An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism

The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open question whether individualization-refinement algorithms can achieve upper bounds on the running time similar to the more theoretical techniques based on a group theoretic approach. In this work we give a negative answer to this question and construct a family of graphs on which algorithms based on the individualization-refinement paradigm require exponential time. Contrary to a previous construction of Miyazaki, that only applies to a specific implementation within the individualization-refinement framework, our construction is immune to changing the cell selector, or adding various heuristic invariants to the algorithm. Furthermore, our graphs also provide exponential lower bounds in the case when the $k$-dimensional Weisfeiler-Leman algorithm is used to replace the standard color refinement operator and the arguments even work when the entire automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page

Using Canonical Forms for Isomorphism Reduction in Graph-based Model Checking

Graph isomorphism checking can be used in graph-based model checking to achieve symmetry reduction. Instead of one-to-one comparing the graph representations of states, canonical forms of state graphs can be computed. These canonical forms can be used to store and compare states. However, computing a canonical form for a graph is computationally expensive. Whether computing a canonical representation for states and reducing the state space is more efficient than using canonical hashcodes for states and comparing states one-to-one is not a priori clear. In this paper these approaches to isomorphism reduction are described and a preliminary comparison is presented for checking isomorphism of pairs of graphs. An existing algorithm that does not compute a canonical form performs better that tools that do for graphs that are used in graph-based model checking. Computing canonical forms seems to scale better for larger graphs

Adaptive search techniques in AI planning and heuristic search

State-space search is a common approach to solve problems appearing in artificial intelligence and other subfields of computer science. In such problems, an agent must find a sequence of actions leading from an initial state to a goal state. However, the state spaces of practical applications are often too large to explore exhaustively. Hence, heuristic functions that estimate the distance to a goal state (such as straight-line distance for navigation tasks) are used to guide the search more effectively. Heuristic search is typically viewed as a static process. The heuristic function is assumed to be unchanged throughout the search, and its resulting values are directly used for guidance without applying any further reasoning to them. Yet critical aspects of the task may only be discovered during the search, e.g., regions of the state space where the heuristic does not yield reliable values. Our work here aims to make this process more dynamic, allowing the search to adapt to such observations. One form of adaptation that we consider is online refinement of the heuristic function. We design search algorithms that detect weaknesses in the heuristic, and address them with targeted refinement operations. If the heuristic converges to perfect estimates, this results in a secondary method of progress, causing search algorithms that are otherwise incomplete to eventually find a solution. We also consider settings that inherently require adaptation: In online replanning, a plan that is being executed must be amended for changes in the environment. Similarly, in real-time search, an agent must act under strict time constraints with limited information. The search algorithms we introduce in this work share a common pattern of online adaptation, allowing them to effectively react to challenges encountered during the search. We evaluate our contributions on a wide range of standard benchmarks. Our results show that the flexibility of these algorithms makes them more robust than traditional approaches, and they often yield substantial improvements over current state-of-the-art planners.Die Zustandsraumsuche ist ein oft verwendeter Ansatz um verschiedene Probleme zu lösen, die in der Künstlichen Intelligenz und anderen Bereichen der Informatik auftreten. Dabei muss ein Akteur eine Folge von Aktionen finden, die einen Pfad von einem Startzustand zu einem Zielzustand bilden. Die Zustandsräume von praktischen Anwendungen sind häufig zu groß um sie vollständig zu durchsuchen. Aus diesem Grund leitet man die Suche mit Heuristiken, die die Distanz zu einem Zielzustand abschätzen; zum Beispiel lässt sich die Luftliniendistanz als Heuristik für Navigationsprobleme einsetzen. Heuristische Suche wird typischerweise als statischer Prozess angesehen. Man nimmt an, dass die Heuristik während der Suche eine unveränderte Funktion ist, und die resultierenden Werte werden direkt zur Leitung der Suche benutzt ohne weitere Logik darauf anzuwenden. Jedoch könnten kritische Aspekte des Problems erst im Laufe der Suche erkannt werden, wie zum Beispiel Bereiche des Zustandsraums in denen die Heuristik keine verlässlichen Abschätzungen liefert. In dieser Arbeit wird der Suchprozess dynamischer gestaltet und der Suche ermöglicht sich solchen Beobachtungen anzupassen. Eine Art dieser Anpassung ist die Onlineverbesserung der Heuristik. Es werden Suchalgorithmen entwickelt, die Schwächen in der Heuristik erkennen und mit gezielten Verbesserungsoperationen beheben. Wenn die Heuristik zu perfekten Werten konvergiert ergibt sich daraus eine zusätzliche Form von Fortschritt, wodurch auch Suchalgorithmen, die sonst unvollständig sind, garantiert irgendwann eine Lösung finden werden. Es werden auch Szenarien betrachtet, die schon von sich aus Anpassung erfordern: In der Onlineumplanung muss ein Plan, der gerade ausgeführt wird, auf Änderungen in der Umgebung angepasst werden. Ähnlich dazu muss sich ein Akteur in der Echtzeitsuche unter strengen Zeitauflagen und mit eingeschränkten Informationen bewegen. Die Suchalgorithmen, die in dieser Arbeit eingeführt werden, folgen einem gemeinsamen Muster von Onlineanpassung, was ihnen ermöglicht effektiv auf Herausforderungen zu reagieren die im Verlauf der Suche aufkommen. Diese Ansätze werden auf einer breiten Reihe von Benchmarks ausgewertet. Die Ergebnisse zeigen, dass die Flexibilität dieser Algorithmen zu erhöhter Zuverlässigkeit im Vergleich zu traditionellen Ansätzen führt, und es werden oft deutliche Verbesserungen gegenüber modernen Planungssystemen erzielt.DFG grant 389792660 as part of TRR 248 – CPEC (see https://perspicuous-computing.science), and DFG grant HO 2169/5-1, "Critically Constrained Planning via Partial Delete Relaxation

