4,111 research outputs found
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication
Understanding the query complexity for testing linear-invariant properties
has been a central open problem in the study of algebraic property testing.
Triangle-freeness in Boolean functions is a simple property whose testing
complexity is unknown. Three Boolean functions , and are said to be triangle free if there is no such that . This property
is known to be strongly testable (Green 2005), but the number of queries needed
is upper-bounded only by a tower of twos whose height is polynomial in 1 /
\epsislon, where \epsislon is the distance between the tested function
triple and triangle-freeness, i.e., the minimum fraction of function values
that need to be modified to make the triple triangle free. A lower bound of for any one-sided tester was given by Bhattacharyya and
Xie (2010). In this work we improve this bound to .
Interestingly, we prove this by way of a combinatorial construction called
\emph{uniquely solvable puzzles} that was at the heart of Coppersmith and
Winograd's renowned matrix multiplication algorithm
Move-minimizing puzzles, diamond-colored modular and distributive lattices, and poset models for Weyl group symmetric functions
The move-minimizing puzzles presented here are certain types of one-player
combinatorial games that are shown to have explicit solutions whenever they can
be encoded in a certain way as diamond-colored modular and distributive
lattices. Such lattices can also arise naturally as models for certain
algebraic objects, namely Weyl group symmetric functions and their companion
semisimple Lie algebra representations. The motivation for this paper is
therefore both diversional and algebraic: To show how some recreational
move-minimizing puzzles can be solved explicitly within an order-theoretic
context and also to realize some such puzzles as combinatorial models for
symmetric functions associated with certain fundamental representations of the
symplectic and odd orthogonal Lie algebras
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