31
votes

### Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

This is an instance of Holte's Amazing matrix. Consider addition of binary digits. Start with a carry of $c \in \{0,1,\ldots,2(m-1)\}$. Choose $2m-1$ bits uniformly at random, and add their sum to $c$....

26
votes

### Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

The Lindstrom-Gessel-Viennot lemma says that the number of families of nonintersecting lattice paths can be counted by a determinant. Let $a_i = (2m-i,i)$. Let $b_j = (2m-2j,-2m+2j)$. Then the number ...

25
votes

Accepted

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Consider the polytope in $\mathbb{R}^3$ with $8$ vertices at coordinates $(\pm 1, \pm 2, 1), (\pm 2, \pm 1, -1)$. Geometrically this looks like a cube where the top face is stretched in the direction ...

24
votes

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Your question is essentially about extension complexity. In general, the extension complexity of a polytope $P$ is the minimum number of facets over all polytopes $Q$ which project to $P$. You are ...

21
votes

### Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

Here is a very low-brow answer to the original question.
Consider the lower-triangular matrix
\begin{equation*}
V = [V_{ij}] = \left[\binom{i-1}{j-1}\right]\quad \text{for}\quad i \ge j.
\end{...

16
votes

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Another, combinatorially minimal, counterexample of such a polytope $P$ (with only five facets) is the convex hull of the six vertices $(\pm2, 0, 0)$, $(\pm1, \pm1, 1)$. Its projection to the $xy$-...

15
votes

### Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

Let $A_n(x,\lambda)$ be the $n\times n$ matrix
$$\left[\binom{x}{2j-i+\lambda}\right]_{i,j=1}^n.$$
Let's "generalize to trivialize". Sometimes, generalizations offer more elbow room to maneuver, such ...

14
votes

### Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

The matrix $A_n(x,\lambda)$ is obtained from the dual Jacobi matrix for the partition $\mu=(n+\lambda,n-1+\lambda,...,1+\lambda)$ by setting $x$ variables equal to 1 and the remaining variables equal ...

13
votes

Accepted

### Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

This is a tough question, and I don’t think there’s a definitive answer yet. For some mathematical details, see the following survey articles:
https://arxiv.org/abs/1611.01685
https://arxiv.org/abs/...

12
votes

Accepted

### The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$

It is quite likely. At least, the proof for the case $d\mid n$ is easy. First of all, the restriction $i\ne j$ does not matter: adding $n$ ones changes nothing in the problem. Now notice that $|\...

11
votes

### Formula for volume of a convex polytope

Here is one paper, whose introduction will lead you to others:
Lasserre, Jean B., and Eduardo S. Zeron. "A new algorithm for the volume of a convex polytope." arXiv math/0106168 (2001).
If you ...

11
votes

### Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

In my understanding, the connection to modular forms came via a result of Cohn and Elkies in their paper "New upper bounds on sphere packings I," Ann. of Math. 157 (2003) 689-714, also on the arxiv.
...

10
votes

Accepted

### Applications of linear programming duality in combinatorics

How about
Boosting1
and the Hardcore Lemma, as described in this paper?
Trevisan, Luca, Madhur Tulsiani, and Salil Vadhan. "Regularity, boosting, and efficiently simulating every high-entropy ...

10
votes

### How did they come up with the MRRW bound?

For linear programming type bounds it is sometimes only possible to give effective bounds (that is bounds that work and are manageable) and what is surprising is that the primitive method often gives ...

8
votes

### Applications of linear programming duality in combinatorics

So, this is not an example of using linear programming duality within a proof of a theorem but, rather, an example of using linear programming duality to search for a proof.
The discharging method ...

8
votes

Accepted

### Definition of packing property

That Def 1 and Def 2 are equivalent is a well-known Conjecture, still open as far as I know. Curiously, you can translate the whole conjecture to the language of commutative algebra, see for example ...

7
votes

Accepted

### Linear programming is continuous

The answer is no. Consider the probem
$$x+y\to\max,$$
$$x\geq 0,\; y\geq 0,$$
$$x+y\leq 1.$$
It has infinitely many solutions. One of them is $(0,1)$.
Now change the last inequality to
$$x+(1+\...

7
votes

### The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$

In addition to fedja's clever argument for the case $d|n$, let me prove this for $d=2$ (and $n$ of arbitrary parity).
We have $|\cos x|\geqslant 1-\frac2\pi x$ for $x\in [0,\pi/2]$ by concavity of ...

7
votes

### Is Binary Integer Linear Programming solvable in polynomial time?

Often called Binary Integer Programming (BIP).
Wikipedia:
Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and ...

7
votes

### Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers

Here is a proof of Conjecture 2.
First, we have
\begin{split}
\sum_{k=1}^n M(n,k) &= \sum_{k=1}^n \sum_{m=k}^n \sum_{d|\gcd(m,k)} d\cdot\mu(d) \\
&= \sum_{m=1}^n \sum_{k=1}^m \sum_{d|\gcd(m,k)...

7
votes

Accepted

### Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment

You can solve this as a minimum-weight perfect matching problem in a graph with a node for each point and an edge for each pair of points. Because the distances satisfy the triangle inequality, an ...

6
votes

### The cone of positive semidefinite matrices is self-dual? (reference needed)

I know you are looking for a reference and you probably know how to prove it (and that the post is old). However, I want to include a short form of the proof for those coming to the post for this ...

6
votes

### Applications of linear programming duality in combinatorics

In Examples of combinatorial duality Garth Isaak demonstrates how Farkas' Lemma (which is LP duality) can be used to prove Landau's characterization of the possible score sequences in round robin ...

5
votes

Accepted

### Better tactics for removing redundant constraints than Linear Programming?

I don't know why you find this surprising. Saying that a particular linear inequality $a x \le b$ is non-redundant is exactly the statement that the linear programming problem, maximize $a x$ subject ...

5
votes

### Applications of linear programming duality in combinatorics

Here is a fairly recent example. Davies, Jenssen, Perkins and Roberts use linear programming duality to obtain a tight upper bound on the number of independent sets and matchings in a $d$-regular ...

5
votes

### Linear programming is continuous

You can get this kind of continuity if the optimal solution is nondegenerate in the following sense. Let the coefficient matrix be $m \times n$, and suppose there is a subset $B$ of $[1,\ldots,n]$ ...

5
votes

### The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$

In fact, the minimizers of the sum $\sum \langle v_i, v_j \rangle^2$ are precisely tight frames, i.e. sets such that for some $C>0$and for each $x\in \mathbb R^d$, one has $\| x \|^2 = C \sum \...

5
votes

Accepted

### a linear programming problem

The statement in the gray box is an immediate consequence of Helly's theorem: for each $i$, the set $\{ v\in\mathbb{R}^n\colon\ f_i(v)\geq 0\}$ is convex, and because of $r>n+1$, the condition ...

5
votes

Accepted

### Sampling uniformly from the vertices of a polytope

Here is one efficient approach, performing a random walk with a rapid mixing time, that has been implemented for a particular class of polytopes, but which might well be adaptable to a more general ...

5
votes

Accepted

### Famous theorems that are special cases of linear programming (or convex) duality

To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make ...

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