The packing lemma of Haussler states that given a set system (X, R) with bounded VC dimension, if every pair of sets in R are 'far apart' (i.e., have large symmetric difference), then R cannot contain too many sets. This has turned out to be the technical foundation for many results in geometric discrepancy using the entropy method (see [Mat99] for a detailed background) as well as recent work on set systems with bounded VC dimension [FPS + ar]. Recently it was generalized to the shallow packing lemma [DEG15, Mus16], applying to set systems as a function of their shallow cell complexity. In this paper we present several new results and applications related to packings: 1. an optimal lower bound for shallow packings, thus settling the open question in Ezra (SODA 2016) and Dutta et al. (SoCG 2015). 2. improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry (Annals of Mathematics, 1952). 3. simplifying and generalizing the main technical tool in Fox et al. (J. of the EMS, 2016).Besides using the packing lemma and a combinatorial construction, our proofs combine tools from polynomial partitioning and the probabilistic method.

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We introduce and study conflict-free colourings of t-subsets in hypergraphs. In such colourings, one assigns colours to all subsets of vertices of cardinality t such that in any hyperedge of cardinality at least t there is a uniquely coloured t-subset.The case t = 1, i.e., vertex conflict-free colouring, is a well-studied notion that has been originated in the context of frequency allocation to antennas in cellular networks. It has caught the attention of researchers both from the combinatorial and algorithmic points of view. A special focus was given to hypergraphs arising in geometry.Already the case t = 2 (i.e., colouring pairs) seems to present a new challenge. Many of the tools used for conflict-free colouring of geometric hypergraphs rely on hereditary properties of the underlying hypergraphs. When dealing with subsets of vertices, the properties do not pass to subfamilies of subsets. Therefore, we develop new tools, which might be of independent interest.(i) For any fixed t, we show that the n t t-subsets in any set P of n points in the plane can be coloured with O(t 2 log 2 n) colours so that any axis-parallel rectangle that contains at least t points of P also contains a uniquely coloured t-subset.(ii) For a wide class of "well behaved" geometrically defined hypergraphs, we provide near tight upper bounds on their t-subset conflict-free chromatic number.For t = 2 we show that for each of those "well -behaved" hypergraphs H, the hypergraph H obtained by taking union of two hyperedges from H, admits a 2-subset conflict-free colouring with roughly the same number of colours as H. For example, we show that the n 2 pairs of points in any set P of n points in the plane can be coloured with O(log n) colours such that for any two discs d1, d2 in the plane with |(d1 ∪ d2) ∩ P | ≥ 2 there is a uniquely (in d1 ∪ d2) coloured pair.(iii) We also show that there is no general bound on the t-subset conflict-free chromatic number as a function of the standard conflict-free chromatic number already for t = 2.

The last decade has seen the resolution of several basic NP-complete problems in geometric combinatorial optimisation-interestingly, all with the same algorithm: local search. This includes the existence of polynomial-time approximation schemes (PTASs) for hitting set, set cover, dominating set, independent set, and other problems for some basic geometric objects. More precisely, it was shown that for many of these problems, local search with radius λ gives aOn the other hand, hardness results suggest that there do not exist PTASs for these problems with running time poly(n) • f (ε) for any arbitrary computable f . Thus the main question left open in previous work is in improving the exponent of n to o(ε −2 ).Our main result is that the approximation guarantee of the standard local search algorithm cannot be improved for any of these problems, which we show by constructing instances with poor "locally optimal solutions". The key ingredient, of independent interest, is a new lower bound on locally expanding planar graphs. Our construction extends to other graph families with small separators.

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel).Not all directed graphs contain kernels, and computing a kernel or deciding that none exist is NP-complete even on low-degree planar digraphs. The existing polynomial-time algorithms for this problem all restrict both the undirected structure and the edge orientations of the input: for example, to chordal graphs without bidirectional edges (Pass-Lanneau, Igarashi and Meunier, Discrete Appl Math 2020) or to permutation graphs where each clique has a sink (Abbas and Saoula, 4OR 2005).By contrast, we count the kernels of a fuzzy circular interval graph in polynomial time, regardless of its edge orientations, and return a kernel when one exists. (Fuzzy circular graphs were introduced by Chudnovsky and Seymour in their structure theorem for claw-free graphs.)We also consider kernels on cographs, where we establish NP-hardness in general but linear running times on the subclass of threshold graphs.