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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 [7] 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 [12], and our proof is based on a lemma of [10]., Comment: 14 pages, 1 figure

In traditional machine teaching, a teacher wants 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 for some representational languages (DNF expressions and Turing-complete P3 programs). Our theoretical and experimental results indicate that there are various types of redundancy, handled better by the Greedy schema introduced here than by the Eager schema, although both can be arbitrarily far away from the Optimal. For P3 programs we found that witness sets are usually smaller than the programs they identify, which is an illuminating justification of why machine teaching from examples makes sense at all., Comment: 16 pages, 3 figures, 3 tables

Imagine a learner L who tries to infer a hidden concept from a collection of observations. Building on the work [4] 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: We show that this teaching model has some desirable monotonicity properties. 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.

In a graph, a perfect matching cut is an edge cut that is a perfect matching. Perfect Matching Cut (PMC) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and show that PMC remains NP-complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which PMC is polynomial time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chordal graphs. Assuming the Exponential Time Hypothesis, we show there is no $O^*(2^{o(n)})$-time algorithm for PMC even when restricted to $n$-vertex bipartite graphs, and also show that PMC can be solved in $O^*(1.2721^n)$ time by means of an exact branching algorithm.

A $k$-leaf power of a tree $T$ is a graph $G$ whose vertices are the leaves of $T$ and whose edges connect pairs of leaves whose distance in $T$ is at most $k$. A graph is a leaf power if it is a $k$-leaf power for some $k$. Over 20 years ago, Nishimura et al. [J. Algorithms, 2002] asked if recognition of leaf powers was in P. Recently, Lafond [SODA 2022] showed an XP algorithm when parameterized by $k$, while leaving the main question open. In this paper, we explore this question from the perspective of two alternative models of leaf powers, showing that both a linear and a star variant of leaf powers can be recognized in polynomial-time.

The minimum height of vertex and edge partition trees are well-studied graph parameters known as, for instance, vertex and edge ranking number. While they are NP-hard to determine in general, linear-time algorithms exist for trees. Motivated by a correspondence with Dasgupta's objective for hierarchical clustering we consider the total rather than maximum depth of vertices as an alternative objective for minimization. For vertex partition trees this leads to a new parameter with a natural interpretation as a measure of robustness against vertex removal. As tools for the study of this family of parameters we show that they have similar recursive expressions and prove a binary tree rotation lemma. The new parameter is related to trivially perfect graph completion and therefore intractable like the other three are known to be. We give polynomial-time algorithms for both total-depth variants on caterpillars and on trees with a bounded number of leaf neighbors. For general trees, we obtain a 2-approximation algorithm., Comment: Full version, 19 pages

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.

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., Comment: 19 pages, 7 figures. Extended version of conference paper, to appear in proceedings from MFCS 2020

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 $2^{O(rw^3)}\cdot n^{4}$, $2^{O(q^2\log(q))}\cdot n^{4}$, and $n^{O(k^2)}$ where $rw$ is the rank-width, $q$ the $\mathbb{Q}$-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, 2019) concerning an XP 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 provide an $O(n \log n)$ algorithm computing the linear maximum induced matching width of a tree and an optimal layout., Comment: 19 pages, 7 figures, full version of WG19 paper of same name

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 $\mathcal{O}(n^2)$ time., Comment: 30 pages, 10 figures; accepted at STACS 2020

We investigate the teaching of infinite concept classes through the effect of the learning bias (which is used by the learner to prefer some concepts over others and by the teacher to devise the teaching examples) and the sampling bias (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 biased teaching dimension when the learning bias is derived from a complexity measure (Kolmogorov complexity and minimal number of states respectively) and analyse the sampling distributions that lead to finite expected biased teaching dimensions. We highlight the existing trade-off between the bound and the representativeness of the sample, and its implications for the understanding of what teaching rich concepts to machines entails.

We generalize the family of $(\sigma, \rho)$-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) $(\sigma, \rho)$-problems are W[1]-hard parameterized by mim-width + solution size., Comment: Accepted at IPEC 2018

We complement the recent algorithmic result that Feedback Vertex Set is XP-time solvable parameterized by the mim-width of a given branch decomposition of the input graph [3] by showing that the problem is W[1]-hard in this parameterization. The hardness holds even for linear mim-width, as well as for H-graphs, where the parameter is the number of edges in H. To obtain this result, we adapt a reduction due to Fomin, Golovach and Raymond [2], following the same line of reasoning but adding a new gadget., Comment: 7 pages, 3 figures

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 $n^{\mathcal{O}(w)}$-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, 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 $n^{\mathcal{O}(w)}$. 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., Comment: 26 pages, 3 figures; accepted at STACS 2018

