All Questions
Tagged with computational-complexity integer-programming
25 questions
8
votes
0
answers
231
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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
6
votes
1
answer
861
views
Is Binary Integer Linear Programming solvable in polynomial time?
The paper Solving the Binary Linear Programming Model in Polynomial Time claims that Binary Integer Linear Programming is in P. However, it seems that no subsequent literature in the mainstream has ...
- 221
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
1
answer
2k
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Under what conditions does an Integer Programming problem run in polynomial time?
Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in ...
- 13.9k
4
votes
2
answers
2k
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Simplified knapsack problem
There is a problem that I can not solve.
Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
- 41
4
votes
0
answers
104
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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
1
answer
329
views
Nonexistence of short integer program sequence which generates squares
Is there a way to show within an integer program with constant number of variables and constraints of length $poly(\log B)$ (say length $\leq10^{1000000}\log B$), it is not possible for a variable to ...
- 13.9k
3
votes
1
answer
198
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Separation of Anti-Hole Inequality
Given an undirected graph $G=(V,E)$ with no loops or multiple edges, a stable set is a set of vertices for which no two vertices are adjacent.
An induced subgraph $H$ of $G$ is called an odd-antihole ...
- 41
3
votes
0
answers
50
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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
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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
1
answer
340
views
Bit complexity of Barvinok's algorithm
I have seen many references which state Barvinok's algorithm has polynomial time complexity for counting integer points of polytopes in fixed dimension.
What exactly is this arithmetic complexity?
...
- 13.9k
2
votes
1
answer
69
views
Clarification on FPTAS optimization in a paper
In the abstract of this paper by Hildebrand, Weismantel & Zemmer it is stated that they provide an FPTAS for $$\min x'Qx$$ over a fixed dimension polyhedron when $Q$ has at most one negative or ...
- 13.9k
2
votes
2
answers
202
views
Combinatorial optimization problem involving infinite spin system
In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity.
...
- 867
2
votes
0
answers
95
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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
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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
1
answer
237
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Gröbner basis via integer programming
I have studied some papers related to solving integer programs via Gröbner bases. I wonder if the other way is possible or not — i.e., given any ideal, can we find the Gröbner basis by translating ...
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
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$\mathsf{NP}$ 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
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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)
$$
\begin{split}
f^* = &\max \; ...
- 6,853
1
vote
0
answers
132
views
Complexity :: Integer Programming :: Non-Poly Example [closed]
When learning about computational complexity I find that when discussing the NP-Complete problems authors always give examples of such problems that can in fact be solved in poly time.
I understand ...
- 19
1
vote
0
answers
493
views
Complexity of Nested Linear Optimization
My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
$$\max(\...
- 13.2k
0
votes
1
answer
128
views
What is the computational complexity of the calculation of $ \Psi(x) $?
What is the computational complexity of the calculation of $ \Psi(x) $ described below:
Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
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
0
votes
1
answer
270
views
Generalized assignment problem with no integrality gap
Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{...
