Fast Consistency Checking of Very Large Real-World RCC-8 Constraint Networks Using Graph Partitioning

We present a new reasoner for RCC-8 constraint networks, called gp-rcc8, that is based on the patchwork property of path-consistent tractable RCC-8 networks and graph partitioning. We compare gp-rcc8 with state of the art reasoners that are based on constraint propagation and backtracking search as well as one that is based on graph partitioning and SAT solving. Our evaluation considers very large realworld RCC-8 networks and medium-sized synthetic ones, and shows that gp-rcc8 outperforms the other reasoners for these networks, while it is less efficient for smaller networks. Introduction, motivation, and related work The fundamental reasoning problem in RCC-8 is deciding the consistency of a set of constraints Θ, i.e., whether there is a spatial configuration where the relations between the regions can be described by Θ. Traditionally in qualitative spatial reasoning (QSR) consistency of such sets is decided by a backtracking algorithm which optionally uses a pathconsistency algorithm as a preprocessing step for forward checking. In general, this problem is NP-complete (Renz and Nebel 1999). However it has been shown in (Renz 1999) that there are tractable subsets of RCC-8 for which the consistency problem can be decided by path-consistency. Table 1 depicts the characteristics of some real-world RCC-8 networks recording the topological relations between administrative regions in Europe (networks nuts, adm1, and adm2) and the world (networks gadm1 and gadm2), and the performance of the following reasoners regarding consistency checking: Renz-Nebel01 (Renz and Nebel 2001), GQR-1500 (Gantner, Westphal, and Woelfl 2008; Westphal and Hué 2012), PPyRCC8 (Sioutis and Koubarakis 2012), and rcc8sat (Huang, Li, and Renz 2013). All reasoners but rcc8sat follow the standard methods developed in QSR and CSP for consistency checking, namely constraint propagation techniques in combination with a backtracking search algorithm, whereas rcc8sat follows the SAT paradigm according to which the problem of consistency is reduced to the satisfiability of a Boolean formula using appropriate encodings (Pham, Thornton, and Sattar 2008). Copyright c © 2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Table 1: Characteristics of some real-world networks and performance of consistency (in seconds) by state of the art reasoners (dashes denote abrupt termination due to memory allocation or a bug) C ha ra ct er is tic s nuts adm1 gadm1 gadm2 adm2 nodes 2,236 11,762 42,750 276,728 1,732,999 avg. degree 2.84 7.62 7.46 4.26 6.04 avg. labels 1.99 1 1 1.99 1.98 relation set tract. tract. tract. tract. tract. 2D array (GB) 0.004 0.135 1.78 74.78 2,932 R ea so ne rs Renz-Nebel01 12.25 16,783.47 1,975.04 GQR-1500 10.04 8,540.48 176.15 PPyRCC8 0.99 1,604.87 621.53 rcc8sat gp-rcc8 0.03 0.47 4.04 33.83 18,275 In contrast to the synthetic RCC-8 networks that have been used in the literature for evaluating the aforementioned reasoners, the real-world networks of Table 1 are very sparse and one to two orders of magnitude larger. The labels on their edges contain 1 or 2 base RCC-8 relations forming a disjunction. This kind of networks have not been employed in any experimental evaluation of RCC-8 reasoners with the exception of (Sioutis and Koubarakis 2012) in which the network adm1 has been used. Typically, the literature focuses on quite smaller networks (20 to 1000 nodes) with an average of 4 base RCC-8 relations per edge, and an average node degree ranging from 4 to 20. Deciding the consistency of real-world networks is a very important task. Inconsistencies might arise because their RCC-8 relations are computed based on the geometries of geographical objects which often have not been captured correctly (e.g., overlapping geometries between two regions that in principle are externally connected). This is the case for the networks gadm1 and gadm2. The characteristics of the networks of Table 1 are sufficient to stress the current reasoners on their implementations of the path-consistency algorithm which is traditionally employed by a backtracking algorithm for pruning the search space. The implementation of path-consistency has always been an integral part of a RCC-8 reasoner also due to its ability of being a very good approximation to the consistency problem, especially for networks that do not contain relations from the NP8 subset. This subset contains the socalled “hard” relations (Renz and Nebel 2001), i.e., relations that make consistency NP-complete. In addition, since realProceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence