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 ...

5
votes

Accepted

### What are the definable sets in Skolem arithmetic?

$\def\mr{\mathrm}$As it happens, quantifier elimination for Skolem arithmetic came up recently in my research. The concise description is that every formula $\phi(x_1,\dots,x_k)$ is in $(\mathbb N^{&...

5
votes

Accepted

### Correct way to conduct equilibrium scaling of linear/integer/MIP program

Regarding integer columns, it means the columns that correspond to integer variables. The $j\in \mathcal{I}$ notation makes that explicit. The idea is that scaling continuous variables preserves ...

5
votes

### Maximally sparse integer solutions

Minimizing the sum of absolute values was covered in the comments. To instead minimize the number of nonzero variables, let $[\ell_j,u_j]$ be finite constant bounds on $x_j$, introduce binary ...

4
votes

Accepted

### Sign Enumeration

The g.f. equals
$$\frac1{1-y}\prod_{i=1}^n \left(xy^i + (xy^i)^{-1}\right).$$
That is, the number of solutions is given by the coefficient of $x^cy^b$.

4
votes

### Integer programming of free energy

Evaluating the minimum $\delta_k:=\min_{x\neq y}|f(y)-f(x)|$ over distinct strings $x, y\in \{0,1\}^k$ seems a hard diophantine problem. It is quite cheap to prove that $\delta_k$ is $O(1/k)$ and ...

4
votes

### Under what conditions does an Integer Programming problem run in polynomial time?

The way I think about Kannan's algorithm (or, for that matter, any of the several "Lenstra-type" algorithms which present similar bounds as the original Lenstra algo from 1983) is that it's designed ...

4
votes

### Bit complexity of Barvinok's algorithm

I looked into that at some point. The original paper says $N^{O(d^2)}$, where $d$ is the dimension and $N$ is the size of the input (sum of bit lengths of matrix entries in $A\overline x\le \overline ...

4
votes

Accepted

### Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way

Copying from http://mathworld.wolfram.com/SumofSquaresFunction.html:
To find in how many ways a positive integer $n>1$ can be expressed as a sum of $k=2$ squares ignoring order and signs, ...

3
votes

Accepted

### Detection of gaps in binary vector through linear methods

OK, sorry for the delay. I just don't have too much free time nowadays.
Let $a_0,\dots,a_{n-1}$ be your string. Note that for any $z=e^{it}$, we have
$$
P_a(z)=a_0+a_1z+a_2z^2+\dots+a_{n-1}z^{n}=\...

3
votes

### Detection of gaps in binary vector through linear methods

Introduce binary variables $d_1,\dots,d_{N-1}$ and constraints:
$$-d_i \leq a_i - a_{i+1} \leq d_i,\quad i=1,\dots,N-1$$
and
$$a_1 + a_{N} + d_1 + \dots + d_{N-1} \leq 2.$$

3
votes

Accepted

### On necessary condition for no integer points in polytope

The answer is $no$.
In the plane, the circle of diameter $d$ greater than $1$ but smaller than $\sqrt2$ and centered at $({1\over2},{1\over2})$ contains no point with integer coordinates, and the ...

3
votes

### An interesting problem which I think only needs elementary number theory

I am not sure about only using elementary number theory but here is an approach using the Bateman-Horn conjecture. Maybe one can get by with a bit less using sieve methods but I will leave that to the ...

3
votes

### Maximize the determinant of Boolean combinations of positive definite matrices

Given $n$ symmetric positive definite matrices $\mathrm W_1, \mathrm W_2, \dots, \mathrm W_n \in \mathbb R^{m \times m}$ and $s \in \mathbb N$, where $s < n$,
$$\begin{array}{ll} \text{maximize} &...

3
votes

Accepted

### programming to compute kernel quotient image of a $\mathbb{Z}$-module endomorphism

Ok, not really beautiful, but the lines below are a simple SAGE implementation of the map $\partial$, computing both the representing matrix and the elementary divisors. In the implementation I ...

3
votes

### Algorithm that solves every Mixed Integer Linear Program (to optimality)?

I sometimes read the claim that Gomory cuts alone (without any branch-and-X) are sufficient for finding integer solutions to linear programs in a finite number of steps. E.g. at the bottom of page 2:
...

