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In general though, if $n$ gets too large, you'd want to use a greedy heuristic where at each step you pick the set that covers the most number of "unfilled" spots. That is in a sense the best possible …

Let's say we consider the set of strings in which we see $n$ zeros first (by symmetry, this should be half the total number). Fix the number of 1s also encountered to be $k$. then the total remaining …

Seth Pettie has done some fascinating work on this topic and generalizations. In his setting, the goal is to upper bound the number of 1s in a 0-1 matrix that excludes a particular submatrix pattern. …

Here's a sketch of an idea. Consider first the problem of computing the number of permutations of $k$ elements given the number of elements $n_i$ in each class. This is easy: it works out to $$\frac{k …

Along the lines of what Thomas Bloom points out, it's unlikely that any NP-Complete property of a graph will have a local-global structure, because such a structure would imply an algorithm that might …

I have the standard lattice L defined over partitions of $1\ldots n$ under the split-merge relation. I also have an antimonotone function from L to R that's submodular, and so gives me a metric on L v …

There are a few different ways of approaching the problem. A good reference for this precise problem is 'Rank Aggregation methods For The Web', by Dwork, Kumar, Naor and Sivakumar from the WWW confere …

If you're willing to tolerate an approximate answer, there are some options, but there's nothing direct. One idea would be to write the packing problem as an (exponential-sized) integer program and th …

Abstractly, tree/path decompositions of a graph $G$ can be thought of as doing the following: fix a "skeleton" class of graphs (tree or path) Pick a member of this class $H$. Associate with each nod …

If I understand your question correctly, you're trying to find a permutation of the bit lines so the maximum gate "length" is as small as possible. This is called the bandwidth problem: Given a graph …

Consider a hypergraph (of rank 3) $H = (V, E)$ (where the rank of $H$ is the maximum cardinality of a hyperedge). $H$ is said to be planar if we can construct a planar graph $G = (V, A)$, and a mappin …

By triangle inequality, preserving the property you wish for means that you can find "representatives" for each $v$ so that the $\ell_1$ distances between any $v, v'$ are preserved to within 2$\epsilo …

I don't know of any definite answer to your question, but one idea would be to sort each level by increasing degree ? that gets you a little closer (though not that much) to isomorphism

The result you're trying to prove is a "circular" version of the colored Helly theorem for the line (see the first exercise on page 3 of these notes by Matousek). Although I haven't verified it, it se …

I'm not sure if this is a soft question, or should be community wiki. I was explaining to a student how to prove that two sets were equal using what I called the 'oldest trick in the book': to show …