Search Results

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to mini …

The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any othe …

If I sample a large number of uniform points in the unit square and take a traveling salesman tour of them, is there anything at all that can be said/is known about the distribution of angles at each …

Suppose I have a convex polygon $C$ and a radius $r>0$, and I seek a path $P$ that "covers" $C$, in the sense that any point $C$ is within distance $r$ of $P$: $$d(x,P)\leq r~\forall x\in C~,$$ where …

A while ago I saw an image like the one below in a lecture, which was supposed to represent a rail network in a (square) city: The circles represent trains that are moving either North/South or Eas …

Let $\Pi$ denote the space of all permutations of $\{1,\dots,n\}$, and let $d(\cdot,\cdot)$ be a metric on $\Pi$. Suppose I am given a large integer $K$ and I have to select $K$ permutations $\pi_1,\ …

Suppose I have $n$ points $x_1,\dots,x_n$ that are all independent uniform samples in the unit square, and I'd like to find a short path (in terms of Euclidean length) that touches all of them (a trav …

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical distributio …

This is not a homework problem, although I fear it may turn out to be at that level. For any nonnegative $x\in\mathbb{R}^n$, let $f_k(x)$ be the sum of the $k$ largest values in $x$, and define $$f(x …

Can anyone point me to an example of a problem that (more or less) originated in computational geometry whose solution requires the use of Lagrange multipliers (or Kuhn-Tucker conditions, or dual vari …

Given a path $P$ in the unit square, and two points $p_{1},p_{2}$ located on $P$, let $d_{P}(p_{1},p_{2})$ denote the distance from $p_{1}$ to $p_{2}$ traversed along $P$. Given $a>1$, I am looking f …

Can anyone direct me to any algorithms or theorems that describe the difficulty of solving a non-smooth convex optimization problem for the special case where the full subdifferential is available? A …

Suppose I am given $n$ points $p_1,\dots,p_n\in \mathbb{R}^2$, as well as two positive coefficients $a_1$ and $a_2$, and I am trying to select $n$ points $x_1,\dots,x_n$ to solve the following optimiz …

Suppose I have a lattice grid of $m \times n$ points in the plane, with $m\leq n$. I want to traverse this grid in such a way as to minimize the total amount of turning that occurs. I am pretty sure …

Is there any global constant $C$ such that $$C<\frac{\sum_{i=1}^{n}x_{i}\log x_{i}-x_{i}+(1-x_{i})\log(1-x_{i})}{\sum_{i=1}^{n}x_{i}}+\log(\sum_{i=1}^{n}x_{i})-\log(\sum_{i=1}^{n}x_{i}^{2})$$ for all …