In many situations, an explanation of the reasons behind inconsistency in an overconstrained CSP is required. This explanation can be given in terms of minimal unsatisfiable subsets (MUSes) of constraints. This paper presents algorithms for finding minimal unsatisfiable subsets (MUSes) of constraints in overconstrained CSPs with finite domains and binary constraints. The approach followed is to generate subsets in the subset space, test them for consistency and record the inconsistent subsets found. We present three algorithms as variations of this basic approach. Each algorithm generates subsets in the subset space in a different order and curtails search by employing various search pruning mechanisms. The proposed algorithms are anytime algorithms: a time limit can be set on an algorithm's search and the algorithm can be made to find a subset of MUSes. Experimental evaluation of the proposed algorithms demonstrates that they perform two to three orders of magnitude better than the existing indirect algorithms. Furthermore, the algorithms are able to find MUSes in large CSP benchmarks.

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In the context of smart cities, ridesharing in urban areas is gaining researchers’ interest and is considered to be a sustainable transportation solution. In this paper, we present SRide (Shared Ride), a multi-hop ridesharing system as a mode of sustainable transportation. Multi-hop ridesharing is a type of ridesharing in which a rider travels in multiple hops to reach a destination, transferring from one driver to another between hops. The key problem in multi-hop ridesharing is to find an optimal itinerary or route plan for a rider from an origin to a destination in a dynamic, online setting. SRide adopts a novel approach to finding itineraries for riders suited to the online nature of the problem. The system represents ride offers as a time-dependent directed graph and finds itineraries dynamically by updating the graph incrementally and decrementally as ride offers are updated in the system. The system’s distinguishing feature is its incremental and decremental operation, which is enabled by employing dynamic single-source shortest-path algorithms. We conducted two extensive simulation studies to evaluate its performance. Metrics, including the matching rate, savings in total system-wide vehicle-miles, and total system-wide driving times were measured. In the first study, SRide’s dynamic update algorithms were compared with their non-dynamic versions. Results show that SRide’s algorithms run up to thirteen times faster than their non-dynamic versions. In the second study, we used data from the travel demand model for metropolitan Atlanta in the US state of Georgia, to assess the benefits of multi-hop ridesharing. Results show that matching rates increase up to 68%, saving in total system-wide vehicle-miles of up to 12%, and reduction in the total system-wide driving time of up to 12.86% is achieved.

Minimal Unsatisfiable Subsets (MUSes) are the subsets of constraints of an overconstrained constraint satisfaction problem (CSP) that cannot be satisfied simultaneously and therefore are responsible for the conflict in the CSP. In this paper, we present a hybrid algorithm for finding MUSes in overconstrained CSPs. The hybrid algorithm combines the direct and the indirect approaches to finding MUSes in overconstrained CSPs. Experimentation with random CSPs reveals that the hybrid approach is not only quite efficient but when operating under a time bound it finds a more representative set of MUSes. C 2011 Wiley Periodicals, Inc.

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