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 $k$., Comment: 20 pages, 4 figures; accepted at IPEC 2017

We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs $K_t \boxminus K_t$ and $K_t \boxminus S_t$ are graphs obtained from the disjoint union of two cliques of size $t$, and one clique of size $t$ and one independent set of size $t$ respectively, by adding a perfect matching. We prove that : (1) interval graphs are $(K_3\boxminus S_3)$-free chordal graphs; and $(K_t\boxminus S_t)$-free chordal graphs have mim-width at most $t-1$, (2) permutation graphs are $(K_3\boxminus K_3)$-free co-comparability graphs; and $(K_t\boxminus K_t)$-free co-comparability graphs have mim-width at most $t-1$, (3) chordal graphs and co-comparability graphs have unbounded mim-width in general. We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time $n^{\mathcal{O}(t)}$ on $(K_t\boxminus S_t)$-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless $P=NP$. We generalize these ideas to bigger graph classes. We introduce a new width parameter sim-width, of stronger modelling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding $K_t\boxminus K_t$ and $K_t\boxminus S_t$ as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width., Comment: 24 pages, 5 figures; An extended abstract appeared in the proceedings of WALCOM2017

It was recently shown \cite{STV} 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 that it holds 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 $\operatorname{mmw}(G) \leq k$ 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 $|A|,|B|,|C| \leq k$ 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 $O^*(3^{\operatorname{tw}(G)})$ whenever $\operatorname{tw}(G) > \log_3{8} \times k \approx 1.893 k$. Note that $\operatorname{mmw}(G) \leq \operatorname{tw}(G)+1 \leq 3 \operatorname{mmw}(G)$ 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 $\operatorname{tw}(G) > 1.549 \times \operatorname{mmw}(G)$.

Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that are all FPT parameterized by treewidth but none of which can be FPT parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7, 8]. We therefore seek a structural graph parameter that shares some of the generality of clique-width without paying this price. Based on splits, branch decompositions and the work of Vatshelle [18] on Maximum Matching-width, we consider the graph parameter sm-width which lies between treewidth and clique-width. Some graph classes of unbounded treewidth, like distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized by sm-width.

In this paper we propose a structural parameter of CNF formulas and use it to identify instances of weighted MaxSAT and #SAT that can be solved in polynomial time. Given a CNF formula we say that a set of clauses is precisely satisfiable if there is some complete assignment satisfying these clauses only. Let the ps-value of the formula be the number of precisely satisfiable sets of clauses. Applying the notion of branch decompositions to CNF formulas and using ps-value as cut function, we define the ps-width of a formula. For a formula given with a decomposition of polynomial ps-width we show dynamic programming algorithms solving weighted MaxSAT and #SAT in polynomial time. Combining with results of 'Belmonte and Vatshelle, Graph classes with structured neighborhoods and algorithmic applications, Theor. Comput. Sci. 511: 54-65 (2013)' we get polynomial-time algorithms solving weighted MaxSAT and #SAT for some classes of structured CNF formulas. For example, we get $O(m^2(m + n)s)$ algorithms for formulas $F$ of $m$ clauses and $n$ variables and size $s$, if $F$ has a linear ordering of the variables and clauses such that for any variable $x$ occurring in clause $C$, if $x$ appears before $C$ then any variable between them also occurs in $C$, and if $C$ appears before $x$ then $x$ occurs also in any clause between them. Note that the class of incidence graphs of such formulas do not have bounded clique-width.

Given a graph $G=(V,E)$ and a set of terminal vertices $T$ we say that a superset $S$ of $T$ is $T$-connecting if $S$ induces a connected graph, and $S$ is minimal if no strict subset of $S$ is $T$-connecting. In this paper we prove that there are at most ${|V \setminus T| \choose |T|-2} \cdot 3^{\frac{|V \setminus T|}{3}}$ minimal $T$-connecting sets when $|T| \leq n/3$ and that these can be enumerated within a polynomial factor of this bound. This generalizes the algorithm for enumerating all induced paths between a pair of vertices, corresponding to the case $|T|=2$. We apply our enumeration algorithm to solve the {\sc 2-Disjoint Connected Subgraphs} problem in time $O^*(1.7804^n)$, improving on the recent $O^*(1.933^n)$ algorithm of Cygan et al. 2012 LATIN paper., Comment: 13 pages, 1 figure

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs., Comment: 11 pages

Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and a linear number of bags. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded tree-width has linear convex crossing number, and every $K_{3,3}$-minor-free graph with bounded degree has linear rectilinear crossing number., Comment: An extended abstract of this paper was published in the Proceedings of the Graph Drawing Conference (GD 2006), Lecture Notes in Computer Science 4372:150-161, Springer, 2007