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On the Computational Complexity of Defining Sets
Suppose we have a family of sets. For every , a
set is a {\sf defining set} for if is the
only element of that contains as a subset. This concept has been
studied in numerous cases, such as vertex colorings, perfect matchings,
dominating sets, block designs, geodetics, orientations, and Latin squares.
In this paper, first, we propose the concept of a defining set of a logical
formula, and we prove that the computational complexity of such a problem is
-complete.
We also show that the computational complexity of the following problem about
the defining set of vertex colorings of graphs is -complete:
{\sc Instance:} A graph with a vertex coloring and an integer .
{\sc Question:} If be the set of all -colorings of
, then does have a defining set of size at most ?
Moreover, we study the computational complexity of some other variants of
this problem
Invariance of generalized wordlength patterns
The generalized wordlength pattern (GWLP) introduced by Xu and Wu (2001) for
an arbitrary fractional factorial design allows one to extend the use of the
minimum aberration criterion to such designs. Ai and Zhang (2004) defined the
-characteristics of a design and showed that they uniquely determine the
design. While both the GWLP and the -characteristics require indexing the
levels of each factor by a cyclic group, we see that the definitions carry over
with appropriate changes if instead one uses an arbitrary abelian group. This
means that the original definitions rest on an arbitrary choice of group
structure. We show that the GWLP of a design is independent of this choice, but
that the -characteristics are not. We briefly discuss some implications of
these results.Comment: To appear in: Journal of Statistical Planning and Inferenc
Blocked regular fractional factorial designs with minimum aberration
This paper considers the construction of minimum aberration (MA) blocked
factorial designs. Based on coding theory, the concept of minimum moment
aberration due to Xu [Statist. Sinica 13 (2003) 691--708] for unblocked designs
is extended to blocked designs. The coding theory approach studies designs in a
row-wise fashion and therefore links blocked designs with nonregular and
supersaturated designs. A lower bound on blocked wordlength pattern is
established. It is shown that a blocked design has MA if it originates from an
unblocked MA design and achieves the lower bound. It is also shown that a
regular design can be partitioned into maximal blocks if and only if it
contains a row without zeros. Sufficient conditions are given for constructing
MA blocked designs from unblocked MA designs. The theory is then applied to
construct MA blocked designs for all 32 runs, 64 runs up to 32 factors, and all
81 runs with respect to four combined wordlength patterns.Comment: Published at http://dx.doi.org/10.1214/009053606000000777 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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