Explaining unsolvability of planning problems is of significant research interest in Explainable AI Planning. AI planning literature has reported several research efforts on generating explanations of solutions to planning problems. However, explaining the unsolvability of planning problems remains a largely open and understudied problem. A widely practiced approach to plan generation and automated problem solving, in general, is to decompose tasks into sub-problems that help progressively converge towards the goal. In this paper, we propose to adopt the same philosophy of sub-problem identification as a mechanism for analyzing and explaining unsolvability of planning problems in hybrid systems. In particular, for a given unsolvable planning problem, we propose to identify common waypoints, which are universal obstacles to plan existence; in other words, they appear on every plan from the source to the planning goal. This work envisions such waypoints as sub-problems of the planning problem and the unreachability of any of these waypoints as an explanation for the unsolvability of the original planning problem. We propose a novel method of waypoint identification by casting the problem as an instance of the longest common subsequence problem, a widely popular problem in computer science, typically considered as an illustrative example for the dynamic programming paradigm. Once the waypoints are identified, we perform symbolic reachability analysis on them to identify the earliest unreachable waypoint and report it as the explanation of unsolvability. We present experimental results on unsolvable planning problems in hybrid domains.

We consider the problem of learning the structure underlying a Gaussian graphical model when the variables (or subsets thereof) are corrupted by independent noise. A recent line of work establishes that even for tree-structured graphical models, only partial structure recovery is possible and goes on to devise algorithms to identify the structure up to an (unavoidable) equivalence class of trees. We extend these results beyond trees and consider the model selection problem under noise for non tree-structured graphs, as tree graphs cannot model several real-world scenarios. Although unidentifiable, we show that, like the tree-structured graphs, the ambiguity is limited to an equivalence class. This limited ambiguity can help provide meaningful clustering information (even with noise), which is helpful in computer and social networks, protein-protein interaction networks, and power networks. Furthermore, we devise an algorithm based on a novel ancestral testing method for recovering the equivalence class. We complement these results with finite sample guarantees for the algorithm in the high-dimensional regime.

On an assigned graph, the problem of Multi-Agent Pathfinding (MAPF) consists in finding paths for multiple agents, avoiding collisions. Finding the minimum-length solution is known to be NP-hard, and computation times grows exponentially with the number of agents. However, in industrial applications, it is important to find feasible, suboptimal solutions, in a time that grows polynomially with the number of agents. Such algorithms exist for undirected and biconnected directed graphs. Our main contribution is to generalize these algorithms to the more general case of strongly connected directed graphs. In particular, given a MAPF problem with at least two holes, we present an algorithm that checks the problem feasibility in linear time with respect to the number of nodes, and provides a feasible solution in polynomial time.

We introduce a novel approach to Compressed Path Databases, space efficient oracles used to very quickly identify the first edge on a shortest path. Our algorithm achieves query running times on the 100 nanosecond scale, being significantly faster than state-of-the-art first-move oracles from the literature. Space consumption is competitive, due to a compression approach that rearranges rows and columns in a first-move matrix and then performs run length encoding (RLE) on the contents of the matrix. One variant of our implemented system was, by a convincing margin, the fastest entry in the 2014 Grid-Based Path Planning Competition. We give a first tractability analysis for the compression scheme used by our algorithm. We study the complexity of computing a database of minimum size for general directed and undirected graphs. We find that in both cases the problem is NP-complete. We also show that, for graphs which can be decomposed along articulation points, the problem can be decomposed into independent parts, with a corresponding reduction in its level of difficulty. In particular, this leads to simple and tractable algorithms with linear running time which yield optimal compression results for trees.

Many problems in computational sustainability involve constraints on connectivity. When designing a new wildlife corridor, we need it to be geographically connected. When planning the harvest of a forest, we need new areas to harvest to be connected to areas that have already been harvested so we can access them easily. And when town planning, we need to connect new homes to the existing utility infrastructure. To reason about connectivity, we propose a new family of global connectivity constraints. We identify when these constraints can be propagated tractably, and give some efficient, typically linear time propagators for when this is the case. We report results on several benchmark problems which demonstrate the efficiency of our propagation algorithms and the promise offered by reasoning globally about connectivity.

In this work we give a first tractability analysis of Compressed Path Databases, space efficient oracles used to very quickly identify the first arc on a shortest path. We study the complexity of computing an optimal compressed path database for general directed and undirected graphs. We find that in both cases the problem is NP-complete. We also show that, for graphs which can be decomposed along articulalion points, the problem can be decomposed into independent parts, with a corresponding reduction in its level of difficulty. In particular, this leads to simple and tractable algorithms which yield optimal compression results for trees.