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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5 votes
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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^{&...
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5 votes
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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 ...
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  • 3,770
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 ...
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  • 3,770
4 votes
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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 $x^cy^b$.
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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 ...
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  • 51.3k
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 ...
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  • 141
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 ...
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  • 15.3k
4 votes
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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, ...
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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 ...
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3 votes
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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 ...
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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: ...
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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\...
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3 votes
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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 ...
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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 $$a_1 + a_{N} + d_1 + \dots + d_{N-1} \leq 2.$$
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3 votes
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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}=\...
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  • 52.9k
3 votes
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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 ...
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  • 722
3 votes
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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 ...
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3 votes
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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{...
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2 votes
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Find base of kernel with as many 0 as possible

It sounds like you're describing an instance of the "sparse null basis problem" – this was the title of what (I believe is) the seminal work on this problem, by Coleman and Pothen. You can find their ...
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  • 1,493
2 votes
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Algorithm to minimally connect line segments in Euclidean plane

Another way to solve the problem might be to reduce it to a traditional version of the TSP and then use a free, off-the-shelf solver like one of these. At the very least, it should be a relatively ...
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  • 3,306
2 votes

Sufficient condition for solvability of linear diophantine system

This can certainly be done in polynomial time using the Smith normal form (SNF) of $A$, which is a diagonal matrix $D = SAT$ for some integer matrices $S,T$ of determinant $\pm 1$ (and thus whose ...
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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)...
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2 votes
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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_{...
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  • 21.5k
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 ...
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  • 2,610
2 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} &...
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2 votes

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

This is only a partial answer. $\mathcal{L}_\mathbb{Q}=\mathcal{L}_\mathbb{R}$ for all $B$. Decompose $\mathbb{R}$ as a $\mathbb{Q}$-vector space into $\mathbb{Q}\oplus V$. Any $u\in\mathbb{R}^k$ can ...
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  • 1,563
2 votes
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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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2 votes

Clarification on FPTAS optimization in a paper

Unfortunately, no. The form we use is very restrictive. The key is that the function $f(x)$ can be decomposed in a sign compatible way as $f(x) = g(x) h(x)$, where $g(x)$ is convex (or concave) and ...
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2 votes

Feasibility Mixed integer Linear programming with quadratic constraints?

Mixed integer Linear Program (MILP), i.e., with no quadratic constraints, is already in general NP hard, even without quadratic constraints. If the problem had only continuous variables, it could be ...
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