Unlabeled sample compression schemes and corner peelings for ample and maximum classes

We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that all previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous. On the positive side we present a new construction of an unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabeled sample compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes

Lagrangian dynamical geography of the Gulf of Mexico

We construct a Markov-chain representation of the surface-ocean Lagrangian dynamics in a region occupied by the Gulf of Mexico (GoM) and adjacent portions of the Caribbean Sea and North Atlantic using satellite-tracked drifter trajectory data, the largest collection so far considered. From the analysis of the eigenvectors of the transition matrix associated with the chain, we identify almost-invariant attracting sets and their basins of attraction. With this information we decompose the GoM's geography into weakly dynamically interacting provinces, which constrain the connectivity between distant locations within the GoM. Offshore oil exploration, oil spill contingency planning, and fish larval connectivity assessment are among the many activities that can benefit from the dynamical information carried in the geography constructed here.Comment: Submitted to Scientific Report

On the Extremal Functions of Acyclic Forbidden 0--1 Matrices

The extremal theory of forbidden 0--1 matrices studies the asymptotic growth of the function $\mathrm{Ex}(P,n)$, which is the maximum weight of a matrix $A\in\{0,1\}^{n\times n}$ whose submatrices avoid a fixed pattern $P\in\{0,1\}^{k\times l}$. This theory has been wildly successful at resolving problems in combinatorics, discrete and computational geometry, structural graph theory, and the analysis of data structures, particularly corollaries of the dynamic optimality conjecture. All these applications use acyclic patterns, meaning that when $P$ is regarded as the adjacency matrix of a bipartite graph, the graph is acyclic. The biggest open problem in this area is to bound $\mathrm{Ex}(P,n)$ for acyclic $P$. Prior results have only ruled out the strict $O(n\log n)$ bound conjectured by Furedi and Hajnal. It is consistent with prior results that $\forall P. \mathrm{Ex}(P,n)\leq n\log^{1+o(1)} n$, and also consistent that $\forall \epsilon>0.\exists P. \mathrm{Ex}(P,n) \geq n^{2-\epsilon}$. In this paper we establish a stronger lower bound on the extremal functions of acyclic $P$. Specifically, we give a new construction of relatively dense 0--1 matrices with $\Theta(n(\log n/\log\log n)^t)$ 1s that avoid an acyclic $X_t$. Pach and Tardos have conjectured that this type of result is the best possible, i.e., no acyclic $P$ exists for which $\mathrm{Ex}(P,n)\geq n(\log n)^{\omega(1)}$

From matchings to independent sets

In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyze these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs

Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width

Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ? := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures ? under consideration have relational width exactly (2, ?_?) where ?_? is the maximal size of a forbidden subgraph of ?, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_? may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3)