Many real-world applications require the successful combination of spatial and temporal reasoning. In this paper, we study the general framework of the Traveling Salesman Problem with Simple Temporal Constraints. Representationally, this framework subsumes the Traveling Salesman Problem, Simple Temporal Problems, as well as many of the frameworks described in the literature. We analyze the theoretical properties of the combined problem providing strong inapproximability results for the general problem, and positive results for some special cases.
We consider an orienteering problem (OP) where an agent needs to visit a series (possibly a subset) of depots, from which the maximal accumulated profits are desired within given limited time budget. Different from most existing works where the profits are assumed to be static, in this work we investigate a variant that has arbitrary time-dependent profits. Specifically, the profits to be collected change over time and they follow different (e.g., independent) time-varying functions. The problem is of inherent nonlinearity and difficult to solve by existing methods. To tackle the challenge, we present a simple and effective framework that incorporates time-variations into the fundamental planning process. Specifically, we propose a deterministic spatio-temporal representation where both spatial description and temporal logic are unified into one routing topology. By employing existing basic sorting and searching algorithms, the routing solutions can be computed in an extremely efficient way. The proposed method is easy to implement and extensive numerical results show that our approach is time efficient and generates near-optimal solutions.
Given a set of obstacles and two designated points in the plane, the Minimum Constraint Removal problem asks for a minimum number of obstacles that can be removed so that a collision-free path exists between the two designated points. It is a well-studied problem in both robotic motion planning and wireless computing that has been shown to be NP-hard in various settings. In this work, we extend the study of Minimum Constraint Removal. We start by presenting refined NP-hardness reductions for the two cases: (1) when all the obstacles are axes-parallel rectangles, and (2) when all the obstacles are line segments such that no three intersect at the same point. These results improve on existing results in the literature. As a byproduct of our NP-hardness reductions, we prove that, unless the Exponential-Time Hypothesis (ETH) fails, Minimum Constraint Removal cannot be solved in subexponential time 2 o ( n ) , where n is the number of obstacles in the instance. This shows that significant improvement on the brute-force 2 O ( n ) -time algorithm is unlikely. We then present a subexponential-time algorithm for instances of Minimum Constraint Removal in which the number of obstacles that overlap at any point is constant; the algorithm runs in time 2 O (√ N ) , where N is the number of the vertices in the auxiliary graph associated with the instance of the problem. We show that significant improvement on this algorithm is unlikely by showing that, unless ETH fails, Minimum Constraint Removal with bounded overlap number cannot be solved in time 2 o (√ N ) . We describe several exact algorithms and approximation algorithms that leverage heuristics and discuss their performance in an extensive empirical simulation.
Bachrach, Yoram (Microsoft Research Cambridge) | Kohli, Pushmeet (Microsoft Research Cambridge) | Kolmogorov, Vladimir (nstitute of Science and Technology) | Zadimoghaddam, Morteza (Massachusetts Institute of Technology)
Representation languages for coalitional games are a key research area in algorithmic game theory. There is an inherent tradeoff between how general a language is, allowing it to capture more elaborate games, and how hard it is computationally to optimize and solve such games. One prominent such language is the simple yet expressive Weighted Graph Games (WGGs) representation (Deng and Papadimitriou, 1994), which maintains knowledge about synergies between agents in the form of an edge weighted graph. We consider the problem of finding the optimal coalition structure in WGGs. The agents in such games are vertices in a graph, and the value of a coalition is the sum of the weights of the edges present between coalition members. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that finding the optimal coalition structure is not only hard for general graphs, but is also intractable for restricted families such as planar graphs which are amenable for many other combinatorial problems. We then provide algorithms with constant factor approximations for planar, minor-free and bounded degree graphs.
The problem of multiagent patrol has gained considerable attention during the past decade, with the immediate applicability of the problem being one of its main sources of interest. In this paper we concentrate on frequency-based patrol, in which the agents' goal is to optimize a frequency criterion, namely, minimizing the time between visits to a set of interest points. We consider multiagent patrol in environments with complex environmental conditions that affect the cost of traveling from one point to another. For example, in marine environments, the travel time of ships depends on parameters such as wind, water currents, and waves. We demonstrate that in such environments there is a need to consider a new multiagent patrol strategy which divides the given area into parts in which more than one agent is active, for improving frequency. We show that in general graphs this problem is intractable, therefore we focus on simplified (yet realistic) cyclic graphs with possible inner edges. Although the problem remains generally intractable in such graphs, we provide a heuristic algorithm that is shown to significantly improve point-visit frequency compared to other patrol strategies. For evaluation of our work we used a custom developed ship simulator that realistically models ship movement constraints such as engine force and drag and reaction of the ship to environmental changes.