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 $c$....
- 10.6k
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 ...
- 27.7k
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 ...
- 1,982
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 ...
- 30.2k
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{...
- 28.2k
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 $xy$-...
- 7,240
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 ...
- 41k
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 ...
- 47.9k
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/...
- 16.4k
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 $|\...
- 57.8k
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 ...
- 148k
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.
...
- 2,115
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 ...
- 148k
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 ...
- 3,147
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 ...
- 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 ...
- 30.2k
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+\...
- 86.4k
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 ...
- 96.9k
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 ...
- 148k
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)...
- 29.1k
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 ...
- 4,874
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 ...
- 11.7k
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 ...
- 2,936
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 ...
- 52.8k
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 ...
- 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 $[1,\ldots,n]$ ...
- 52.8k
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 \...
- 141
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 ...
- 5,911
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 ...
- 168k
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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