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
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Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting.