The sample compressibility of concept classes plays an important role in learning theory, as a sufficient condition for PAC learnability, and more recently as an avenue for robust generalisation in adaptive data analysis. Whether compression schemes of size $O(d)$ must necessarily exist for all classes of VC dimension $d$ is unknown, but conjectured to be true by Warmuth. Recently Chalopin, Chepoi, Moran, and Warmuth (2018) gave a beautiful unlabelled sample compression scheme of size VC dimension for all maximum classes: classes that meet the Sauer-Shelah-Perles Lemma with equality. They also offered a counterexample to compression schemes based on a promising approach known as corner peeling. In this paper we simplify and extend their proof technique to deal with so-called extremal classes of VC dimension $d$ which contain maximum classes of VC dimension $d-1$. A criterion is given which would imply that all extremal classes admit unlabelled compression schemes of size $d$. We also prove that all intersection-closed classes with VC dimension $d$ admit unlabelled compression schemes of size at most $11d$.

Under the prediction model of learning, a prediction strategy is presented with an i.i.d. sample of n - 1 points in X and corresponding labels from a concept f F , and aims to minimize the worst-case probability of erring on an nth point. By exploiting the structure of F , Haussler et al. achieved a VC(F )/n bound for the natural one-inclusion prediction strategy, improving on bounds implied by PAC-type results by a O(log n) factor. The key data structure in their result is the natural subgraph of the hypercube--the one-inclusion graph; the key step is a d = VC(F ) bound on one-inclunion graph density. The first main result of this s /n -1 paper is a density bound of n d-1 ( d ) < d, which positively resolves a conjecture of Kuzmin & Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved mistake bound for the randomized (deterministic) one-inclusion strategy for all d (for d (n)). The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is a k -class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This bound on expected risk improves on known PAC-based results by a factor of O(log n) and is shown to be optimal up to a O(log k ) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout.

Abstract We give a polynomial‐time algorithm for finding a solution to the Traveling Salesman Problem when the points given are constrained to lie on a fixed set of smooth curves of finite length. © 2001 John Wiley & Sons, Inc.

We give a polynomial time algorithm for solving the Euclidean Steiner tree problem when the terminals are constrained to lie on a fixed finite set of disjoint finite-length compact simple smooth curves. The problem is known to be NP-hard in general. We also show it to be NP-hard if the terminals lie on two parallel infinite lines or on a bent line segment provided the bend has an angle of less than $120^\circ$.

Abstract We show that a degree‐five Steiner point can never appear in a least‐cost planar network; that is we show that a degree‐five Steiner point actally increases the cost.