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A Multivariate Complexity Analysis of Qualitative Reasoning Problems
Qualitative reasoning is an important subfield of artificial intelligence
where one describes relationships with qualitative, rather than numerical,
relations. Many such reasoning tasks, e.g., Allen's interval algebra, can be
solved in time, but single-exponential running times
are currently far out of reach. In this paper we consider
single-exponential algorithms via a multivariate analysis consisting of a
fine-grained parameter (e.g., the number of variables) and a coarse-grained
parameter expected to be relatively small. We introduce the classes FPE and
XE of problems solvable in , respectively , time,
and prove several fundamental properties of these classes. We proceed by
studying temporal reasoning problems and (1) show that the Partially Ordered
Time problem of effective width is solvable in time and is thus
included in XE, and (2) that the network consistency problem for Allen's
interval algebra with no interval overlapping with more than others is
solvable in time and is included in FPE. Our
multivariate approach is in no way limited to these to specific problems and
may be a generally useful approach for obtaining single-exponential algorithms