A multistage linear array assignment problem

The implementation of certain algorithms on parallel processing computing architectures can involve partitioning contiguous elements into a fixed number of groups, each of which is to be handled by a single processor. It is desired to find an assignment of elements to processors that minimizes the sum of the maximum workloads experienced at each stage. This problem can be viewed as a multi-objective network optimization problem. Polynomially-bounded algorithms are developed for the case of two stages, whereas the associated decision problem (for an arbitrary number of stages) is shown to be NP-complete. Heuristic procedures are therefore proposed and analyzed for the general problem. Computational experience with one of the exact problems, incorporating certain pruning rules, is presented with one of the exact problems. Empirical results also demonstrate that one of the heuristic procedures is especially effective in practice

Search Problems in Trees with Symmetries: Near Optimal Traversal Strategies for Individualization-Refinement Algorithms

We define a search problem on trees that closely captures the backtracking behavior of all current practical graph isomorphism algorithms. Given two trees with colored leaves, the goal is to find two leaves of matching color, one in each of the trees. The trees are subject to an invariance property which promises that for every pair of leaves of equal color there must be a symmetry (or an isomorphism) that maps one leaf to the other. We describe a randomized algorithm with errors for which the number of visited nodes is quasilinear in the square root of the size of the smaller of the two trees. For inputs of bounded degree, we develop a Las Vegas algorithm with a similar running time. We prove that these results are optimal up to logarithmic factors. For this, we show a lower bound for randomized algorithms on inputs of bounded degree that is the square root of the tree sizes. For inputs of unbounded degree, we show a linear lower bound for Las Vegas algorithms. For deterministic algorithms we can prove a linear bound even for inputs of bounded degree. This shows why randomized algorithms outperform deterministic ones. Our results explain why the randomized "breadth-first with intermixed experimental path" search strategy of the isomorphism tool Traces (Piperno 2008) is often superior to the depth-first search strategy of other tools such as nauty (McKay 1977) or bliss (Junttila, Kaski 2007). However, our algorithm also provides a new traversal strategy, which is theoretically near optimal and which has better worst case behavior than traversal strategies that have previously been used

A Global Optimisation Toolbox for Massively Parallel Engineering Optimisation

A software platform for global optimisation, called PaGMO, has been developed within the Advanced Concepts Team (ACT) at the European Space Agency, and was recently released as an open-source project. PaGMO is built to tackle high-dimensional global optimisation problems, and it has been successfully used to find solutions to real-life engineering problems among which the preliminary design of interplanetary spacecraft trajectories - both chemical (including multiple flybys and deep-space maneuvers) and low-thrust (limited, at the moment, to single phase trajectories), the inverse design of nano-structured radiators and the design of non-reactive controllers for planetary rovers. Featuring an arsenal of global and local optimisation algorithms (including genetic algorithms, differential evolution, simulated annealing, particle swarm optimisation, compass search, improved harmony search, and various interfaces to libraries for local optimisation such as SNOPT, IPOPT, GSL and NLopt), PaGMO is at its core a C++ library which employs an object-oriented architecture providing a clean and easily-extensible optimisation framework. Adoption of multi-threaded programming ensures the efficient exploitation of modern multi-core architectures and allows for a straightforward implementation of the island model paradigm, in which multiple populations of candidate solutions asynchronously exchange information in order to speed-up and improve the optimisation process. In addition to the C++ interface, PaGMO's capabilities are exposed to the high-level language Python, so that it is possible to easily use PaGMO in an interactive session and take advantage of the numerous scientific Python libraries available.Comment: To be presented at 'ICATT 2010: International Conference on Astrodynamics Tools and Techniques