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 $c$....
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  • 10.4k
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 ...
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  • 27.5k
24 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 ...
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  • 1,849
23 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 ...
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  • 28.7k
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{...
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  • 27.9k
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 $xy$-...
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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 ...
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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 ...
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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/...
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  • 15.9k
12 votes
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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 $|\...
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  • 53k
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 ...
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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. ...
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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 ...
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8 votes
Accepted

Multiplicative gradient descent?

The most general form of such algorithms are named Mirror-Descent. This algorithm is an extension of gradient descent for non-Euclidean geometries. For a formal explanation on how multiplicative ...
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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 ...
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  • 761
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 ...
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  • 29.8k
8 votes
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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 ...
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  • 3,094
7 votes
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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+\...
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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 ...
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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)...
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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 ...
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  • 3,820
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 ...
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  • 10k
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 ...
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6 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 ...
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  • 89.5k
5 votes

convex polytope integer points

Edit: the following argument is true when the polyhedron $P$ is the intersection of finitely many linear constraints: $a_1 x_1 + \ldots + a_n x_n \leq b$. The coefficient of linear constraints must ...
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  • 1,961
5 votes
Accepted

Examples of Polyhedra with Large Shadows

If I understood your question right, you are looking for a polygon that has a small so-called extended formulation. This problem is studied for example here: http://arxiv.org/abs/1107.0371.
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  • 17.3k
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 ...
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  • 761
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 $[1,\ldots,n]$ ...
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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 ...
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  • 5,823
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 ...
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