The recent developments in computational geometry and graph theory have seen significant advancements in the efficiency and precision of algorithms for various geometric and topological problems. Notably, there has been a shift towards constant workspace algorithms for computing relative hulls in the plane, which represent a novel approach to solving such problems without the need for extensive memory resources. Additionally, the field has seen innovative methods for triangulating unknown smooth surfaces based on finite point sets, with algorithms that simplify alpha-complexes through new types of collapses, offering more efficient and compartmentalized proofs.
Another area of innovation lies in the study of geometry-preserving reductions between constraint satisfaction problems, which has been motivated by phase transitions in combinatorial optimization. This research introduces new quantitative methods for the lambda-calculus, leveraging partial metrics to provide insights into equational theories and higher-order Scott topologies. Furthermore, the study of multipacking problems for geometric point sets has been initiated, with significant findings on the computational complexity of such problems in the Euclidean plane.
In the realm of graph modifications, parameterized algorithms and kernels for geometric graph modification through disk scaling have been proposed, addressing the limitations of traditional combinatorial modifications. This approach has broader implications for topology control and parameterized complexity. Lastly, there has been a focus on minimal and minimum cylindrical algebraic decompositions, with new algorithms for computing these decompositions more efficiently, particularly in lower dimensions.
Noteworthy papers include one on constant workspace algorithms for computing relative hulls, which represents a pioneering effort in this area, and another on triangulating unknown smooth surfaces using alpha-complexes, which offers a practical and theoretical advancement in surface triangulation. Additionally, the paper on geometry-preserving reductions between constraint satisfaction problems provides a foundational framework for future research in combinatorial optimization.
