Efficient motion planning for problems lacking optimal substructure

We consider the motion-planning problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We suggest a natural cost function that balances path length and risk-exposure time. Specifically, we consider the discrete setting where we are given a graph, or a roadmap, and we wish to compute the minimal-cost path under this cost function. Interestingly, paths defined using our cost function do not have an optimal substructure. Namely, subpaths of an optimal path are not necessarily optimal. Thus, the Bellman condition is not satisfied and standard graph-search algorithms such as Dijkstra cannot be used. We present a path-finding algorithm, which can be seen as a natural generalization of Dijkstra's algorithm. Our algorithm runs in $O\left((n_B\cdot n) \log( n_B\cdot n) + n_B\cdot m\right)$ time, where~$n$ and $m$ are the number of vertices and edges of the graph, respectively, and $n_B$ is the number of intersections between edges and the boundary of the risk zone. We present simulations on robotic platforms demonstrating both the natural paths produced by our cost function and the computational efficiency of our algorithm

Geometric Secluded Paths and Planar Satisfiability

We consider paths with low exposure to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between integral exposure (when we care about how long the path sees every point of the domain) and 0/1 exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT

Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Finding conserved patterns in biological sequences, networks and genomes

Biological patterns are widely used for identifying biologically interesting regions within macromolecules, classifying biological objects, predicting functions and studying evolution. Good pattern finding algorithms will help biologists to formulate and validate hypotheses in an attempt to obtain important insights into the complex mechanisms of living things. In this dissertation, we aim to improve and develop algorithms for five biological pattern finding problems. For the multiple sequence alignment problem, we propose an alternative formulation in which a final alignment is obtained by preserving pairwise alignments specified by edges of a given tree. In contrast with traditional NPhard formulations, our preserving alignment formulation can be solved in polynomial time without using a heuristic, while having very good accuracy. For the path matching problem, we take advantage of the linearity of the query path to reduce the problem to finding a longest weighted path in a directed acyclic graph. We can find k paths with top scores in a network from the query path in polynomial time. As many biological pathways are not linear, our graph matching approach allows a non-linear graph query to be given. Our graph matching formulation overcomes the common weakness of previous approaches that there is no guarantee on the quality of the results. For the gene cluster finding problem, we investigate a formulation based on constraining the overall size of a cluster and develop statistical significance estimates that allow direct comparisons of clusters of different sizes. We explore both a restricted version which requires that orthologous genes are strictly ordered within each cluster, and the unrestricted problem that allows paralogous genes within a genome and clusters that may not appear in every genome. We solve the first problem in polynomial time and develop practical exact algorithms for the second one. In the gene cluster querying problem, based on a querying strategy, we propose an efficient approach for investigating clustering of related genes across multiple genomes for a given gene cluster. By analyzing gene clustering in 400 bacterial genomes, we show that our algorithm is efficient enough to study gene clusters across hundreds of genomes

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Using GRASP and GA to design resilient and cost-effective IP/MPLS networks

The main objective of this thesis is to find good quality solutions for representative instances of the problem of designing a resilient and low cost IP/MPLS network, to be deployed over an existing optical transport network. This research is motivated by two complementary real-world application cases, which comprise the most important commercial and academic networks of Uruguay. To achieve this goal, we performed an exhaustive analysis of existing models and technologies. From all of them we took elements that were contrasted with the particular requirements of our counterparts. We highlight among these requirements, the need of getting solutions transparently implementable over a heterogeneous network environment, which limit us to use widely standardized features of related technologies. We decided to create new models more suitable to fit these needs. These models are intrinsically hard to solve (NP-Hard). Thus we developed metaheuristic based algorithms to find solutions to these real-world instances. Evolutionary Algorithms and Greedy Randomized Adaptive Search Procedures obtained the best results. As it usually happens, real-world planning problems are surrounded by uncertainty. Therefore, we have worked closely with our counterparts to reduce the fuzziness upon data to a set of representative cases. They were combined with different strategies of design to get to scenarios, which were translated into instances of these problems. Finally, the algorithms were fed with this information, and from their outcome we derived our results and conclusions