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Instances of SAT induce a bipartite graph between clauses vertices and variable vertices, and for planar 3SAT, the resulting bipartite graph is planar. It would be very convenient if there was a pla …

The trick is to use a transform called the Steinhaus Transform. Given a metric $(X, d)$ and a fixed point $a \in X$, you can define a new distance $D'$ as $$D'(x,y) = \frac{2D(x,y)}{D(x,a) + D(y,a) + …

The spectrum of a graph is the (multi)set of eigenvalues of its adjacency matrix (or Laplacian, depending on what you're interested in). In general, two non-isomorphic graphs might have the same spect …

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 …

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 …

There's a standard trick to convert the min cost matching problem on a balanced bipartite graph to one on an unbalanced bipartite graph. Let $G = (X \cup Y, E, w)$ be the bipartite graph where $E \sub …

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 …

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 …

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 …

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 …

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. …

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 …

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 …

In an indirect way, this related to graphical models and the problem of estimating events in a given model. Your matrix yields a digraph with parallel edges between each pair of nodes $(i,j)$ labelled …

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 …