Fingerprint-based indoor localization methods are promising due to the high availability of deployed access points and compatibility with commercial-off-the-shelf user devices. However, to train regression models for localization, an extensive site survey is required to collect fingerprint data from the target areas. In this paper, we consider the problem of informative path planning (IPP) to find the optimal walk for site survey subject to a budget constraint. IPP for location fingerprint collection is related to the well-known orienteering problem (OP) but is more challenging due to edge-based non-additive rewards and revisits. Given the NP-hardness of IPP, we propose two heuristic approaches: a Greedy algorithm and a genetic algorithm. We show through experimental data collected from two indoor environments with different characteristics that the two algorithms have low computation complexity, can generally achieve higher utility and lower localization errors compared to the extension of two state-of-the-art approaches to OP.
The Recursive Greedy (RG) algorithm is an approximate algorithm for the submodular orienteering problem (SOP), first proposed by Chekuri in 2005 @cite_2 . The algorithm considers all possible combinations of intermediate vertices and budget, and then it is recursively applied on the smaller sub-problems. The greedy aspect of the algorithm comes from the divide-and-conquer approach by simply concatenating the paths returned from sub-problems into a single one as the final solution. The algorithm can be summarized informally in the following steps: enumerate all possible combinations of intermediate vertices and budget splits recursively find the first half of the path within the budget split find the second half of the path within the remaining budget return the concatenation of the two sub-paths which have the biggest reward
@cite_2
2737130609 2151690061 2177862760 2605027943
Maximizing information gathered within a budget is a relevant problem for information gathering tasks for robots with cost or operating time constraints. This problem is also known as the informative path planning (IPP) problem or correlated orienteering. It can be formalized as that of finding budgeted routes in a graph such that the reward collected by the route is maximized, where the reward at nodes can be dependent. Unfortunately, the problem is NP-Hard and the state of the art methods are too slow to even present an approximate solution online. Here we present Randomized Anytime Orienteering (RAOr) algorithm that provides near optimal solutions while demonstrably converging to an efficient solution in runtimes that allows the solver to be run online. The key idea of our approach is to pose orienteering as a combination of a Constraint Satisfaction Problem and a Traveling Salesman Problem. This formulation allows us to restrict the search space to routes that incur minimum distance to visit a set of selected nodes, and rapidly search this space using random sampling. The paper provides the analysis of asymptotic near-optimality, convergence rates for RAOr algorithms, and present strategies to improve anytime performance of the algorithm. Our experimental results suggest an improvement by an order of magnitude over the state of the art methods in relevant simulation and in real world scenarios. The greedy sequential algorithm for maximal independent set (MIS) loops over the vertices in an arbitrary order adding a vertex to the resulting set if and only if no previous neighboring vertex has been added. In this loop, as in many sequential loops, each iterate will only depend on a subset of the previous iterates (i.e. knowing that any one of a vertex's previous neighbors is in the MIS, or knowing that it has no previous neighbors, is sufficient to decide its fate one way or the other). This leads to a dependence structure among the iterates. If this structure is shallow then running the iterates in parallel while respecting the dependencies can lead to an efficient parallel implementation mimicking the sequential algorithm. In this paper, we show that for any graph, and for a random ordering of the vertices, the dependence length of the sequential greedy MIS algorithm is polylogarithmic (O(log^2 n) with high probability). Our results extend previous results that show polylogarithmic bounds only for random graphs. We show similar results for greedy maximal matching (MM). For both problems we describe simple linear-work parallel algorithms based on the approach. The algorithms allow for a smooth tradeoff between more parallelism and reduced work, but always return the same result as the sequential greedy algorithms. We present experimental results that demonstrate efficiency and the tradeoff between work and parallelism. Greedy algorithms are practitioners’ best friends—they are intuitive, are simple to implement, and often lead to very good solutions. However, implementing greedy algorithms in a distributed setting is challenging since the greedy choice is inherently sequential, and it is not clear how to take advantage of the extra processing power. Our main result is a powerful sampling technique that aids in parallelization of sequential algorithms. Armed with this primitive, we then adapt a broad class of greedy algorithms to the MapReduce paradigm; this class includes maximum cover and submodular maximization subject to p-system constraint problems. Our method yields efficient algorithms that run in a logarithmic number of rounds while obtaining solutions that are arbitrarily close to those produced by the standard sequential greedy algorithm. We begin with algorithms for modular maximization subject to a matroid constraint and then extend this approach to obtain approximation algorithms for submodular maximization subject to knapsack or p-system constraints. Copyright © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GREEDY-MATCHING. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GREEDYMATCHING is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GREEDYMATCHING is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider three natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and computational tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GREEDYMATCHING on a special class of weighted graphs, called bush graphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is θ, then for every ϵ > 0 the randomized MRG algorithm of ( 1995) gives a (ρ - ϵ)-approximation for the maximum cardinality matching. We conjecture that a tight bound for ρ is 2 3; we prove our conjecture true for four subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for ρ) is a long-standing open problem (Poloczek and Szegedy 2012). This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem.