Questions tagged [integer-programming]
248 questions
81
votes
3 answers
Express boolean logic operations in zero-one integer linear programming (ILP)
I have an integer linear program (ILP) with some variables $x_i$ that are intended to represent boolean values. The $x_i$'s are constrained to be integers and to hold either 0 or 1 ($0 \le x_i \le 1$).
I want to express boolean operations on these…
D.W.
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39
votes
1 answer
Is it NP-hard to fill up bins with minimum moves?
There are $n$ bins and $m$ type of balls.
The $i$th bin has labels $a_{i,j}$ for $1\leq j\leq m$, it is the expected number of balls of type $j$.
You start with $b_j$ balls of type $j$. Each ball of type $j$ has weight $w_j$, and want to put the…
Chao Xu
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17
votes
2 answers
Poly-time reduction from ILP to SAT?
So, as is known, ILP's 0-1 decision problem is NP-complete. Showing it's in NP is easy, and the original reduction was from SAT; since then, many other NP-Complete problems have been shown to have ILP formulations (which function as reductions from…
codetaku
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15
votes
3 answers
Cast to boolean, for integer linear programming
I want to express the following constraint, in an integer linear program:
$$y = \begin{cases}
0 &\text{if } x=0\\
1 &\text{if } x\ne 0.
\end{cases}$$
I already have the integer variables $x,y$ and I'm promised that $-100 \le x \le 100$. How can I…
D.W.
- 167,959
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15
votes
1 answer
Fastest known complexity for combinatorial ILP algorithm?
I'm wondering, what is the best known algorithm, in terms of Big-$O$ notation, to solve Integer Linear Programming?
I know that the problem is $NP$-complete, so I'm not expecting anything polynomial. And I know there are lots of heuristics and such…
Joey Eremondi
- 30,277
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15
votes
5 answers
Are all Integer Linear Programming problems NP-Hard?
As I understand, the assignment problem is in P as the Hungarian algorithm can solve it in polynomial time - O(n3). I also understand that the assignment problem is an integer linear programming problem, but the Wikipedia page states that this is…
Matt
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12
votes
4 answers
Are there competitions for integer programming?
Are there competitions for integer programming like there are for SAT and MAXSAT?
Omar Shehab
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10
votes
1 answer
Hardness of approximating 0-1 integer programs
Given a $0,1$ (binary) integer program of the form:
$$
\begin{array}{lll}
\text{min} & f(x) & \\
\text{s.t.} & A x = b \\
& x_i \ge 0 & \quad \forall i\\
& x_i \in \{0,1\} & \quad \forall i
\end{array}
$$
Note that the size of $A$ is not fixed in…
Jonas Anderson
- 203
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10
votes
3 answers
Finding all solutions to an integer linear programming (ILP) problem
My problem is to find all integer solutions to an ILP. As an example, I'm using an ILP with two variables, but I may have more than two variables. I describe the method I currently use to solve this problem near the end, but I'm interested in…
resyst
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10
votes
1 answer
Is weighted XOR-SAT NP-hard?
Given $n$ boolean variables $x_1,\ldots,x_n$ each of which is assigned a positive cost $c_1,\ldots,c_n\in\mathbb{Z}_{>0}$ and a boolean function $f$ on these variables given in the…
Alexander Klauer
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9
votes
3 answers
Boolean variable true iff equation is satisfied in ILP
Assuming $y$ is a boolean variable in an ILP program (that is $y \in Z$, s.t. $0 <= y <= 1$) and $x_1$, $x_2$ are bounded integer variables between $0$ and $M$. I want to encode the following high level constraint:
$$y = 1 \iff x_1 \le x_2$$
So far…
Setzer22
- 327
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9
votes
1 answer
How to partition a set into a given number of disjoint subsets subject to some conditions?
I am given a set $A\triangleq\{1,\ldots,k\}$, an integer $s\leqslant k$, and non-negative integers $a_{ij}$. My problem is to find $s$ disjoint subsets $S_j$ of $\{1,\ldots,k\}$ such that:
$\bigcup_{j=1}^s S_j=A$; and
$|S_j|\leqslant a_{ij}$ for…
drzbir
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9
votes
3 answers
Linear programming with absolute values
I know that sometimes we can use absolute values into the objective functions or constraints. Is it always possible to use them, anywhere ?
Example of use of absolute values:
Minimize |a+b+c| + |a-c| s.t.
|a| + b > 3
| |a| - |b| | <= 5
| |b| - 3…
permanganate
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8
votes
2 answers
Can we compute in polynomial time, an upper bound on an optimal solution of an integer linear program?
Consider the following integer linear program (where $A$ is an integer matrix, $b$ an integer vector, and $c$ a positive integer vector):
$$
\text{minimize}~~~ c\cdot x
\\
\text{subject to}~~~ A\cdot x \geq b, ~~~ x\geq 0, ~~~ x~\text{is an integer…
Samuel Bismuth
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7
votes
3 answers
Why can't we round results of linear programming to get integer programming?
If linear programming suggests that we need $2.5$ trucks to deliver goods, why can't we round up and say $3$ trucks are needed?
If linear programming suggests we can afford only $3.7$ workers, then why can't we just round down to $3$ workers?
When…
Souradeep Nanda
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