This paper describes new algorithms for processing quantitative and qualitative Temporal Constraint Satisfaction Problems (TCSP). In contrast to discrete Constraint Satisfaction Problems (CSP), enforcing path-consistency on quantitative TCSP is exponential, due to the fragmentation problem. Identifying the fragmentation problem allows us to design several efficient polynomial algorithms that are effective for detecting inconsistencies and explicating some implicit constraints. We discuss the tradeoffs between their effectiveness and efficiency both theoretically and empirically. 1
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