3
votes

### Minimal solution of simultaneous congruences

Let $A_i$ be the set of allowed values for $a_i$. Then $x$ satisfying the system of congruences represents a zero of the following polynomial modulo $M$:
$$f(x) := \sum_{i=1}^n \frac{M}{p_i} \prod_{a\...

3
votes

Accepted

### Gröbner basis via integer programming

ILP is $NP$-complete, while computing a Grobner Basis is $EXPSPACE$-complete. As one has the containments $NP\subseteq PSPACE\subsetneq EXPSPACE$, one should expect that you can reduce ILP to ...

3
votes

Accepted

### Reliability of ILP approach to number-theoretic optimization

Well, the question is broad, but let us address at least some of it.
Q1. What is the cause of failure for ILP solvers in this problem?
Let's concentrate in the GLPK failure with $n=27$. Here is a ...

3
votes

Accepted

### Constructing an integer with small residues for two distinct primes in polynomial time

If such $m$ exists, then $m=up+a=vq+b$ for some $u,v\in O(T^{1/2+\epsilon})$ and $a,b\in O(\mathrm{polylog}(T))$. Then $up-vq=b-a$ and thus $\frac{p}q - \frac{v}u=\frac{b-a}{uq}$, implying that $\frac{...

3
votes

### Fastest way to solve non-negative linear diophantine equations

Consider constraint programming for this. On my laptop, the constraint programming solver in SAS finds all the solutions in one second.
Code:
...

3
votes

### Fastest way to solve non-negative linear diophantine equations

Such problems are naturally expressed as finding integral points within a bounded polyhedron. There is a bunch of software available for this, e.g., Normaliz. Although my initial attempt to employ ...

3
votes

### Only trivial solutions to system of linear diophantine equations possibly related to hamiltonian cycles in graphs

Below I show that $m$ cannot be constant.
The given system of $m$ equations can be reduced to the case of $m=1$, that is, to a single equation:
$$\sum_{j=1}^n y_j a'_{j} = \sum_{j=1}^n a'_{j}$$
where $...

3
votes

### How quickly can this IQP or its MILP relaxation be solved

For binary $P$, we have $\min\{P_{k,i},P_{l,j}\} = P_{k,i} P_{l,j}$. In your linearization, you have introduced $r_{i,k,l,j}$ to represent this product. Because of the linear constraints $$\sum_k P_{...

3
votes

Accepted

### Benefit of adding a trivial constraint to ILPs

This question is addressed in a few OR StackExchange questions:
https://or.stackexchange.com/questions/419/feeding-known-lower-bounds-to-solvers
https://or.stackexchange.com/questions/3777/how-to-...

3
votes

Accepted

### Existence of some lattice path connecting all given lattice paths

Let us draw a checkered table with $N$ columns and $n$ rows, such that the cell $(i,j)$ (that is, the one in the $i$th row and the $j$th column) contains the number $q_i^j$. Paint all cells with ...

2
votes

### basis of the lattice generated by the integer points inside a subspace of R^L

Although the brute force solution of Alex is certainly possible, it is not very efficient.
It would be easier in practice to use Hermite's normal form:
First, find a linear map $A \in \mathbb{Z}^{(L-K)...

2
votes

Accepted

### On linear integer inequalities with infinitely many solutions

Let the constraints be numbered 1 to $m$. Let the $i$th constraint be $a^{(i)}\cdot x\le b$. Let $S$ be the set of solutions. Inductively refine the constraints as follows: either there are $\limsup_{...

2
votes

### Integer programming of free energy

I'm confused. Why not $x = 0$ and $y=(0, 0, \ldots , 0, 1)$? This has $f(y) - f(x) = e -1$, which is clearly best possible [though not unique]. (As you note, $f$ is an injection, so different ...

2
votes

Accepted

### Integer points spanned by real, rational and integer combination of integer vectors

We always have $\mathcal L_\Bbb Q=\mathcal L_\Bbb R:$ For any particular $v \in\Bbb Z^n$ consider how we determine $\{u \in \Bbb R^n \mid uB=v\}.$ The method, if the set is nonempty, will yield one or ...

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