All Questions
Tagged with integer-programming computational-complexity
13 questions with no upvoted or accepted answers
8
votes
0
answers
231
views
Complexity of integer programming with added predicates
A classical theorem in Integer Programming by Lenstra says that any integer system
$$A x \le b$$
can be solved in polynomial time, where $A \in \mathbb{Z}^{m \times n}, x \in \mathbb{Z}^n, b \in \...
- 343
5
votes
0
answers
240
views
Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$
Crossposted at Theoretical Computer Science SE
A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$....
- 123
4
votes
0
answers
104
views
Questions in number theory related to $NC$ and $P$-completeness
Given $a,b\in\mathbb N$ find $\operatorname{GCD}(a,b)$.
Given $a,b,c\in\mathbb N$ find $x,y\in\mathbb Z$ such that $ax+by=c$.
Euclidean algorithm solves both.
My question is if either 1 or 2 is in ...
- 13.9k
4
votes
0
answers
242
views
Domination in Nice Lattices
Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...
- 1,288
3
votes
0
answers
50
views
Modular counting of integral points under sparse non-negativity
Given a polyhedron
$$Ax\geq b$$
where every entry of $A,b$ are non-negative and $A\in\{0,1\}^{m\times n}$ and there are $O(1)$ (say $\leq8$) non-negative entries per row of $A$ is it possible to ...
- 13.9k
3
votes
0
answers
125
views
Does Barvinok's algorithm apply to convex integer program?
Barvinok provided a counting algorithm to count number of integer solutions to integer linear program that runs in polynomial time if the number of integer variables is fixed.
If we have convex ...
- 1,826
2
votes
0
answers
95
views
Why cannot we adapt Barvinok type counting techniques to general convex integer programs?
Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program ...
- 13.9k
2
votes
0
answers
221
views
Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
- 13.9k
1
vote
0
answers
92
views
Proof for non-existence of short integer program for squares
We do not know if $P=NP$ or not or if there is a superfast integer mutiplication algorithm. But I do not think either assumption is necessary to answer this question.
Is there a way to show within an ...
- 13.9k
1
vote
0
answers
78
views
$\mathsf{NP}$ complete version of Skolem arithmetic
Definable subsets of $\mathbb N$ in the language of Presburger arithmetic are exactly the eventually periodic sets and quantifier free part corresponds to Integer Programming with linear inequalities.
...
- 13.9k
1
vote
0
answers
37
views
Fast certficate of negativity for objective value of mixed-integer linear program
Let $c \in \mathbb R^n$, $A \in \mathbb R^{m \times n}$, $b \in \mathbb R^m$, and $I \subseteq \{1,2,\ldots,n\}$. Consider the Mixed integer linear program (MILP)
$$
\begin{split}
f^* = &\max \; ...
- 6,853
1
vote
0
answers
493
views
Complexity of Nested Linear Optimization
My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
$$\max(\...
- 13.2k
0
votes
0
answers
122
views
Will an integer program to deterministically factor integers help derandomize $\mathbb F_q[x]$ factoring?
There are many analogies between the objects $\mathbb F_q[x]$ and $\mathbb Z$.
Supposing there is a fixed (say $10^9$) dimension linear integer program (describable without any objective function) in ...
- 13.9k