9
Maybe the keyword you are looking for is "Implicit Complexity". It is more general than Curry-Howard correspondence, but several lines of research investigate along the axis you are interested in. You can check for instance the publications of Patrick Baillot for many references and pointers.
For a little self-promotion, here are for instance two ...
6
If $G$ is $2k$-regular, then a relaxed edge coloring with exactly $k$ colors is the same thing as a 2-factorization, and is known to always exist by results of Petersen 1891.
Otherwise, let $k=\lceil\Delta/2\rceil$ where $\Delta$ is the maximum degree of $G$. Then obviously, at least $k$ colors are needed in any relaxed coloring of $G$. But $G$ can be ...
4
There are a few points to consider here:
If we wish to show that $CC(f) = \omega(\log n)$, we should show that it is not possible to cover the domain of $f$ using $\mathrm{poly}(n)$ monochromatic rectangles (rather than $n$ rectangles). Showing it for $n$ rectangles would only establish that $CC(f) > \log n$.
If we wish to prove a lower bound on $D(EQ)$,...
3
This paper by Régis Barbanchon might be of interest. From the abstract:
We prove that the Satisfiability (resp. planar Satisfiability) problem
is parsimoniously P-time reducible to the 3-Colorability (resp. Planar
3-Colorability) problem, that means that the exact number of solutions
is preserved by the reduction, provided that 3-colorings are counted
...
2
It is unlikely to have a "name" because it is trivial: it can be solved with a hashtable, array, self-balanced binary search tree, or any other data structure that maps $x$ to $X_i$.
2
After discussing it further with a3nm, I propose an algorithm that is
different than Neal's algorithm and works in a more general
setting. The approach gives polynomial delay algorithm but it uses
exponential space.
The only two properties that we will be using are :
(1) The problem is self-reducible, in the sense that if I am given an
automaton $A$ and $u \...
1
The minimum spanning tree problem has an algorithm that was proven to be optimal by Pettie and Ramachandran, and they did so by essentially brute forcing decision trees corresponding to MSTs, and using them to construct the actual MST. However, the runtime is unknown, other than the fact that it is optimal.
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