that our algorithms extend all previous results for MaxSAT and #SAT achieved by dynamic programming along structural decompositions of the incidence graph of the input formula. We present some limited experimental results, comparing implementations of our algorithms to state-of-the-art #SAT and MaxSAT solvers, as a proof of concept that warrants further research.

We present an algorithm with runtime $O(k^{2k}n^3m)$ for the following NP-complete problem [M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, 1979, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the long-standing open question [H. Kaplan, R. Shamir, and R. E. Tarjan, SIAM J. Comput., 28 (1999), pp. 1906–1922; R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999; M. Serna and D. Thilikos, Bull. Eur. Assoc. Theory Comput. Sci. EATCS, 86 (2005), pp. 41–65; G. Gutin, S. Szeider, and A. Yeo, in Proceedings IWPEC 2006, Lecture Notes in Comput. Sci. 4169, Springer-Verlag, Berlin, 2006, pp. 60–71], first posed by Kaplan, Shamir, and Tarjan, of whether this problem was fixed parameter tractable. The problem has applications in profile minimization for sparse matrix computations [J. A. George and J. W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981; R. E. Tarjan, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, 1976, pp. 3–22], and our results show tractability for the case of a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search.

A survey of the most important and general techniques in parameterized algorithm design is given. Each technique is explained with a meta-algorithm, its use is illustrated by examples, and it is placed in a taxonomy under the four main headings of branching, kernelization, induction and win/win.

We present an algorithm with runtime O(k(2k)n3 * m) for the following NP-complete problem: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the long-standing open question, first posed by Kaplan, Shamir and Tarjan, of whether this problem could be solved in time f(k) * n(O(1)).The problem has applications in Physical Mapping of DNA and in Profile Minimization for Sparse Matrix Computations. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope.

The problem of computing minimal triangulations of graphs, also called minimal fill, was introduced and solved in 1976 by Rose, Tarjan, and Lueker [SIAM J. Comput., 5 (1976), pp. 266-283] in time $O(nm)$ and thus $O(n^3)$ for dense graphs. Although the topic has received increasing attention since then and several new results on characterizing and computing minimal triangulations have been presented, this first time bound has remained the best. In this paper we introduce an $O(n^\alpha \log n)$ time algorithm for computing minimal triangulations, where $O(n^\alpha)$ is the time required to multiply two $n \times n$ matrices. The current best known $\alpha$ is less than $2.376$, and thus our result breaks the longstanding asymptotic time complexity bound for this problem. To achieve this result, we introduce and combine several techniques that are new to minimal triangulation algorithms, such as working on the complement of the input graph, graph search for a vertex set A that bounds the size of the connected components when A is removed, and matrix multiplication.

For an arbitrary filled graph G+ of a given original graph G, we consider the problem of removing fill edges from G+ in order to obtain a graph M that is both a minimal filled graph of G and a subgraph of G+. For G+ with f fill edges and e original edges, we give a simple O(f(e+f)) algorithm which solves the problem and computes a corresponding minimal elimination ordering of G. We report on experiments with an implementation of our algorithm, where we test graphs G corresponding to some real sparse matrix applications and apply well-known and widely used ordering heuristics to find G+. Our findings show the amount of fill that is commonly removed by a minimalization for each of these heuristics, and also indicate that the runtime of our algorithm on these practical graphs is better than the presented worst-case bound.

In this paper, we consider a large class of vertex partitioning problems and apply to them the theory of algorithm design for problems restricted to partial k-trees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications. We give a precise characterization of vertex partitioning problems, which include domination, coloring and packing problems, and their variants. Several new graph parameters are introduced as generalizations of classical parameters. This characterization provides a basis for a taxonomy of a large class of problems, facilitating their common algorithmic treatment and allowing their uniform complexity classification. We present a design methodology of practical solution algorithms for generally $\NP$-hard problems when restricted to partial k-trees (graphs with treewidth bounded by k). This "practicality" accounts for dependency on the parameter k of the computational complexity of the resulting algorithms. By adapting the algorithm design methodology on partial k-trees to vertex partitioning problems, we obtain the first algorithms for these problems with reasonable time complexity as a function of treewidth. As an application of the methodology, we give the first polynomial-time algorithm on partial k-trees for computation of the Grundy number.

We address the problem of mapping divide-and-conquer programs to mesh connected multicomputers with wormhole or store-and-forward routing. We propose the binomial tree as an efficient model of parallel divide-and-conquer and present two mappings of the binomial tree to the 2D mesh. Our mappings exploit regularity in the communication structure of the divide-and-conquer computation and are also sensitive to the underlying flow control scheme of the target architecture. We evaluate these mappings using new metrics which are extensions of the classical notions of dilation and contention. We introduce the notion of communication slowdown as a measure of the total communication overhead incurred by a parallel computation. We conclude that significant performance gains can be realized when the mapping is sensitive to the flow control scheme of the target architecture.