New Multiobjective Shortest Path Algorithms

The main problem studied in this thesis is the Multiobjective Shortest Path (MOSP) problem. We focus on the problem variant with three or more objectives. It is a generalization of the classical Shortest Path problem in which arcs in the input graph are weighted with vectors instead of scalars. The main contribution is the Multiobjective Dijkstra Algorithm (MDA), a label- setting algorithm that achieves state of the art performance in theory and in practice. Motivated by this result, the thesis goes on with the design of variants of the MDA for different variants of the MOSP problem. For the One-to-One MOSP problem the Targeted MDA (T-MDA) has the same asymptotic running time than the MDA but trades in memory for speed in practice. The additional paths stored during the T-MDA are managed in a pseudo-lazy way. This is a novel way to organize explored paths that ensures that the algorithm’s priority queue stores at most one path per node in the input graph simultaneously. The paths that are held back from the queue are organized in lists that are kept sorted by just prepending or appending paths to them, i.e., using constant time insertions. The resulting implementation of the T-MDA solves bigger instances than the MDA and is also faster. We study the generalization of the Time-Dependent Shortest Path problem to the multiobjective case. We provide a detailed analysis of the generalization’s limitations and discuss when the MDA is applicable. For MOSP instances with large sets of optimal paths, good approximation algorithms are important. We combine the MDA with an outcome space partition technique from the literature to obtain a new FPTAS for the MOSP problem. The resulting MD-FPTAS works also for multiobjective instances of the Time-Dependent Shortest Path problem, which is a novelty. Finally, we use the MDA and its biobjective version, the BDA, to solve the Multiobjective Minimum Spanning Tree (MO-MST) problem and the k-Shortest Simple Path (k-SSP) problem, respectively. For the solution of instances of these two problems, the MDA and the BDA are used as subroutines. An MO-MST instance is solved applying the MDA on a so called transition graph. In this graph paths have equivalent costs to trees in the original graph and thus, the optimal solutions computed by the MDA in the transition graph correspond to optimal trees in the original graph. Since the transition graph has an exponential size w.r.t. the size of the original graph, we discuss new pruning techniques to reduce its number of arcs effectively. The solution of the k-SSP, which is a scalar optimization problem, using a biobjective subroutine is surprising but our new algorithm is state of the art in theory and in practice.In dieser Arbeit beschäftigen wir uns hauptsächlich mit dem Multikriterielle Kürzeste Wege (MOSP) Problem. Es ist eine Verallgemeinerung des klassischen Kürzeste Wege Problems, bei der Kanten im Eingangsgraphen mit d-dimensionalen Vektoren anstelle von Skalaren gewichtet sind. Der Hauptbeitrag der Arbeit ist der Multiobjective Dijkstra Algorithmus (MDA), ein label-setting Algorithmus, der in der Theorie und in der Praxis eine Verbesserung der in der Literatur vorhandenen Ergebnisse darstellt. Motiviert durch dieses Ergebnis entwickeln wir im weiteren Verlauf der Arbeit Varianten des MDAs für verschiedene Varianten des MOSP Problems. Für das Punkt-zu-Punkt MOSP Problem hat der Targeted MDA (T-MDA) die gleiche asymptotische Laufzeit wie der MDA. Durch einen erhöhten Speicherverbrauch ist er in der Praxis aber deutlich schneller. Die zusätzlich gespeicherten Pfade werden auf eine pseudo-lazy Art verwaltet. Dies ist eine neuartige Methode zur Organisation von explorierten Pfaden, die sicherstellt, dass die Größe der Prioritätswarteschlange des Algorithmus auf höchstens einen Pfad pro Knoten im Graphen beschränkt bleibt. Die Pfade, die aus der Warteschlange zurückgehalten werden, werden in Listen organisiert, die durch Voranstellen oder Anhängen von Pfaden sortiert bleiben. Das heißt, dass nur Konstantzeit-Operationen benötigt werden. Der T-MDA löst größere Instanzen als der MDA und ist auch schneller. Wir untersuchen die Verallgemeinerung des zeitabhängigen Kürzeste Wege Problems auf den multikriteriellen Fall. Wir bieten eine detaillierte Analyse der Grenzen und Möglichkeiten dieser Verallgemeinerung und diskutieren, wann der MDA hierauf anwendbar ist. Für MOSP Instanzen mit großen Mengen optimaler Pfade sind gute Approximationsalgorithmen wichtig. Wir kombinieren den MDA mit einer Technik zur Partitionierung des Ergebnisraums aus der Literatur, um einen neuen FPTAS für das MOSP Problem zu erhalten. Der resultierende MD-FPTAS funktioniert auch für Instanzen mit verallgemeinert zeitabhängigen Kostenfunktionen, was neuartig ist. Schließlich verwenden wir den MDA und seine bikriterielle Version als Unterroutinen, um jeweils das Multiobjective Minimum Spanning Tree (MOMST) Problem und das k-Shortest Simple Path (k-SSP) Problem zu lösen. Eine MO-MST Instanz wird gelöst, indem der MDA auf einen sogenannten Übergangsgraphen angewendet wird. In diesem Graphen haben Pfade äquivalente Kosten zu den Bäumen im ursprünglichen Graphen und somit entsprechen die optimalen Lösungen, die vom MDA im Übergangsgraphen berechnet werden, den optimalen Bäumen im ursprünglichen Graphen. Da der Übergangsgraph eine exponentielle Größe im Verhältnis zur Größe des ursprünglichen Graphen hat, diskutieren wir neue Techniken zur Reduzierung seiner Anzahl von Kanten. Die Lösung des k-SSP Problems, ein skalares Optimierungsproblem, unter Verwendung einer bikriteriellen Unterroutine ist überraschend, aber unser neuer Algorithmus ist sowohl in der Theorie als auch in der Praxis auf dem neuesten Stand

