Approximability of (Simultaneous) Class Cover for Boxes

Bereg et al. (2012) introduced the Boxes Class Cover problem, which has its roots in classification and clustering applications: Given a set of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes whose union covers the red points without covering any blue point. In this paper we give an alternative proof of APX-hardness for this problem, which also yields an explicit lower bound on its approximability. Our proof also directly applies when restricted to sets of points in general position and to the case where so-called half-strips are considered instead of boxes, which is a new result. We also introduce a symmetric variant of this problem, which we call Simultaneous Boxes Class Cover and can be stated as follows: Given a set S of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes which together cover S such that all boxes cover only points of the same color and no box covering a red point intersects a box covering a blue point. We show that this problem is also APX-hard and give a polynomial-time constant-factor approximation algorithm

Orthogonal dissection into few rectangles

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis-parallel polygon into a simple polygon with the minimum possible number of edges. When rotations or reflections are allowed, we can approximate the minimum number of rectangles to within a factor of two.Comment: 18 pages, 8 figures. This version adds results on dissection with rotations and reflection

Maximum Clique in Geometric Intersection Graphs

An intersection graph is a graph that represents some geometric objects as vertices, and joins edges between the nodes corresponding to the items that intersect. The maximum clique in a geometric intersec- tion graph is the largest mutually intersecting set of objects. In this thesis, the primary focus is to study the maximum clique in various geometric intersection graphs. We develop three results motivated by the maximum clique problem in the intersection graph of disks in the Euclidean plane. First, we improve the time complexity of calculating the maximum clique in unit disk graphs from O(n3 log n) to O(n2.5 log n). Second, we introduce a new technique called pair-oriented labelling. This method is used to show the NP- hardness of finding a maximum clique in various geometric intersection graphs, acting as a way to augment the commonly used co-2-subdivision approach. Finally, finding maximum clique in two classes of geometric intersection graphs are proven to be NP-hard. These are the intersection graph of disks and axis-aligned rectangles, and the outer triangle graph. The former is previously known to be NP-hard, and so this proof represents the use of pair-oriented labelling in a problem that was otherwise considered difficult to prove NP-hard using a co-2-subdivision approach. The outer triangle graph is a novel intersection graph, which therefore provides new NP-hardness results for finding a maximum clique in geometric intersection graphs

Induction and interaction in the evolution of language and conceptual structure

Languages evolve in response to various pressures, and this thesis adopts the view that two pressures are especially important. Firstly, the process of learning a language functions as a pressure for greater simplicity due to a domain-general cognitive preference for simple structure. Secondly, the process of using a language in communicative scenarios functions as a pressure for greater informativeness because ultimately languages are only useful to the extent that they allow their users to express – or indeed represent – nuanced meaning distinctions. These two fundamental properties of language – simplicity and informativeness – are often, but not always, in conflict with each other. In general, a simple language cannot be informative and an informative language cannot be simple, resulting in the simplicity–informativeness tradeoff. Typological studies in several domains, including colour, kinship, and spatial relations, have demonstrated that languages find optimal solutions to this tradeoff – optimal solutions to the problem of balancing, on the one hand, the need for simplicity, and on the other, the need for informativeness. More specifically, the thesis explores how inductive reasoning and communicative interaction contribute to simple and informative structure respectively, with a particular emphasis on how a continuous space of meanings, such as the colour spectrum, may be divided into discrete labelled categories. The thesis first describes information-theoretic perspectives on learning and communication and highlights the fact that one of the hallmark feature of conceptual structure – which I term compactness – is not subject to the simplicity–informativeness tradeoff, since it confers advantages on both learning and use. This means it is unclear whether compact structure derives from a learning pressure or from a communicative pressure. To complicate matters further, some researchers view learning as a pressure for simplicity, as outlined above, while others have argued that learning might function as a pressure for informativeness in the sense that learners might have an a-priori expectation that languages ought to be informative. The thesis attempts to resolve this by formalizing these different perspectives in a model of an idealized Bayesian learner, and this model is used to make specific predictions about how these perspectives will play out during individual concept induction and also during the evolution of conceptual structure over time. Experimental testing of these predictions reveals overwhelming support for the simplicity account: Learners have a preference for simplicity, and over generational time, this preference becomes amplified, ultimately resulting in maximally simple, but nevertheless compact, conceptual structure. This emergent compact structure remains limited, however, because it only permits the expression of a small number of meaning distinctions – the emergent systems become degenerate. This issue is addressed in the second part of the thesis, which compares the outcomes of three experiments. The first replicates the finding above – compact categorical structure emerges from learning; the second and third experiments compare artificial and genuine pressures for expressivity, and they show that it is only in the presence of a live communicative task that higher level structure – a kind of statistical compositionality – can emerge. Working together, the low-level compact categorical structure, derived from learning, and the high-level compositional structure, derived from communicative interaction, provide a solution to the simplicity–informativeness tradeoff, expanding on and lending support to various claims in the literature

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum