First Order Logic and Twin-Width in Tournaments

We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments T, first-order model checking either is fixed parameter tractable, or is AW[*]-hard. This dichotomy coincides with the fact that T has either bounded or unbounded twin-width, and that the growth of T is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: T has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs. The key for these results is a polynomial time algorithm which takes as input a tournament T and computes a linear order < on V(T) such that the twin-width of the birelation (T, <) is at most some function of the twin-width of T. Since approximating twin-width can be done in FPT time for an ordered structure (T, <), this provides a FPT approximation of twin-width for tournaments

Big Step Normalisation for Type Theory

Big step normalisation is a normalisation method for typed lambda-calculi which relies on a purely syntactic recursive evaluator. Termination of that evaluator is proven using a predicate called strong computability, similar to the techniques used to prove strong normalisation of ?-reduction for typed lambda-calculi. We generalise big step normalisation to a minimalist dependent type theory. Compared to previous presentations of big step normalisation for e.g. the simply-typed lambda-calculus, we use a quotiented syntax of type theory, which crucially reduces the syntactic complexity introduced by dependent types. Most of the proof has been formalised using Agda

Maximum Independent Set when excluding an induced minor: K1+tK2K_1 + tK_2 and tC3⊎C4tC_3 \uplus C_4

Dallard, Milani\v{c}, and \v{S}torgel [arXiv '22] ask if for every class excluding a fixed planar graph $H$ as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when $H$ is any planar complete bipartite graph, or the 5-vertex clique minus one edge, or minus two disjoint edges. A positive answer would constitute a far-reaching generalization of the state-of-the-art, when we currently do not know if a polynomial-time algorithm exists when $H$ is the 7-vertex path. Relaxing tractability to the existence of a quasipolynomial-time algorithm, we know substantially more. Indeed, quasipolynomial-time algorithms were recently obtained for the $t$-vertex cycle, $C_t$ [Gartland et al., STOC '21] and the disjoint union of $t$ triangles, $tC_3$ [Bonamy et al., SODA '23]. We give, for every integer $t$, a polynomial-time algorithm running in $n^{O(t^5)}$ when $H$ is the friendship graph $K_1 + tK_2$ ($t$ disjoint edges plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm running in $n^{O(t^2 \log n)+t^{O(1)}}$ when $H$ is $tC_3 \uplus C_4$ (the disjoint union of $t$ triangles and a 4-vertex cycle). The former extends a classical result on graphs excluding $tK_2$ as an induced subgraph [Alekseev, DAM '07], while the latter extends Bonamy et al.'s result.Comment: 15 pages, 2 figure

A tamed family of triangle-free graphs with unbounded chromatic number

We construct a hereditary class of triangle-free graphs with unbounded chromatic number, in which every non-trivial graph either contains a pair of non-adjacent twins or has an edgeless vertex cutset of size at most two. This answers in the negative a question of Chudnovsky, Penev, Scott, and Trotignon. The class is the hereditary closure of a family of (triangle-free) twincut graphs $G_1, G_2, \ldots$ such that $G_k$ has chromatic number $k$. We also show that every twincut graph is edge-critical

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

A graph is $O_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in $O_k$-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $O_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of $O_2$-free graphs without $K_{3,3}$-subgraph and whose treewidth is (at least) logarithmic. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $O_k$-free graphs, and that deciding the $O_k$-freeness of sparse graphs is polynomial time solvable.Comment: 28 pages, 6 figures. v3: improved complexity result

First Order Logic and Twin-Width in Tournaments

International audienceWe characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments T, first-order model checking either is fixed parameter tractable, or is AW[*]-hard. This dichotomy coincides with the fact that T has either bounded or unbounded twin-width, and that the growth of T is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: T has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs. The key for these results is a polynomial time algorithm which takes as input a tournament T and computes a linear order < on V(T) such that the twin-width of the birelation (T, <) is at most some function of the twin-width of T. Since approximating twin-width can be done in FPT time for an ordered structure (T, <), this provides a FPT approximation of twin-width for tournaments