New Techniques and Algorithms for Multiobjective and Lexicographic Goal-Based Shortest Path Problems

Shortest Path Problems (SPP) are one of the most extensively studied problems in the fields of Artificial Intelligence (AI) and Operations Research (OR). It consists in finding the shortest path between two given nodes in a graph such that the sum of the weights of its constituent arcs is minimized. However, real life problems frequently involve the consideration of multiple, and often conflicting, criteria. When multiple objectives must be simultaneously optimized, the concept of a single optimal solution is no longer valid. Instead, a set of efficient or Pareto-optimal solutions define the optimal trade-off between the objectives under consideration. The Multicriteria Search Problem (MSP), or Multiobjective Shortest Path Problem, is the natural extension to the SPP when more than one criterion are considered. The MSP is computationally harder than the single objective one. The number of label expansions can grow exponentially with solution depth, even for the two objective case. However, with the assumption of bounded integer costs and a fixed number of objectives the problem becomes tractable for polynomially sized graphs. A wide variety of practical application in different fields can be identified for the MSP, like robot path planning, hazardous material transportation, route planning, optimization of public transportation, QoS in networks, or routing in multimedia networks. Goal programming is one of the most successful Multicriteria Decision Making (MCDM) techniques used in Multicriteria Optimization. In this thesis we explore one of its variants in the MSP. Thus, we aim to solve the Multicriteria Search Problem with lexicographic goal-based preferences. To do so, we build on previous work on algorithm NAMOA*, a successful extension of the A* algorithm to the multiobjective case. More precisely, we provide a new algorithm called LEXGO*, an exact label-setting algorithm that returns the subset of Pareto-optimal paths that satisfy a set of lexicographic goals, or the subset that minimizes deviation from goals if these cannot be fully satisfied. Moreover, LEXGO* is proved to be admissible and expands only a subset of the labels expanded by an optimal algorithm like NAMOA*, which performs a full Multiobjective Search. Since time rather than memory is the limiting factor in the performance of multicriteria search algorithms, we also propose a new technique called t-discarding to speed up dominance checks in the process of discarding new alternatives during the search. The application of t-discarding to the algorithms studied previously, NAMOA* and LEXGO*, leads to the introduction of two new time-efficient algorithms named NAMOA*dr and LEXGO*dr , respectively. All the algorithmic alternatives are tested in two scenarios, random grids and realistic road maps problems. The experimental evaluation shows the effectiveness of LEXGO* in both benchmarks, as well as the dramatic reductions of time requirements experienced by the t-discarding versions of the algorithms, with respect to the ones with traditional pruning

An efficient method for multiobjective optimal control and optimal control subject to integral constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures. Since the previous version: typos fixed, formatting improved, one mistake in bibliography correcte

Improving Bi-Objective Shortest Path Search with Early Pruning.

Bi-objective search problems are a useful generalization of shortest path search. This paper reviews some recent contributions for the solution of this problem with emphasis on the efficiency of the dominance checks required for pruning, and introduces a new algorithm that improves time efficiency over previous proposals. Experimental results are presented to show the performance improvement using a set of standard problems over bi-objective road maps.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Financiado por Plan Propio de Investigación de la Universidad de Málaga (UMA), Campus de Excelencia Internacional Andalucía Tech. Work supported by the Spanish Ministry of Science and Innovation, European Regional Development Fund (FEDER), Junta de Andalucía, and Universidad de Málaga through the research projects with reference IRIS PID2021-122812OB-I00, PID2021-122381OB-I00 and UMA20-FEDERJA-065

Enhanced Multi-Objective A* with Partial Expansion

The Multi-Objective Shortest Path Problem (MO-SPP), typically posed on a graph, determines a set of paths from a start vertex to a destination vertex while optimizing multiple objectives. In general, there does not exist a single solution path that can simultaneously optimize all the objectives and the problem thus seeks to find a set of so-called Pareto-optimal solutions. To address this problem, several Multi-Objective A* (MOA*) algorithms were recently developed to quickly compute solutions with quality guarantees. However, these MOA* algorithms often suffer from high memory usage, especially when the branching factor (i.e. the number of neighbors of any vertex) of the graph is large. This work thus aims at reducing the high memory consumption of MOA* with little increase in the runtime. By generalizing and unifying several single- and multi-objective search algorithms, we develop the Runtime and Memory Efficient MOA* (RME-MOA*) approach, which can balance between runtime and memory efficiency by tuning two user-defined hyper-parameters.Comment: 8 pages, 4 figure

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

Computational intelligence approaches to robotics, automation, and control [Volume guest editors]

No abstract available

A benchmark test problem toolkit for multi-objective path optimization

Due to the complexity of multi-objective optimization problems (MOOPs) in general, it is crucial to test MOOP methods on some benchmark test problems. Many benchmark test problem toolkits have been developed for continuous parameter/numerical optimization, but fewer toolkits reported for discrete combinational optimization. This paper reports a benchmark test problem toolkit especially for multi-objective path optimization problem (MOPOP), which is a typical category of discrete combinational optimization. With the reported toolkit, the complete Pareto front of a generated test problem of MOPOP can be deduced and found out manually, and the problem scale and complexity are controllable and adjustable. Many methods for discrete combinational MOOPs often only output a partial or approximated Pareto front. With the reported benchmark test problem toolkit for MOPOP, we can now precisely tell how many true Pareto points are missed by a partial Pareto front, or how large the gap is between an approximated Pareto front and the complete one